Non-EquilibriumStatisticalPhysicsof Queueing-Networks ...web.sg.ethz.ch/users/rpfitzner/material/diplomarbeit.pdf · Non-EquilibriumStatisticalPhysicsof Queueing-Networks: Theory,Numericsand

Non-Equilibrium Statistical Physics of

Queueing-Networks: Theory, Numerics and

Application

vorgelegt von

Rene Pfitzner

April 2011

Contents

0. How to read this thesis 15

I. Theory 17

1. Non-Equilibrium Statistical Physics 19

1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2. Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.3. Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.4. The Birth-Death Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2. Queueing Theory 29

2.1. A short glance on history . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2. Theory of Single Queues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3. The Single Markovian Queue as Birth-Death System . . . . . . . . . . . . 332.4. Networks of M/M/1/∞ queues . . . . . . . . . . . . . . . . . . . . . . . . 362.5. Operator technique to solve ME for Queueing Networks . . . . . . . . . . 39

3. Zero-Range Processes and Exclusion Processes 43

3.1. The Zero-Range Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2. The Exclusion Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

II. Application and Numerics 49

4. A Queueing Network based description of General Zero-Range Processes 51

4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2. Mapping between queueing networks and zero-range processes . . . . . . . 514.3. The general n-dimensional Zero-Range Process . . . . . . . . . . . . . . . 524.4. Condensation and renormalization . . . . . . . . . . . . . . . . . . . . . . 55

5. Numerical and analytical results for the n-dimensional ZRP 59

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2. Description and general behavior of a global µi-model . . . . . . . . . . . 59

6. Conclusion and Further Studies 75

3

Before I start...

Und jedem Anfang wohnt ein Zauber inne,Der uns beschutzt und der uns hilft, zu leben.- Hermann Hesse

The history of this thesis is somewhat unusual and its existence due to a coupleof people, who I want to thank before I dive into the material of this work. Offi-cially this diploma thesis is submitted to the Faculty of Physics and Astronomy atFriedrich-Schiller-University Jena, Germany under the official supervision of Prof. Ger-hard Schafer. I am inexpressibly thankful to Prof. Schafer for the freedom he gave mein my thesis studies and for being a great mentor in the last years.

The research presented in this work was conducted at Los Alamos National Labora-tory in Los Alamos, NM, USA from April 2010 to April 2011 under the guidance of Dr.Michael ”Misha” Chertkov and Dr. Konstantin ”Kostya” Turitsyn. I am enormouslythankful to Dr. Chertkov for providing me with a very inspiring, diverse and excellentresearch environment as well as the necessary funding via NFS grant ”Harnessing Statis-tical Physics for Computing and Communication”. I am grateful to both, Dr. Chertkovand Dr. Turitsyn, for their countless hours of discussion, inspiration and expertise whichalways led me the right way. My experience in Los Alamos and the USA in general ispriceless and influenced my thinking and world view in a profound way.

There have been several other people in the past who influenced my thinking as well aspersonal and professional development strongly. I especially want to thank Prof. ErikAurell, who basically was the seed for my professional development and made my stayin Los Alamos possible. He was the first one to teach me what interdisciplinarity andbroadness in theoretical research means.

At the end I want to thank my parents, who always encouraged me in my path, providedme with the necessary resources and always have been and will be there for me.

5

Abstract

Science is not an end in itself, but is there to improve society by improving knowledge.Hence modern theoretical physics is interdisciplinary - and it has to be. This thesis is anattempt to make this thinking clear. Not only are a lot of tools and concepts developedin theoretical physics used in other context, but also concepts from other disciplines canproof enormously useful when talking about ”native” physics issues. In this thesis I ambuilding exactly such a connection. The concept of queueing networks and its establishedmathematical theory is transferred into a physics context and is shown to be useful inextending the non-equilibrium statistical physics framework of zero-range processes. Indetail, the 1-dimensional zero-range process is extended to the n-dimensional case. Thebehavior of a special n = 2 model is studied analytically and numerically. In such asetting new effects, not known in the 1-dimensional case, emerge. This study is far fromexhausted and offers material for further research.

7

Zu aller erst...

Und jedem Anfang wohnt ein Zauber inne,Der uns beschutzt und der uns hilft, zu leben.- Hermann Hesse

Die Entstehungsgeschichte dieser Arbeit ist etwas ungewohnlich und ihre Existenzverschiedenen Menschen zu verdanken, denen ich an dieser Stelle danken mochte. DieseDiplomarbeit ist offiziell an der Physikalisch-Astronomischen Fakultat der Friedrich-Schiller Universitat Jena, unter der Betreuung von Professor Gerhard Schafer, eingere-icht worden. Ich bin Herrn Professor Schafer unsaglich dankbar fur die Freiheiten, dieer mir in meinen Studien gelassen hat, sowie dafur in den letzten Jahren ein wertvollerMentor gewesen zu sein.

Diese Arbeit wurde am Los Alamos National Laboratory, NM, USA von April 2010bis April 2011 unter Anleitung von Dr. Michael ”Misha” Chertkov und Dr. Konstantin”Kostya” Turitsyn angefertigt. Ich bin Dr. Chertkov uberaus dankbar dafur, dass ich inseiner Arbeitsgruppe in Los Alamos eine sehr inspirierende, vielfaltige und schlichtwegexzellente Forschungsumgebung finden durfte, sowie die Bereitstellung finanzieller Mittelunter US National Science Foundation grant ”Harnessing Statistical Physics for Compu-tation and Communication”. Ich bin beiden, Dr. Chertkov und Dr. Turitsyn dankbarfur die unzahligen Stunden wissenschaftlicher Diskussionen, Inspirationen und das Weit-ergeben ihrer Expertise, die mich stets in die richtige Richtung gefuhrt hat. Meine Er-fahrung in Los Alamos und den USA ganz allgemein ist unbezahlbar und hat mich tiefbeeinflusst.

Naturlich haben verschiedene andere Menschen in der Vergangenheit mein Denken sowiemeine personliche und akademische Entwicklung positiv beeinflusst. An dieser Stellemochte ich ganz besonders Professor Erik Aurell danken. Er war der Grundstein furmeine akademische Entwicklung in den letzten zwei Jahren und hat meinen Aufenthaltin Los Alamos moglich gemacht. Er war der erste von dem ich erfahren durfte, wasInterdisziplinaritat und Breite in der theoretischen Forschung bedeuten kann.

Zum Schluss mochte ich auch, und vor allem, meinen Eltern danken. Sie haben michstets in meinem Weg unterstutzt, waren immer fur mich da und werden es immer sein.

9

Zusammenfassung

Wissenschaft hat keinen Selbstzweck. Ihre Aufgabe ist es, unsere Gesellschaft durch dieErweiterung des menschlichen Wissens voranzutreiben. Daher ist moderne theoretischePhysik interdisziplinar - und sie muss es auch sein. Diese Arbeit ist ein Versuch genaudieses Denken von Interdisziplinaritat exemplarisch darzustellen. Es ist nicht nur so, dassviele in der theoretischen Physik angesiedelte Methoden in vielen anderen Disziplinennutzbar und nutzlich sind, sondern gibt es auch Methodiken aus anderen Fachbereichen,die in der theoretischen Physik angewendet werden konnen. In dieser Arbeit baue ichgenau eine solche Verbindung. Das Konzept der Warteschlangennetzwerke (queueingnetworks) und ihre etablierte mathematische Theorie wird in einen physikalischen Kon-text gesetzt. Es wird gezeigt, dass mit dieser Symbiose die Erweiterung der Theorieder zero-range Prozesse aus der nicht-Gleichgewichts statistischen Physik moglich ist.Um genau zu sein, die 1-dimensionale Theorie der zero-range Prozesse wird auf eine n-dimensionale Theorie erweitert. das Verhalten des speziellen n = 2 Falls wird analytischund numerisch untersucht. Neue Effekte, bislang unbekannt in der n = 1 Theorie, bildensich heraus. Diese Arbeit ist weit davon entfernt vollstandig zu sein, sondern bietetMaterial fur weitere Studien dieser generellen Theorie der n-dimensionalen zero-rangeProzesse.

11

Notations and Conventions

P usually denotes a probability or probability distribution.

Wkk′ denotes the transition rate from state k′ to k.

Y capital roman letters usually denote a random variable.

〈x〉, x denotes the average of x.

X ∼ P if X is a random variable, this denotes that X is distributed according to distribution P .

P ∼ f denotes that P is (up to a constant) equal to f . This is often used when P is a probabilitydistribution and f is not yet normalized.

a ≈ b denotes that a is approximately b.

N bold letters denote vectors or matrices.

∂i denotes the graph neighbor of i.

(a)i = ai denotes the i-th element of vector a.

a denotes that a is an operator.

|s〉 denotes the ket-vector s.

13

0. How to read this thesis

This thesis consists of two parts and is divided into six consecutive chapters. Part oneprovides the necessary theoretical foundations to understand the material presented inpart two.Part one is mostly a repetition of known concepts and results. We will either show

short proofs/derivations of important results or otherwise cite appropriate references.

• Chapter one begins with a short introduction into the theory of non-equilibriumstatistical physics and proceeds by presenting important results in the field ofstochastic processes. To understand this chapter basic knowledge of probabilitytheory is necessary.

• Chapter two is devoted to the mathematical theory of queues. Building on thematerial presented in chapter one, we will introduce the basic concept thoroughlyand proceed with the more advanced theory of queueing networks. We put specialemphasis on Burke’s theorem, which will provide us with a strong concept to findthe steady-state solution of a queueing network without the necessity to solve theMaster Equation.

• Chapter three is an introduction into the concepts of (1-dimensional) zero-rangeprocesses and exclusion processes, which are famous in non-equilibrium statisticalphysics.

Part two presents new ideas and original research conducted for this thesis.

• Chapter four presents the main contribution of this research, which is to establisha strong connection between the theory of queueing networks and zero-range pro-cesses. Building on that we suggest a possible queueing-network-based treatmentof a general n-dimensional zero-range process.

• Chapter five provides, based on the material of chapter four, numerical studies ofthe general 2-dimensional zero-range process.

• Chapter six provides a summary as well as a path forward.

15

Part I.

Theory

17

1. Non-Equilibrium Statistical Physics

1.1. Introduction

The classical theory of thermodynamics is concerned with providing a phenomenologicalunderstanding and mathematical framework to deal with systems of many particles. Theprime example here, and the motivation of developing such a theory, was to understandthe behavior of gases using their main characteristics: temperature, pressure and volume.Dating back to the early 19th century and Sadi Carnot’s (1796-1832) theory of theprinciples and fundamental limits of the newly emerging steam engine, thermodynamicswas from the beginning a very applied theory, almost an engineering discipline. Itwas only with Ludwig Boltzmann (1844-1906), Josiah Willard Gibbs (1839-1903) andJames Clerk Maxwell (1831-1879) that phenomenological thermodynamics got a moremathematical flavor. Statistical mechanics was used to describe the gas as a many-particle system. This newly emerged theory was able to reproduce the phenomenologicalresults of thermodynamics from a statistics based point of view and is the foundation ofwhat today is called statistical physics. Statistical physics is a very general theory (ormaybe more precise: a set of tools) to describe many-particle systems of every kind ina statistical/probabilistic fashion where, due to the vast number of degrees of freedomand interaction, a classical (mechanical) Hamiltonian approach is not feasible. Mostfamously, the Boltzmann-Gibbs distribution links the variables Energy of a state andTemperature of a system in thermodynamic equilibrium to its microscopic probabilitydistribution

P (x) =1

Z(β)e−E(x)β, (1.1.1)

where x denotes the state of the system, P (x) the probability that the system will befound in that state, Z(β) is the partition function (the normalization factor) and β isthe inverse temperature, scaled by the Boltzmann constant kB

β =1

kBT. (1.1.2)

This result is remarkable, since it builds the connection between phenomenological ther-modynamics and microscopic statistical physics. Even more remarkable is, that thereexists one distribution which is able to describe the statistics of every system in ther-modynamic equilibrium. It is also this one distribution which is at the very heart ofequilibrium thermodynamics and statistical physics. From it, and specifically the parti-tion function Z(β), every thermodynamic quantity can be derived. This is, because thepartition function is directly linked to the free energy F = −β−1ln(Z(β)), which is athermodynamic potential and hence contains every thermodynamical information about

19

Chapter 1: Non-Equilibrium Statistical Physics

the system.In equilibrium settings the Boltzmann distribution provides the correct statistical de-scription of the system. In systems out of equilibrium (and here we will be mainlyconcerned with systems out of ”chemical” equilibrium, i.e. systems with changing parti-cle number) this elegant approach does not hold - in general there is no energy functionwhich can be assigned to the system and no universal statistical description exists. In-stead, in such settings one must employ other methods. One such method is to solve forthe Master Equation of the system directly1, which often is an exceptionally hard prob-lem and does not guarantee analytical feasibility[25]. The concept of Master Equationswill play an important role in this thesis and will be outlined in more details later.The boundary between equilibrium and non-equilibrium phenomena is often very loose

so that we feel to elaborate on this issue in more details. In classical thermodynamics asystem is said to be in equilibrium, if it is in

• thermal equilibrium, i.e. the temperature of the system is constant

• mechanical equilibrium, i.e. the system is mechanically stable

• chemical equilibrium, i.e. the concentration of its compounds is constant

simultaneously. Every completely isolated system will eventually arrive at such an equi-librium state when time t → ∞. Those are however phenomenological measures. Whatdoes this mean from a statistical point of view? Let’s assume that the process we areinterested in is Markovian, which is true for most ”physical” processes. Then the Mas-ter Equation2 of the system provides a set of differential-difference equations for theprobability distribution for every state:

dPk(t)

dt=∑

k′

[Wkk′Pk′(t)−Wk′kPk(t)] . (1.1.3)

Informally, as an easy analogy to classical mechanics, one would expect to find an equi-librium solution via solving the Master Equation for

dPk(t)

dt= 0. (1.1.4)

However, this is not a sufficient equilibrium condition but merely the weaker conditionof stationarity. Solving this equation one finds a time-independent, stationary solution,at which a real-world system eventually arrives in the large t limit. In this case it clearlyholds:

∑

k

Wkk′Pk′(t) =∑

k

Wk′kPk(t) (1.1.5)

1The Master Equation exists for systems whose underlying microscopic description as a stochasticprocesses shows the Markov property, which is the case for the vast majority of physical systems.

2For a detailed treatment and derivation of the Master Equation, see the section on Markov Processes.We merely state it here to be able to introduce the notion of detailed balance in an early stage of thisthesis.

20


To arrive at an equilibrium solution, in the sense defined above, one needs to impose thestronger condition of detailed balance:

Wkk′Pk′(t) = Wk′kPk(t) (1.1.6)

on the underlying process. Clearly stationarity is a necessary condition for detailedbalance and an equilibrium state is a special steady state. It can be shown rigorously(see e.g. [36]) that detailed balance holds in closed and isolated physical systems, whichmatches our phenomenological definition of equilibrium. This also means that for allprocesses which show detailed balance (and even stronger: only for those) equilibriumstatistical physics can be applied and the Boltzmann-Gibbs distribution is the correctprobability measure. Again, in a non-equilibrium setting, i.e. a setting with detailedbalance being broken, no such universal measure exists, but the Master Equation has tobe solved for its steady state. It is rather rare that in such cases the Master Equationcan be solved exactly. In this work however, we will deal with exactly such a solvablenon-equilibrium system, which hence is often called the Ising-model of non-equilibriumstatistical physics.

Due to its generality, modern Statistical Physics is highly interdisciplinary. Its meth-ods are used far beyond solid state physics - basically everywhere where a statisticaldescription of a large number of (interacting) ”particles” or the concept of entropy isneeded. So for example in computer science, the theory of optimization (and all itsapplications), complex network theory or engineering disciplines.

For further reading on equilibrium and non-equilibrium statistical physics in general,see references [27, 28, 25, 16]. For its treatment in modern, interdisciplinary contexts seeespecially [34]. For some applications in different disciplines see e.g. [1] (complex net-works), [32] (optimization), [22, 2] (Belief Propagation and distributed message passing)or [35] (power grid engineering and mitigation of blackouts).

1.2. Stochastic Processes

In this section we will introduce the notion of a stochastic process. This concept will playan important role in this thesis, especially in the form of so called Markov Processes,which will be the subject of the next section.

Before talking about stochastic processes, it is necessary to briefly review the idea ofa stochastic variable. We will not go too much into detail, but rather choose only topresent the basic ideas, which are necessary to understand the material of this thesis.For a very readable (and classic) treatment of probability theory, see e.g. [15].Stochastic variables are mathematical objects used to capture the outcome of a randomexperiment. Associated with a stochastic variable X is a set Ω of possible values x ∈ Ωthe variable can take. Also, for every value (outcome) x there exists a number P (X =x) ∈ [0, 1] (which is called a probability), such that

∫

∑

x∈Ω

P (X = x)dx = 1 (1.2.1)

21


holds3. For example, in a standard throw-a-dice experiment, one could define a randomvariable X =outcome of the throw. In this case, the value x of the stochastic variable Xcan take the values x ∈ 1, 2, 3, 4, 5, 6. If the dice is fair P (x) = 1/6 ∀x, since everyoutcome of the experiment would be equally probable.From a mathematical point of view, every function (mapping) of a stochastic variableis again a stochastic variable: Y = f(X) is a stochastic variable and its probabilitydistribution is naturally given via:

P (Y = y) =

∫

∑

x∈Ω

δ(y − f(x))P (X = x)dx, (1.2.2)

which isP (Y = y) =

∑

x∈Ω:y=f(x)

P (X = x) (1.2.3)

in the discrete case. Especially, every function of a stochastic variable and a parametert, would be a perfectly fine random variable on its own

Y (t) = f(X, t). (1.2.4)

Y (t) is called a random function or a series of random variables (if t is discrete). If thas the notion of being a parameter measuring time, then Y (t) is said to be a stochasticprocess. For every parameter value (t∗), Y (t∗) is a stochastic variable. And of coursethere is a probability distribution associated with each of those stochastic variables

P (Y = y, t), (1.2.5)

which now is a function of parameter t as well. It is exactly this last statement, whichmakes the use of stochastic processes interesting in a (timely evolving) physical world:the concept of a stochastic process captures the idea of randomness and time evolution:P (Y = y, t) is the timely evolving probability distribution of a stochastic variable Y .How one exactly obtains this quantity for a stochastic process Y (t) will be shown in thenext section for a certain type of stochastic processes, so called Markov processes.In this work we will basically deal with stochastic processes over discrete states. Spe-

cializing to the discrete case, we will re-formulate P (Y = y; t) as P (k, t) = Pk(t), denot-ing the probability that a system is in the discrete state k at time t. In some sense, thisis a more general notation, since it allows in an easy way to talk about systems withmore than one stochastic variables at once. A state k is here defined as a particularrealization of the random variable(s) in a system. For example, consider a system whichcan be described by two random variables, Y1 and Y2. Let each of those variables takea value from the finite set Ω = 0, 14. We then define exactly four distinct states ofthe system (table 1.1). Of course, if one wants to talk about a set of random variablesin a system one could (instead of talking about states) just replace Y in (1.2.5) by a

3Here we choose to combine two possibilities: if x is discrete then the summation applies. If x iscontinuous we have to integrate.

4One may think about a system of two spins.

22


k y1 y2 P (Y1 = y1, Y2 = y2; t)

1 0 0 P1(t)2 1 0 P2(t)3 0 1 P3(t)4 1 1 P4(t)

Table 1.1.: A possible mapping of a 2-spin system to a state variable k.

stochastic vector Y = (Y1, Y2), each element Yi being a stochastic variable. However,here we choose to stick (mainly for notation purposes) with the description in terms ofstates.

Stochastic processes are a mathematical tool for modeling a lot of physical (”real-world”)phenomena: Brownian Motion (the Wiener Process), changes in the stock market or pop-ulation dynamics. For a nice textbook on this issue see [36]. For the new emerging fieldof Econophysics, in which the theory of stochastic processes plays a central role, see e.g.[31].

1.3. Markov Processes

We now proceed to describe a special stochastic process and some (for this work) im-portant properties of it: the Markov Process. A stochastic process is called a MarkovProcess, if for the conditional (transition) probabilities the Markov Property holds:

P (X(tn) = xn|X(t1) = x1, ..., X(tn−1) = xn−1) = P (X(tn) = xn|X(tn−1) = xn−1)(1.3.1)

or using the notation of states

P (kn, tn|k1, t1...kn−1, tn−1) = P (kn, tn|kn−1, tn−1) (1.3.2)

with tn > tn−1 > ... > t1. This property is quite remarkable and of big physicalsignificance. It basically says, that in a process holding this property, the value of therandom variable X (or the state) at time tn will only depend on the previous state ofthe process, i.e. the value X obtained at time tn−1. This is a good model for a lot ofsignificant real-world processes without memory, like Brownian motion: the place andmomentum of a particle will only depend on the at time tn sampled perturbation inmomentum, given the place and momentum at time tn−1, and will not depend on itspast.The Markov property also leads directly to what is formally known as memorylessness.

Memorylessness makes a statement about the distribution of times τ a stochastic processspends in a state i, specifically it states that a continuous time, memoryless stochasticprocess has exponentially distributed transitions times τ :

τ ∼1

τe−

τ

τ (1.3.3)

23


where τ is the mean transition time. That an exponentially distributed random variablein a stochastic process is a sign of memorylessness, is due to the fact that the exponentialfunction locally, i.e. in an ǫ-environment, ”looks the same” everywhere on the wholedomain due to the fact, that f ′(x) = f(x) whereas f ′(x) denotes the first derivative off with respect to x. Let me demonstrate this statement and make it mathematicallymore precise:

The exponential distribution is a sign of memorylessness. Consider a random variableT , distributed with P (T = τ) = τ−1e−

τ

τ . If a stochastic process is supposed to bememoryless, the outcome of sampling the random variable describing it needs to beindependent of its past. In plain language, that means that the probability distribu-tion P (T ) has to be independent of an offset τ0. Since for the exponential function thefunctional equation

f(x+ y) = f(x) · f(y) (1.3.4)

holds, one arrives at

P (T = τ + τ0) =1

τe−

τ+τ0τ (1.3.5)

=1

τe−

τ

τ e−τ0τ (1.3.6)

=e−

τ0τ

τe−

τ

τ . (1.3.7)

Normalizing this with help of∫ ∞

τ0

e−τ

τ = τ e−τ0τ (1.3.8)

one arrives at the new probability density P ′(T = τ) function for τ > τ0

P ′(X = τ) =1

τe

τ0τ e−

τ

τ (1.3.9)

But this is exactly the initial probability distribution, only with a new normalizationconstant. This is what we mean by ”looks the same” - the functional relationship betweenthe probability and the argument τ is identical.

In this demonstration, it is exactly the functional equation (1.3.4) which leads tothe statement. Since this functional relation is unique to the exponential function [3],the vice versa statement of the above also holds: it is only the exponential function,which shows the property of memorylessness. This is an important result in the area ofstochastic processes and the direct link between theory and real-world phenomenon.

A well known result (see e.g. [36]) connects the Markov process and the exponentialdistribution of transition times:

Every continuous-time Markov process shows the property of memorylessness. Hencestate-transition times in such processes are exponentially distributed.

24


This is a key result in the theory of stochastic processes and again one reason whyMarkov chains and Markov processes are of interest in the physical sciences.Markov processes have the nice property that there exists a unique differential-differenceequation for every system, which allows to solve for the probability density: the Mas-ter Equation (1.1.3). The Master Equation is a direct consequence of the Chapman-Kolmogorov equation. The Chapman-Kolmogorov equation itself follows directly fromthe Markov property and states:

P (k3, t3|k1, t1) =∑

k2

P (k3, t3|k2, t2)P (k2, t2|k1, t1). (1.3.10)

This is basically a ”path-integral” kind of logic: the probability to be in state k3 at timet3, given that the system was in state k1 at time t1, is equal to the sum over probabilitiesof all possible paths (states) between t1 and t3. Employing Bayes’ theorem5, one findsthat the above equation is equivalent to

P (k3, t3|k1, t1) =∑

k2

P (k3, t3; k2, t2|k1, t1). (1.3.11)

We can use this property of the Markov process to directly derive the Master Equation.

Derivation of the Master Equation. We want to find an expression for

dP (k, t)

dt=

dP (k, t|k0, t0)

dt= lim

∆t→0

P (k, t+∆t|k0, t0)− P (k, t|k0, t0)

∆t(1.3.12)

where we conditioned on a universal beginning state k0 at time t0. Now, using equation(1.3.11) and again Bayes’ theorem one finds

P (k, t+∆t|k0, t0) =∑

k′

P (k, t+∆t|k′, t)P (k′, t|k0, t0) (1.3.13)

P (k, t|k0, t0) =∑

k′

P (k′, t+∆t|k, t)P (k, t|k0, t0). (1.3.14)

Plugging those two identities into equation (1.3.12):

dP (k, t|k0, t0)

dt=

∑

k′

lim∆t→0

[

P (k, t+∆t|k′, t)

∆tP (k′, t|k0, t0)−

P (k′, t+∆t|k, t)

∆tP (k, t|k0, t0)

]

=∑

k′

[

Wkk′P (k′, t|k0, t0)−Wk′kP (k, t|k0, t0)]

dP (k, t)

dt=

∑

k′

[

Wkk′P (k′, t)−Wk′kP (k, t)]

(1.3.15)

5Bayes’ theorem relates the conditional probability of two random variables X and Y with its jointprobability: P (X;Y ) = P (X|Y )P (Y ) = P (Y |X)P (X).

25


0 1 k − 1 k k + 1... ...

µ1 µ2 µk µk+1 µk+2

λ0 λ1 λk−1 λk λk+1λk−2

µk−1

Figure 1.1.: State diagram of a simple birth-death process.

which is exactly the Master Equation. During this derivation we substituted

Wkk′ = lim∆t→0

P (k, t+∆t|k′, t)

∆t(1.3.16)

Wk′k = lim∆t→0

P (k′, t+∆t|k, t)

∆t(1.3.17)

which hence have the meaning of infinitesimal (unit time) transition probabilities (rates).The Master Equation (1.3.15) can be nicely interpreted as an ”equality of probabilityflows”: the temporal change of the probability of being in state k equals the sum ofprobability flows into that state minus the sum of probability flows out of that state.This interpretation provides at the same time a rule for the construction of the MasterEquation of a Markov Process, given the rates (unit time conditional probabilities) forchanging a state. An example will be given in the next section on birth-death processes.

For a nice text book on Markov processes see [36].

1.4. The Birth-Death Process

A special Markov process, and of interest in understanding queueing theory later, isthe Birth-Death process. Figure 1.1 shows the state diagram of such a process. Statediagrams are a very useful tool to visualize a Markov process: it shows every possiblestate of the system and its unit transition probabilities (rates) Wkk′ . Here the rates arelabeled λi = Wi,i+1 and µi = Wi,i−1. As opposed to a general Markov process, there areonly allowed transitions between two neighboring states. If k is the occupation of thestate, then an allowed transition can go to k+1, which is called a birth and k−1, whichis called a death. The transition rates from occupation state k to k ± 1 are in generaldependent on the state of departure. λk is called the birth rate, µk the death rate. Froma physics point of view, the connection to (second quantization) ladder operator tech-niques famous is fairly obvious and will play a role later, when looking at the treatmentof queueing networks by Massey [33] and Chernyak et al. [6].

To answer the question what the density profile/probability distribution of the states kin such a system is, one looks for the Master Equation of the system. Using the probabil-

26


ity flow interpretation of the Master Equation and the state diagram as a visualizationtool, one finds the following Mater Equation for a general Birth-Death process:

dPk(t)

dt= −(λk + µk)Pk(t) + µk+1Pk+1(t) + λk−1Pk−1(t) (1.4.1)

dP0(t)

dt= −λ0P0(t) + µ1P1(t) (1.4.2)

To solve this equation, the technique of z-transforms is very useful [24]. However, sincethis is not really a necessary result for this thesis, we will not do the calculation andrather look at an important special case of the birth-death process.The uniform pure birth-process, i.e. a birth-death process with µk = 0, is of special

interest since it is a model for a Markovian, memoryless process. The Master Equationfor this process reads

dPk(t)

dt= −λPk(t) + λPk−1(t) (1.4.3)

dP0(t)

dt= −λP0(t). (1.4.4)

This system is easily solved iteratively. For k = 0 the solution is (simply via integration)

P0(t) = e−λt. (1.4.5)

For k = 1 the differential equation reads

dP1(t)

dt+ λP1(t) = λe−λt (1.4.6)

which is a non-homogeneous and not-separable ordinary differential equation, which canbe solved by the method of introducing an integrating factor (see e.g. [3]). Thus onearrives at

P1(t) = λte−λt. (1.4.7)

Iterating this procedure, one arrives again at a differential equation of the same typeand, via an integrating factor, yields the iterative equation

Pk(t) = e−λt

∫

λeλtPk−1(t)dt (1.4.8)

with solution

Pk(t) =(λt)k

k!e−λt. (1.4.9)

Hence the occupation numbers of every Markovian birth process are Poisson distributed.An interesting and important feature of the Poisson distribution is, that a Poisson dis-tributed stochastic process, like here the process of births, has exponentially distributedinter-birth times. This means, that the time-span between two consecutive births is anexponentially distributed random variable and can be seen via the following considera-tion.

27


The inter-birth time is exponentially distributed. Let τ be the inter-birth time, i.e. thetime between to consecutive births and let X(τ0, τ0 + τ) define the random variable ofthe number of births between time span (τ0, τ0 + τ). Then, according to the dynamicsof a Markovian birth process with rate λ, it holds:

P (X(τ0, τ0 + τ) = k) =(λτ)k

k!e−λτ (1.4.10)

Hence, letting k be the parameter and τ the free variable, the probability of havingexactly one birth after duration τ is nothing but P (X(τ0, τ0 + τ) = 1). Hence letting Tdenote the random variable of having exactly one birth after time T = τ one finds

P (T = τ) = λe−λτ (1.4.11)

for the distribution of inter-birth time T 6.

Later the birth process will be used to describe any memoryless, Markovian processof arrivals of customers to the queuing system.

In this chapter we introduced the notion of non-equilibrium statistical physics on aformal level. We furthermore provided the concept of stochastic and especially Markovprocesses. In the next chapter we will use these concepts to address the mathematicalconcept of queues and queueing networks.

6The notation of T and τ as inter-birth time seems eventually a bit confusing. But T denotes therandom variable, whereas τ denotes the value this variable can take. In (1.4.10) τ is a parameter andk the free variable, the value the random variable X can take. Whereas in this equation τ is the freevariable denoting the value the random variable T can take.

28

2. Queueing Theory

2.1. A short glance on history

Queueing Theory is traditionally an Operations Research discipline. It was developedand is employed to optimize and analyze processes including waiting lines (queues), so forexample telecommunication processes, production processes or network traffic in appliedcomputer science. For a nice introduction see e.g. [24], for a more advanced treatmentsee e.g. [5].

2.2. Theory of Single Queues

This chapter is devoted to introducing the basic mathematical object queue as well aspresenting general features and theorems. Especially Burke’s theorem will play a centralrole here and later. As stated earlier, one of the goals of this thesis is to establisha connection between the Operations Research native theory of queues and a possibleanalogy in physics. However, in this chapter we will mostly use the Operations Researchterminology and will later build the desired connection.

A queue is a mathematical object combining two ideas:

a) the idea of a service facility. Its purpose is to complete jobs/serve customers whichare arriving at this facility.

b) the idea of a storage, in which arriving jobs can be queued if the facility is busy(Fig. 2.1).

Such a general system is normally notated as a G/G/m/n system. So what does thismean? To describe a queueing system completely one has to specify four parameters:

1. A parameter to quantify how jobs enter the system.

2. A parameter to quantify how jobs are completed in the facility.

3. A parameter to quantify the ”parallelness” of job completion, i.e. how many jobscan a facility complete at the same time.

4. A parameter to specify the capacity/storage space of a system, i.e. how many jobscan be queued before being processed in the facility.

The above notation is used to denote a very general queue: jobs enter the system in aGeneral fashion, are completed in a General fashion, at most m jobs in parallel and the

29

Chapter 2: Queueing Theory

b

b

b

b

b

bA(t)

B(t)

m

n

b bb b b b

Figure 2.1.: Illustration of a single A(t)/B(t)/m/n queue. Red squares denote jobs enteringthe system, green squares denote completed jobs leaving the system.

storage capacity is n jobs.However, the notion of ”how jobs enter the system” and ”how jobs are completed in thefacility” is not very precise. Every model and every specification in a model is basedon the desired result. In the case of queueing systems one is mostly interested in thenumber of total jobs in the system, the time a job spends in the system and the numberof jobs waiting in the queue. Of course, from a physics point of view a Hamiltonianstandpoint would be totally acceptable and a solution of the problem: specify the exacttimes at which jobs arrive to the system as well as the microscopic dynamics of how jobsare completed. Then construct the Hamiltonian of the system and solve a general kindof equations of motion. However, such an approach is certainly very limited. In mostsystems, especially those arising in Operations Research, one has to deal with a largenumber of unpredictable jobs arriving at the system. In such cases a statistical approachis much more preferable. Hence it is useful to specify the distribution of inter-arrivaltimes ti of jobs entering the system

P (ti ≤ t) = A(t) (2.2.1)

and the distribution of service times ts of completing a job in the facility

P (ts ≤ t) = B(t). (2.2.2)

Note that A(t) and B(t) are cumulative distributions, to stick with the standard notationin queueing theory [24]. The inter-arrival time ti is defined as the time interval betweenthe consecutive arrival of two jobs to the system.We specified the system in terms of constant integer-valued variables m and n as wellas real-valued, positive random variables ti and ts, which we will call the free variablesof our system. As is well known, every function of one or more random variables isitself again a random [15]. Hence the results of the queueing network analysis will berandom variables as well and we end up with a completely probabilistic description of ourunderlying real-world phenomenon. This is important to keep in mind for interpretation

30


of results. It also opens the way to a statistical physics treatment of the problem, whichwill be presented later.

After defining the basic quantities of a single queue it is rather straight forward to getsome first interesting results. First let us define a couple of more quantities, followingdirectly from the basic free variables already defined. Denoting the absolute arrival timeto the system of some job n with τn, it is clear that the inter-arrival time tn between jobn− 1 and n is

ti(n) = τn − τn−1. (2.2.3)

Defining the waiting time tw(n) of a job in the system, i.e. the time it waits in thequeue before the service facility will start processing it, the system time tΣ(n) of job n,is defined as

tΣ(n) = ts(n) + tw(n). (2.2.4)

The waiting time tw and its distribution will be a property of the system, following fromthe free variables in a non-trivial way.Calculating some basic average properties, like the average inter-arrival time1

ti =

∫

∑

ti

tiP (ti)dti (2.2.5)

and the average service time1

ts =

∫

∑

ts

tsP (ts)dts (2.2.6)

one defines the average arrival rate

λ = ti−1 (2.2.7)

and the average service rate

µ = ts−1. (2.2.8)

This definition is, from a physics point of view, quite natural. Additionally, it willproof useful later when talking about continuously exponentially distributed inter-arrivaltimes, since in this case the inverse of the distribution parameter exactly equals theaverage in such

t =

∫

tλe−λtdt = λ−1. (2.2.9)

Another interesting quantity is the distribution P (α(t′)) of customers α(t′) arriving tothe system within the time interval [0, t′]. Here α(t′) is a random variable with parametert′. Obviously the probability of arrival of exactly one customer is exactly the probabilityof having inter-arrival time t′

P (α(t′) = 1) = P (ti = t′). (2.2.10)

1Using∑

or∫

depending on whether we talk about discrete or continuous time variables respectively.

31


The probability of having exactly two arrivals in [0, t′] is given via the convolution of theinter-arrival time distribution

P (α(t′) = 2) =

∫

dτ1P (ti = τ1)P (ti = t′ − τ1). (2.2.11)

Thus the probability of having exactly n arrivals in [0, t′] is given by the followingrecursion

P (α(t′) = n) =

∫

dτ1

∫

dτ2...

∫

dτn−1

n−1∏

j=1

P (ti = τj)P (ti = t′ −n−1∑

k=1

τk). (2.2.12)

Note that, in our interpretation, t′ is a parameter and n is the free variable. If theinter-arrival times are exponentially distributed with parameter λ, the average numberof arrivals within [0, t′] is given by

¯α(t′) = λt′, (2.2.13)

The most interesting queueing theoretical quantity for this thesis will be the numberof customers in the system at time t, which we will denote by

N(t). (2.2.14)

Again, this quantity is a random variable. We are most interested in a possible steady-state solution of the system, i.e. the limit

limt→∞

N(t) = N. (2.2.15)

Such a limit does not necessarily exist and even if it does, its calculation can be far fromtrivial, since generally one would have to solve for the Master Equation of the system.An important queueing theorem correlates the average number of customers in the sys-tem (in the t → ∞ limit), N , to the average system time, tΣ:

N = λtΣ (2.2.16)

This result is known as Little’s result in queueing theory and can be translated as”average customers in the system=(average arrival time to the system)×(average systemtime)”. This seems from a physics point of view fairly obvious. However, the proof forthis theorem was only established in 1961 by John D. C. Little’s[30].In a system containing a simple queue with only one facility, the following measure is

an important quantity:

ρ =λ

µ. (2.2.17)

In queueing theory this is called the utilization factor. Its meaning is important fordeciding of whether or not a stable steady-state solution for the system exists. If

ρ > 1 (2.2.18)

32


holds, the system will not be stable, in the sense that the average waiting time of acustomer will go to infinity, since limt→∞N(t) = ∞ will hold. This is easily seen, sincein this case (via (2.2.7) and (2.2.8)) the service rate would exceed the arrival rate andhence congestion in the waiting queue will result2. Another interpretation of ρ is in termsof “fraction of time the facility is busy”. We will show later that (quite remarkably)equation (2.2.18) holds as a stability condition in a much more general setting than theone-queue system.

2.3. The Single Markovian Queue as Birth-Death System

In the previous chapter we said that the number of customers in the system is an impor-tant quantity for this thesis. However, its computation seems not so straight forward.The knowledge of this quantity is also important to be able to make a theoretical state-ment about the number of customers leaving the system, which clearly is a function ofhow many customers are present in the system.Developing an universal description of the general G/G/m/n system of one queue seemsto be a bit too optimistic. Also, from a physics perspective, this generality is mostlynot needed. Instead, a very common queueing system which has been studied a lot isdenoted by M/M/n/m. This system is one with Markovian input, Markovian servicecompletion, n servers and storage of size m. As pointed out earlier, Markovity andmemorylessness are highly coupled and the underlying principles governing a vast num-ber of real-world stochastic processes. From the things pointed out earlier, it is directlypossible to specify the stochastic processes underlying an M/M/n/m system in moredetail using the analogy to Birth-Death processes. In such a mapping the arrival of acustomer to the system is modeled as birth of a customer to the system and the comple-tion of service as death. In a M/M/n/m system with customer arrival rate λ, this rateis directly related to the birth-rate λ of a corresponding pure birth process. Since in thisMarkovian queueing system the arrival of customers is supposed to be Markovian (i.e.independent of the history of arrivals), we find (transferring the results from section 1.4)that the arrival process to the system is a Poisson distributed stochastic process

P (Narr = k, t) =(λt)k

k!e−λt (2.3.1)

with exponentially distributed inter-arrival times

P (ti = t) = λe−λt. (2.3.2)

Also, if we denote the completion rate of every facility in this system with µ, the MasterEquation for the distribution of number of customers in the system for every given time

2For illustration, the reader may think of its last visit to whatever (German) public administrativeoffice.

33


t is in analogy to equation (1.4.2) given by

dPk(t)

dt= −(Θ(m− k)λ+min(n, k)µ)Pk(t) + min(n, k + 1)µPk+1(t)

+Θ(m− k + 1)λPk−1(t) (2.3.3)

dP0(t)

dt= −λP0(t) + min(n, 1)µP1(t)

Here Θ is a Heaviside-like function with Θ(x) = 1, if x ≥ 0 and Θ(x) = 0, otherwise.What distinguishes this Master Equation from the one of the pure birth-death process arethe multiplicities of the facilities n. This basically leads to a linear increase in customerservice completion and is accounted for by multiplying the death-rates by min(n, k).The restriction of storage capacity is here modeled via the Θ-function.We will not solve this Master Equation but rather look at the important case of the

M/M/1/∞ queue, i.e. a system with one facility and infinite storage. In this case theMaster Equation is just (1.4.2) and as mentioned earlier the time-dependent solutionrather complicated. However, considering the steady-state case

0 = −(λ+ µ)P ssk + µP ss

k+1 + λP ssk−1

0 = −λP ss0 + µP ss

1

finding a solution is possible iteratively and the steady-state solution of the distributionof customers N in the M/M/1/∞ system is given by

P ss(N = k) = P ss0

(

λ

µ

)k

, (2.3.4)

as one easily checks via substituting back into the homogeneous Master Equation. Thisresult is a power-law distribution. P0 serves as the normalization constant and can befound, via the convergence of the geometric series, to be

P0 =

(

∞∑

k=0

ρk

)−1

= 1− ρ, (2.3.5)

where

ρ =λ

µ(2.3.6)

is the previously introduced utilization factor. Probability distribution (2.3.4) is onlynormalizable, if the geometric series in (2.3.5) converges. Convergence is assured onlyfor 0 < ρ < 1 and hence a steady-state solution for this system only exists if λ < µ,which matches the criterium pointed out in (2.2.18) and provides for this system astrict mathematical explanation. Having obtained this solution, the average number ofcustomers in the steady-state is given by

N =ρ

1− ρ. (2.3.7)

34


In the steady-state the distribution of customers Nout leaving the facility can also becomputed. On a first thought one might guess that the P (Nout) will be highly coupledto the number of customers N in the system and will be a complicated result. However,the following consideration shows that the solution is in fact rather simple - albeit verysurprising:

Burke’s theorem. Consider a M/M/1/∞ system of a single queue with customer arrivalrate λ and job-completion rate µ. As established earlier, in a Markovian system, theinter-arrival time and facility completion time are exponentially distributed. As seenearlier, if there is a constant Markovian birth process with rate λ, the number of birthsin every time period is Poisson distributed with rate λ. Since the facility completionprocess is such a process, the number of customers leaving the queue would be Poissondistributed with rate µ, iff there would be a constant supply of customers. However,in this model the supply of customers is regulated by the number of customers in thesystem N , which is in the steady state case distributed with (2.3.4). Hence the supplyof customers to the service facility is constrained exactly by this distribution. Let mederive the statistics of the output:

P (Nout = k, t) ∼ e−µt (µt)k

k!· P (N ≥ k)

∼ e−µt (µt)k

k!

∞∑

l=k

(1− ρ)ρk (2.3.8)

∼ e−µt (µt)k

k!ρk.

Using (2.3.6) and normalizing correctly we finally get

P (Nout = k, t) = e−λt (λt)k

k!. (2.3.9)

This rather surprising result says, that in the steady-state case the output of theM/M/1/∞ system is, as the input, Poisson distributed with the arrival-rate λ. Thisresult is often known as Burke’s theorem[4]. Interestingly, there is (in physics terms)something like a first order phase transition in the following sense. As we have shown,a solution for the distribution of the number of customers N in the system only exists ifρ ≤ 1. In a M/M/1/∞ system with infinite storage space of arriving jobs, this meansthat in the case ρ ≥ 1 the number of customers will diverge, i.e. go to N → ∞. Restatingthis fact, there is an infinite number of supply for the facility, keeping it constantly busy.Hence the output of the queueing system will, opposed to the ρ ≤ 1 case, solely dependon the dynamics of the service facility and not anymore on the input. Hence Burke’stheorem will not hold in this scenario. Instead the output of the system will be Poissondistributed with rate µ (since the service time is exponentially distributed with rate µ).So with ρ → 1 there is an abrupt (first order) change (phase transition) in the output ofthe system from a Poisson distribution with rate λ to a Poisson distribution with rateµ.

35


Q1 Q3

Q2

λ12 λ23

λ31

λ01

λ20

λ30

λ13

Figure 2.2.: Illustration of a queueing network with 3 nodes.

2.4. Networks of M/M/1/∞ queues

A straight forward extension of the theory of a single queue is to consider networks ofqueues. Figure 2.2 illustrates a possible queueing network. These systems are moreelaborate and complex than the former one. The input of one queue in the networkmight not only depend on customers arriving from outside of the system, but will alsodepend on transitions of customers from the output of one queue to an adjacent queue3.This leaves us with a highly coupled system which needs to be analyzed. This task is verymuch simplified by Burke’s theorem: even queues, which are being fed by transitioningcustomers, i.e. customers just leaving another queue, will have Poissonian input andhence can be treated as as a single, independent M/M/1/∞ queue, as presented in theprevious chapter:

The sum of two independent Poisson distributed variables is again Poisson distributed.Consider two consecutive M/M/1/∞ queues Q1 and Q2. Assume that Q1 has, withBurke’s theorem, Poissonian output with rate λ12. Furthermore assume that Q2 hascustomers arriving externally with rate λ02 and customers arriving from Q2 with ex-actly its leaving rate λ12. Considering the sum S = K+L of two independently Poissondistributed random variables K and L:

P (S = s) = (P (K) ∗ P (L))(s)

=∞∑

k=0

P (K = k)P (L = s− k) (2.4.1)

=∞∑

k=0

eλ12t (λ12t)k

k!eλ02t (λ02t)

(s−k)

(s− k)!

(2.4.2)

3The term adjacent here is meant in the graph-theoretical way, i.e. neighboring in the sense of beingconnected by an edge.

36


The infinite sum on the rhs is convergent and the sum is given by4

∞∑

k=0

(λ12t)k

k!

(λ02t)(s−k)

(s− k)!=

(λ12t+ λ02t)s

s!(2.4.3)

which leaves us with

P (S = s) = e(λ12+λ02)t(λ12t+ λ02t)

s

s!(2.4.4)

showing that the sum of two independently Poissonian distributed variables is againPoisson distributed with the sum of the single rates.

Having shown that and keeping Burke’s theorem in mind, the steady-state transitionrates λij in a network of queues can be calculated via

λij = (λ0i − λi0)−∑

k∈∂i/j

λik +∑

k∈∂i

λki (2.4.5)

where ∂i/j denotes the set of all neighbors of i without j. This is a very generalexpression which, thinking of systems of flows, basically equates influx and outflux forevery queue in the system. That such an ”average flux” conversation holds is not at allobvious, since our system is equipped with the possibility of storing jobs in a queue. Thatthis result nevertheless holds is due to the double-Markovity in the system (Markovianinput and Markovian service completion) and the resulting Burke’s theorem.In a case as described here, it is clear that transition rates λik for all neighbors k of iand λi0 must be interrelated. Concretely, denoting the effective service rate at queue iwith λi < µi, the following equation must hold:

λi =∑

k∈∂i

λik + λi0. (2.4.6)

Assuming λik = λipik with∑

k pik = 1 and pik being the probability that a servedcustomer is transferred from node i to node k or leaving the network (k = 0), one findsfor the local, effective steady-state service rates

λi = λ0i +∑

k

λkpki. (2.4.7)

This is a well-posed system of n variables and n equations which can in principal besolved using standard methods. However, if the system is large n >> 1, the solutionof such a linear system is far from trivial and special computational algorithms need tobe employed. However, in most cases the underlying network of queues will be weeklyconnected, i.e. the linear system very sparse and efficient sparse-matrix algorithms likethe Conjugate Gradient method or Gaussian Belief Propagation can be used to do thejob.The first one to mathematically describe such kinds of systems was James R. Jackson

4Using e.g. Mathematica.

37


in his seminal 1963 paper [21]. He even generalized the ideas presented above to a lessrestrictive setting, with arrival and completion rates not constant but dependent on thenumber of customers in the queue.After the effective rates of the system have been calculated and (as we have shown

previously) in the steady state every queueing system in the network can be treated asan independent queueing system with the new effective rates, the solution for the systemP (N) has to be a factorized form of the independent random variables Ni:

P (N) =∏

i

P (Ni) (2.4.8)

Again, for the i−th subsystem there is an effective Poissonian arrival rate λi and aninherent service rate µi associated. We define the i−th utilization factor via

ρi =λi

µi. (2.4.9)

Let us note that it is also possible to introduce the nominal transition rate from queue ito queue j as µij = µipij , for which then the following relation to the effective transitionrate holds:

λij = ρiµij (2.4.10)

This leads us directly toP (Ni = ki) = (1− ρi)ρ

kii (2.4.11)

and with 2.4.8 to the joint probability distribution

P (N = k) =∏

i

(1− ρi)ρkii . (2.4.12)

This result is quite remarkable. There are very few non-equilibrium systems for whichthe solution is nicely factorized. However, we want to stress that it is Burke’s theoremand hence the double-Markovian nature of the system which ensures this nice property.We also want to stress that we were able to obtain this solution without solving theMaster Equation for this coupled system but just via employing Burke’s theorem.Using the above introduced utilization factors, one can write (2.4.5) in a convenient

form as:∑

j

Mijρj = λ0i (2.4.13)

with

Mij =

−µji i 6= j

µi0 +∑

k µik i = j.(2.4.14)

If there is no misunderstanding possible, we will use the Einstein-convention and e.g. in(2.4.13) drop the sum sign and assume summation over double indices.As shortly mentioned earlier, the result obtained by Jackson [21] is more general.

More specifically, he does not restrict himself to constant service rates µi, but consider

38


service rates which can be dependent on the current number of jobs present in the singlequeue. However, the result is remarkably similar and again factorized:

P (N = k) ∼∏

i

ρkii

ki∏

l

µ−1i (l). (2.4.15)

with

λ0i = Mij ρj (2.4.16)

Mij =

−pji i 6= j

pi0 +∑

k pik i = j.(2.4.17)

Often it is more convenient to write equation (2.4.16) as a matrix equation:

λλλ = Mρρρ

where we defined the vector if arrival rates λλλ, the vector ρρρ and matrix M. We want tonote that in the standard case the diagonal elements of M will be Mii = 1.

2.5. Operator technique to solve ME for Queueing Networks

In this chapter we want to introduce a more theoretical physics based description of theopen queueing network, suggested by Chernyak et al. [6], based on the so called Doi-Peliti[12] technique and Massey’s operator theory treatment of such systems [33]. Since wealready solved the steady-state of the Queueing network in the previous section5, we willnot go too much into detail here but rather restrict ourselves to present the basic idea.We choose to present this material mainly for purposes of interest and because it againoutlines a very nice connection between a theoretical physics based theory/methodologyand the seemingly far distant field of Operations Research.In general, the so called Doi-Peliti technique is a very useful operator theory to de-

scribe classical many-particle systems and the production/annihilation of particles insuch systems. It is based on the second quantization method in quantum theory, whichis exhaustively used in high-energy particle physics to describe the creation and anni-hilation of elementary particles as described by the standard model of particle physics.The value of such a second quantization technique for classical systems is that one doesnot have to solve the Master Equation by using classical approaches to solve differential-difference equations. Since this is often a hard and eventually not feasible problem usingthe standard methods known in the theory of stochastic processes, the Doi-Peliti tech-nique offers for birth-death processes a second path to the solution. Since a queue canbe described as a birth-death process, this technique can be applied in such systems.In a discrete stochastic system, like the queueing network, the joint probability dis-

tribution of having Ni customers accumulated at queue Qi i = 1...n is the quan-tity of interest. From a quantization point of view, on can hence define a base state

5Albeit not by solving the Master Equation directly but by employing Burke’s theorem.

39


|N〉 = |N1 = k1, N2 = k2, ...NL = kL〉 = |k1, k2, ..., kL〉 of the system to have exactly kicustomers in queue Qi. Since here we are dealing with a stochastic system, the actualstate |s〉 of the system is only given by a stochastic linear combination of the base states(mixed state)

|s〉 =∑

P (N)|N〉 (2.5.1)

with P (N) being the probability of finding the system in state |N〉6. The idea in thisapproach is to transform the Master Equation of a queueing network (here we choose topresent the ME for the M/M/1/∞) network)

dP (k1, ..., kL; t)

dt=

∑

(i,j)∈E

µijΘ(kj)P (..., ki + 1, .., kj − 1, ...; t)− µijΘ(ki)P (..., ki, ..., kj)

+

L∑

i=1

P (..., ki − 1...)µ0iΘ(ki) + P (..., ki + 1...)µi0

−L∑

i=1

P (..., ki, ...)µ0i + P (..., ki, ...)µi0Θ(ki)

(2.5.2)

with Θ(k) being a Heaviside-like function with

Θ(k) =

1 if k > 0

0 else(2.5.3)

and E the set of all edges in the system, into a time-dependent Schrodinger-like equation

d

dt|s〉 = H|s〉. (2.5.4)

In order to find the steady state we are interested in the time-independent solution

0 = 0|sss〉 = H|sss〉 (2.5.5)

which hence constitutes the problem of finding the eigenstate |sss〉 of the HamiltonianH with eigenvalue 07. However, we need to transform the rhs of the Master Equationinto a second-quantization Hamiltonian. Following the notation of [6], this can be donein the following way. Define particle creation (birth) and particle annihilation (death)operators in such a way, that the standard algebra for ladder operators (see e.g. [8]) isfulfilled

a†i |..., ki, ...〉 = |..., ki + 1, ...〉 (2.5.6)

ai|..., ki, ...〉 = ki|..., ki − 1, ...〉, (2.5.7)

6This idea is clearly borrowed from the theory of many-particle quantum systems where e.g. the effectof entanglement arises.

7This would be the state with lowest energy or ”base state”.

40


which is the usual definition. Since in the ME we have to deal with a couple of Heaviside-like functions, it makes sense to introduce a ”skewed”[6] annihilation operator b, whichis in some sense a ”logical” operator:

bi|..., ki, ...〉 = Θ(ki)|..., ki − 1, ...〉. (2.5.8)

Using the so defined operators, one sees that the Master Equation translates into equa-tion (2.5.4) with Hamiltonian

H =∑

(i,j)∈E

µij(a†j − a†i )bi +

L∑

i=1

(

µ0i(a†i − 1) + µi0(1− a†i )bi

)

, (2.5.9)

which then will be used to solve the eigenvalue problem. We will not solve this problemhere but merely state the solution and refer for details to references [6, 33]. One findsthat the eigenstate elements |sss〉i =

∑

kiP (Ni = ki)|Ni = k+ i〉 of the Hamiltonian are

eigenstates of the corresponding annihilation operator bi|sss〉i = ρi|sss〉i to eigenvalue ρiand that they have the form

|sss〉i =1

1− ρia†i

|0〉 =∞∑

n=0

(ρia†i )

n|0〉 (2.5.10)

which translates for the complete state |sss〉 into

|sss〉 =∏

i

|sss〉i (2.5.11)

=∑

k

∏

i

ρkii |k〉 (2.5.12)

which yields, after normalization and comparing with (2.5.1):

P (N = k) =1

Z

∏

i

ρkii (2.5.13)

=∏

i

(1− ρi)ρkii . (2.5.14)

This is is exactly result (2.4.15) from the previous section.For further literature on this technique in connection to queueing networks, see es-

pecially [6] and the Massey paper [33]. For a general introduction into the Doi-Pelititechnique see [12].

In this chapter we have introduced the concept of queues and queueing networks. Wehave shown that it is possible for these kind of systems to obtain a closed analyticalform of the steady-state solution. Quite remarkably this solution turned out to be nicelyfactorized due to Burke’s theorem.

41

3. Zero-Range Processes and Exclusion

Processes

3.1. The Zero-Range Process

As mentioned in the introduction, the purpose of non-equilibrium thermodynamics is tosolve the Master Equation of the system under consideration. Often knowledge of thesteady state distribution is sufficient for the theoretician, hence solving the homogeneousform of the Master Equation is enough. But even obtaining this special solution in closedanalytical form is very often not possible. It is, however, well known that the so calledzero-range process (ZRP) can (at least in its 1 dimensional form) be solved exactly andexhibits an amazingly simple factorized form of the steady-state distribution. The ZRPis defined as follow: consider a 1-dimensional lattice such that there are exactly L sitesin the lattice. There are N =

∑Li=1Ni particles on the lattice, with site i containing Ni

particles. If N is fixed, then the system is said to be closed. If N is variable, due toparticles entering and leaving the network, the system is said to be open. Each site canpotentially hold an infinitely number of particles. Particles can hop from site i to oneof its neighboring sites i+ 1 or i− 1 with probability pi,i+1 and pi,i−1, respectively. Forthe probabilities it is generally assumed that

pi,i+1 + pi,i−1 = 1 ∀i (3.1.1)

holds. In this particular process the hopping dynamics are zero-range, meaning that therate ui of one particle at site i to jump to one of its neighboring sites is solely a functionof the occupation of the departure site:

ui = ui(Ni). (3.1.2)

Figure 3.1 illustrates the dynamics. In the zero-range process literature it is mostlyassumed that every site is identical and hence

ui(Ni) = u(Ni) (3.1.3)

holds. A closed system corresponds to pi0 = u0i = 0 ∀i. Also, a common choice is toassume that in this 1-dimensional system pi,i+1 = p and pi,i−1 = q for all i holds (seee.g. [29]).

If we want to describe this system from a statistical point of view, we need to knowthe underlying stochastic processes, so for example how the number of arriving particles

43

Chapter 3: Zero-Range Processes and Exclusion Processes

site 1 site 2 site 3 site 4

p12u1 p23u2 p34u3 p40u4

p21u2 p32u3 p43u4 u04

u01

p10u1

Figure 3.1.: Illustration of the 1 dimensional zero-range process with associated rates. Theprobability pi0 and the rate u0i represent the probability that a particle will leavethe grid at site i and the rate of particles entering the grid at site i, respectively.

or the number of particles leaving a site is distributed. The rate u of a physical process,as macroscopic property, is defined as average quantity per time:

u =〈n〉

t. (3.1.4)

This however implies < n >= ut and is a property of the Poisson process. Also, it isphysical to assume that the observed processes are Markovian and memoryless, whichagain is a striking property of the Poisson distribution. Hence one chooses to model thenumber of particles ni hopping from site i as a Poisson distributed random variable

ni ∼ Pois(ui) (3.1.5)

and equivalently the number of particles arriving to the system.The Master Equation of a system as shown in figure 3.1 can be seen to be:

dP (N1 = k1, ..., NL = kL; t)

dt=

L−1∑

i=1

P (..., Ni = ki + 1, Ni+1 = ki+1 − 1, ...)pi,i+1uiΘ(ki+1)

−L−1∑

i=1

P (..., Ni = ki, Ni+1 = ki+1, ...)pi,i+1ui

+L−1∑

i=1

P (..., Ni = ki − 1, Ni+1 = ki+1 + 1, ...)pi+1,iui+1Θ(ki)

−

L−1∑

i=1

P (..., Ni = ki, Ni+1 = ki+1, ...)pi+1,iui+1

+L∑

i=1

P (..., Ni = ki − 1...)u0iΘ(ki) + P (..., Ni = ki + 1...)pi0ui

−L∑

i=1

P (..., Ni = ki, ...)u0i + P (..., Ni = ki...)pi0ui.

(3.1.6)

44


and can be straight forwardly generalized to a case with general underlying geometryand L sites:

dP (N1 = k1, ..., NL = kL; t)

dt=

L−1∑

i=1

∑

j∈∂i


−

L−1∑

i=1

∑

j∈∂i

P (..., Ni = ki, Ni+1 = ki+1, ...)pi,i+1ui

+L−1∑

i=1

∑

j∈∂i


−L−1∑

i=1

∑

j∈∂i

P (..., Ni = ki, Ni+1 = ki+1, ...)pi+1,iui+1

+L∑

i=1


−L∑

i=1


(3.1.7)

where ∂i denotes the set of all direct neighbors of site i. Here again, Θ(x) is a Heaviside-like logical function with Θ(x) = 1 iff x > 0 and Θ(x) = 0 else to guarantee that onlystates with occupation numbers k ≥ 0 are counted in the sum.As to our knowledge, a solution to this very general system is in the physics literaturenot known. However, it is known that the closed general-topology system is equivalent toa ”weighted” random-walk process and its steady-state distribution is known and givenby [14, 29]

P (N) =L∏

i

zNi

i

Ni∏

n

ui(n)−1. (3.1.8)

with the fugacities zi

zi =∑

j

pjizj . (3.1.9)

Now, solving a closed ZRP only corresponds to solving the last equation for the fugacitieszi.We will show later, using the connection to queueing networks, that (3.1.8) is the steady-state distribution even for a general n-dimensional, opened or closed zero-range process.The fugacities zi however will in such a general case not be given by (3.1.9) but a moreevolved equation.

45


The open 1-dimensional ZRP

Levine et al. [29] considered the one dimensional, driven, open zero-range process. Inthis setting there exists a preferred direction of particles1 modeled via the followingparameters

pij =

p j = i+ 1

q i = j − 1

β i = L, j = 0

γ i = 1, j = 0

(3.1.10)

u01 = α (3.1.11)

uL0 = δ. (3.1.12)

The authors were able to derive the steady-state solution using a grand-canonical dis-tribution as Ansatz for the Master Equation. The solution again factorizes and is, asexpected, given by (3.1.8) with the fugacities satisfying

zi =[(α+ δ)(p− q)− αβ + γδ]

(

pq

)i−1− γδ + αβ

(

pq

)Ns−1

γ(p− q − β) + β(p− q + γ)(

pq

)L−1. (3.1.13)

Once the steady-state solution is known, every thermodynamic quantity can be derivedvia the partition function.

Some known results

ZRP’s have been studied quite a lot in the last years. Here we quickly review some ofthe previous work and results.

In [29] the authors study a 1-dimensional ZRP model with open boundary conditions.They divide their study into two seemingly important cases: (i) the totally asymmetricZRP (sometimes called a driven system), in which particle transition is only possiblein one direction, e.g. q = γ = δ = 0 and (ii) the partially asymmetric case withp 6= q, p+ q = 1. Both are studied in a special ”condensation model” with

un = 1 +b

n. (3.1.14)

The authors show that in both cases condensation is possible, in the sense that in thelarge-time limit an infinite number of particles gather on at least one site. In this caseno complete steady-state solution exists since at least one fugacity is z ≥ 1. The authorsalso find that in the totally asymmetric case different condensation regimes, dependingon parameter b exist. In most of these condensation regimes the average number ofparticles at a condensate site either increases linearly or as a power law.

1One might think of an external electromagnetic field.

46



pp

qq

q

Figure 3.2.: Illustration of the 1 dimensional exclusion process with associated rates.

In [19] the authors study current fluctuations in the 1-d ZRP. Their findings coincidewith a more general treatment of large deviations functions for currents in queueingnetworks as presented in [6]. These results are inter-transferable between the disciplinesusing the analogy which will be established in this thesis.

In [23] the authors study the influence of ”quenched disorder” on the 1-d ZRP, i.e. theeffect of non-homogeneous, temporally constant hopping rates from site to site. Whereasin the asymmetric cases p 6= q holds for every site, here the authors assign different pi’sand qi’s to every site i, with pi + qi = 1. This treatment will be included in the generaln-dimensional ZRP model, which we propose in this work.

3.2. The Exclusion Process

When talking about the zero-range process, a seemingly different process often studiedin non-equilibrium statistical physics has to be mentioned: the exclusion process (EP).This kind of process has been studied a lot by e.g. Bernard Derrida (see his lecture notes[11] for a nice summary). The EP is defined as follows. Consider ”fermionic” particleson a 1-dimensional lattice, i.e. each site of the lattice can be occupied by at most oneparticle. Assume again that with each time step dt there is a probability pdt associated,that a particle will jump to the site to its right/left, given that this site is empty (seefigure 3.2). These dynamics imply the following:

• The waiting time for a particle leaving the site it occupies is exponentially dis-tributed with parameter p.

• In contrast to the ZRP, this process is not ”zero-range” since the possibility for aparticle of leaving its current site is dependent on the occupation of the neighboringsites.

Because of the last point, this process seems to be much more evolved than the ZRPand one would not expect a factorized solution for the Master Equation of this system.However, as Evans and Hanney point out [14], there exists a unique mapping betweenthe 1-d EP and the 1-d ZRP (see figure 3.3):

PEP(s1, s2, s3, ..., sL) = PZRP(N1 = s1 − 1, N2 = s2 − s1 − 1, ..., NL = sL − sL−1 − 1)(3.2.1)

47



site 1 , particle 1

exclusion process

zero-range process

1 2 3

site 2 , particle 2 site 3 , particle 3

Figure 3.3.: Illustration of the mapping from a 1-d EP to a 1-d ZRP.

The basic idea is to treat particles in the EP case as sites in the ZRP. Then the occupationof site i with ki particles in the ZRP case corresponds to the fact that between particlei and particle i − 1 in the EP case there are ki vacant sites. Denoting the site whichis occupied by particle i in the EP case with si, then the occupation number of site Ni

in the ZRP case will be given via Ni = si − si−1 − 1. This leads to the above statedmapping. Evans and Hanney point out that this mapping only holds for the case wherethe order of particles is reserved. However, we want to stress that this mapping also onlyseems valid, if in the EP case the amount of particles present on the lattice is preserved,otherwise the mapping would have to go to a ZRP model with changing grid-size. Atthis point we do not see a simple way to treat such a case. The general 1-d exclusionprocess, with open boundaries, can be very elegantly solved using Derrida’s ”MatrixAnsatz”[10].The exclusion process, as a certain particle hopping process, has been utilized as a

model for a variety of transport phenomena. Most popular is certainly its use as aagent-based (microscopic) approach for traffic modeling, as nicely reviewed by Helbingin [20]. Also in the biological sciences exclusion process can be employed, e.g. for thedescription of certain phenomena in mRNA translation as suggested by [7].It also has been pointed out, independently by Krug and Ferrari [26] as well as by

Evans [13], that disordered 1-dimensional exclusion processes, i.e. exclusion processeswith non-uniform transition rates, are highly coupled to the theory of Bose-Einsteincondensation.

After having introduced the concept of zero-range processes and exclusion processes,we proceed in the next chapter with the main statement of this thesis. We will showthat the theory of 1-dimensional zero-range processes can be extended to a theory ofgeneral n-dimensional zero-range processes. This will be done via building a connectionto the previously studied concept of queueing networks.

48

Part II.

Application and Numerics

49

4. A Queueing Network based description

of General Zero-Range Processes

4.1. Introduction

So far we have presented the theory of queues and networks of queues, in particular theM/M/1/∞ queue, as applied in Operations Research and the physics-native theory ofzero-range processes. However, these two non-equilibrium processes are highly linked. Tobe precise: the zero-range process is a special queueing system. The underlying stochasticprocesses are the same. The mathematical descriptions as presented are slightly different,due to the different origins. The results are the same: a factorized, universal solutionof the steady-states. Here we will describe the connection in detail and will presentlimitations of this correspondence as well as a straight-forward extension of the theoryof zero-range processes.

4.2. Mapping between queueing networks and zero-range

processes

Taking into account account the things presented so far, the formal connection is easilyestablished: the zero-range process in 1-d corresponds to the M/M/1/∞ queuing net-work. In the queueing system case one talks about customers arriving at a system andgetting processed by the service facilities. In the zero-range terminology customers arereferred to as particles, who enter the grid and proceed along the sites, due to the un-derlying dynamics of a zero-range process. Both cases have in common that there is noprecise (deterministic) microscopic description for the actual processes at the service fa-cilities/ sites provided. Instead the assumption is that the processes acting there can besufficiently described via a stochastic approach. In both models the underlying processesare supposed to be Markovian and memoryless. The rates of customers getting servedcorresponds to the rate particles leave a site and does in both cases only depend on thenumber of customers/particles present in the queue/ at the site. The number of particlesNi waiting in the queue i to get served corresponds to the number of particles Ni site iis occupied by. The possibility of congestion in the queueing system, i.e. the divergenceof the number of customers (achieved when ρi ≥ 1), can be interpreted as a certainkind of condensation in the zero-range process. However, in the queueing network (as avery application-oriented model) congestion is often equivalent to failure of the system,since in the large time limit newly arrived customers will never get processed throughthe complete network. In the zero-range process, considering particles and with no spe-

51

Chapter 4: A Queueing Network based description of General

Zero-Range Processes

Queueing networks Zero-Range process

customers particleslocal Markovian and memoryless process local Markovian and memoryless process

service rate µi transition rate uicongestion condensation

Table 4.1.: Correspondence Queueing - ZRP

cific application in mind, condensation is not an exit-criteria and one will naturally askwhat happens after one of the sites experiences condensation. This question leads us tosuggest a ”renormalization” procedure and will be discussed more formally in one of thenext chapters. Table 4.1 provides a comprehensive list of the formal analogy betweenthe queueing network model and the zero-range process. This very strong formal cor-respondence, as strong as to the point of the underlying stochastic principles, leads ofcourse also to a similar quantitative description and equivalent results.

• Because of the Markovity and memorylessness underlying in both processes, thetimes between the arrival/departure of a customer/particle are exponentially dis-tributed (1.4.11).

• The number of customers/particles arriving at a site during an interval τ = dt isPoisson distributed (1.4.10).

• Burke’s theorem holds in both models. Hence, Poissonian input leads to Poissonianoutput.

• The steady state solutions to the master equations of both systems are factorized.Compare equations (3.1.8) and (2.4.12).

• The fugacities zi in the zero-range case are given by the generalized utilizationfactors ρ.

Hence, results obtained for each of those two models can be easily transferred to theother. It is rather surprising to us that this equivalence has not been explicitly pointedout before. Also, until now the description and research on zero-range processes wasmainly limited to the 1-dimensional case. Here, with the direct correspondence to thequeueing network, it is easily possible to generalize to the n-dimensional zero-rangeprocess.

4.3. The general n-dimensional Zero-Range Process

Here we will generalize the classical notion of a zero-range process (defined in 1-dimension)to a general n-dimensional zero-range process.Let us define the general n-dimensional zero-range process as follows: consider a graph/networkG = (V , E) with V and E denoting the set of vertices and edges of the graph, respec-tively. Allow particles to enter the network at node i in a Poisson distributed fashion

52



1 2 3 4

5 6 7

8 9 10 11

12 13 14 15

bbb

bbb

b b b

bcbcbc

bc bc bc

Figure 4.1.: Illustration of a possible underlying graph structure of the n-dimensional zero-range process. Here the graph structure is a 4×4 grid with one defect. Transitionsof particles are only allowed along the graphs of the edge. Particles can enter thegrid at sites 5, 7 and 14 and can leave the network at sites 3 and 7.

with rate λ0i. We will refer to those rates as the arrival rates. Let also be the timebetween the departure of two particles from node i be exponentially distributed withrate µi (departure rates). Let also be a number 0 ≤ pij ≤ 1 associated to every directededge, representing the probability that a particle departing at node i will transfer to sitej. Here j is a graph-neighbor of site i. Let pi0 be the probability that a particle at nodei will leave the network (figure 4.1). Then the Master Equation of this system is givenby equation (3.1.7):

dP (N1 = k1, ..., NL = kL; t)

dt=

∑

(i,j)∈E


−∑

(i,j)∈E

P (..., Ni = ki, Ni+1 = ki+1, ...)pi,i+1ui

+∑

(i,j)∈E


−∑

(i,j)∈E

P (..., Ni = ki, Ni+1 = ki+1, ...)pi+1,iui+1

+L∑

i=1


−L∑

i=1


(4.3.1)

53



which we quickly introduced in chapter 3.1. This equation is, however, equivalent to theMaster Equation of a queueing-network with site-dependent hopping rates (equation(2.5.2)). To be precise, building the connection to the queueing network with constanthopping rates µi(Ni) = µi, equation (3.1.7) transforms to equation (2.5.2) with

uipij = µijΘ(ki)

= µipijΘ(ki)ui = µiΘ(ki).

That the hopping rate (function) in the ZRP will be given by a function of form (4.3.2)and hence be truncated for the case that the departing site is empty, is a natural as-sumption. Hence we can conclude that a general ZRP process as described above can bedirectly and without loss of generality mapped to a M/M/1/∞ queueing network. Thesteady-state (dPdt = 0) of the general case is hence given by1 equation (2.4.15)

P (N = k) ∼∏

i

ρkii

ki∏

l

µi(l)−1.

with equation (2.4.16):

λ0i = Mij ρj

Mij =

−pji i 6= j

pi0 +∑

k pik i = j.

This is, from a physics point of view, again a remarkable result. Like the 1-dimensionalZRP, the general (open or closed) n-dimensional ZRP (a non-equilibrium system) showsa factorized steady-state with the universal measure (2.4.15). This result also shows thatthe previously known steady-state solution of the closed n-dimensional ZRP (3.1.8) holdsfor the open case as well, with fugacities zi = ρi however given by the above equation.This formalism also allows for the treatment of locally different departure rates µi(l),

which has to our knowledge not been studied in the ZRP-literature. In general, findinga solution to a given setting now ”only” corresponds to solving (2.4.16) for ρρρ, i.e.

ρρρ = M−1λλλ. (4.3.2)

As said earlier, this corresponds to inverting matrix M, which today can be most effec-tively solved using the Coppersmith-Winograd algorithm in O(L2.376) time (when thesystem is of size L) and is in general conjectured to be O(L2) in computational complex-ity [9]. Hence, in a big system this procedure can be very resource consuming. However,often the system will be sparse and sparse-matrix algorithms can be employed.To connect this result, and to justify its correctness, we prove that the fugacities com-puted by Levine et al. [29] and given in (3.1.13) can be obtained using the here presentedformulation.

1Here and henceforth we will stick with the queueing-network notation, but with table 4.1 the transferis straight forward.

54



The 1-d open ZRP can be treated with the general n-dimensional theory. In [29] equation(3.1.13) was derived as the unique solution of the recursion relation

pzk − qzk+1 = α− γz1 = βzL − δ (4.3.3)

In the following we will show that this recursion relation is equivalent to (4.3.2) for the1-d asymmetric setting, which means solving

−zk(p+ q) + zk−1p+ zk+1q = 0 k 6= 1, L, (4.3.4)

−z1p+ z2q + α− z1γ = 0 k = 1, (4.3.5)

−zLq + zL−1p+ δ − zLβ = 0 k = L. (4.3.6)

To show the equality of (4.3.3) and (4.3.4)-(4.3.6) we use a simple proof-by-inductiontechnique.a) Assumption: Let (4.3.3) be valid, i.e. pzk − qzk+1 = α− γz1.b) Induction begin: For k = 1, (4.3.3) obviously is identical to (4.3.5).c) Induction step: For k → k + 1, (4.3.3) yields

pzk+1 − qzk+2 = α− γz1 (4.3.7)

= pzk − qzk+1 (assumption) (4.3.8)

and renaming the indices k + 1 → k leads to (4.3.4).d) Induction end: For k = L (4.3.3) obviously is identical to (4.3.6).

Hence, we have shown that in the n = 1 case, the here presented general n-dimensionaltheory connects to previously known results.

4.4. Condensation and renormalization

If one considers the case of constant µi(l) = µi, the steady state solution reads

P (N = k) ∼∏

i

(

ρiµi

)ki

. (4.4.1)

In such a case, if for at least one i

(

ρiµi

)

≥ 1 (4.4.2)

holds, the system is non-ergodic since the probability distribution is not normalizable.However, keeping in mind that the above equation is a factorized probability distribu-tion, the condition above does not necessarily mean that the whole system diverges.Instead, if (4.4.2) holds only for a finite number of sites i, there will only be condensates(i.e. a diverging number of particles gathering) at those particular sites. The occupationnumbers of other sites might still be given by a proper probability distribution. However,

55



in the current framework it is not obvious how to obtain a ”renormalized”2 steady-stateprobability distribution, given that one or more sites will form a condensate in the abovesense. We propose the following general scheme to obtain such a renormalized solution.

Renormalizing the model

We propose a renormalization procedure to calculate the steady-state distribution in thecase that condensates emerged at a finite number of sites. This procedure builds on theidea that a congested site is occupied by virtually an infinite number of particles in thelarge-time limit, hence acting as a source. This means that the effective rate of particlesleaving site i to arrive at a neighboring site j is independent of the size of the queue (thereservoir is always big enough), hence Burke’s theorem does not apply and the rate ofparticles leaving that site is equivalent to the intrinsic departure rate µi:

λRij = pijµi. (4.4.3)

We use this fact to renormalize the model in the following way:

1. Move the congested site ξ to spatial infinity, i.e. exclude it from the model.

2. Assign to every former neighbor k of the congested site a new external in-link, i.e.increase the external arrival rate at this node to λR

0k = λ0k + µξpξk.

3. Assign to every former neighbor k of the congested site a new external out-link,maintaining the former transition rate µkξ = µkpkξ.

The so constructed model is normalizable by definition (We removed all non-normalizableparts). Figure 4.2 shows an illustration of the renormalization procedure. We want tostress that this procedure is only applicable if

• the rates µi are constants.

• the rates functions µi(ni) are globally defined, i.e. µi(ni) = µ(ni).

Verification of this renormalization procedure will be illustrated by numerical simula-tions in the next chapter.Since the renormalized model is a proper M/M/1/∞ queue again, the probability dis-tribution is factorized. However, the fugacities ρi will have changed, which we denoteby

ρi → ρRi . (4.4.4)

Hence the renormalized solution is given via:

P (N = k)R ∼∏

iR

(

ρRiµi

)ki

. (4.4.5)

2Here we use the term ”renormalization” in its general meaning of finding a new base, such that thereare no infinities in the system anymore.

56



001 2 3

µ1 µ2 µ3

p21

p12

p32

p23 p30

p10

00 1 3

µ1 µ3 pR30 = p30 + p23

pR10 = p10 + p12

µ01

µ03

µR01 = µ01 + µ2p21

µR03 = µ03 + µ2p23

Figure 4.2.: Illustration of the general renormalization procedure for a possible sub-graph, giventhat there is a condensate at site 2. Top figure shows the pure (not-normalizablemodel) with infinite number of particles at site 2. This model is transformedinto the new renormalized model (bottom), without infinity of the congested siteentering the analysis.

Here the product goes over all nodes present in the new model. If it is clear from the con-text, we choose to drop the superscriptR denoting the renormalized solution/parameters.In a model described by a product measure of geometric distributions as the here

presented, the mean number of particles to occupy site i is given by

〈Ni〉 =∞∑

ki=0

P (Ni = ki)ki

=∞∑

ki=0

(

1−ρiµi

)(

ρiµi

)ki

ki

=

(

ρiµi

)(

1−ρiµi

)−1

. (4.4.6)

This can be easily seen using the sum-formula for the geometric series.Similarly one finds that the variance Var(Ni) = σ2(Ni) holds:

Var(Ni) =

(

ρiµi

)(

1−ρiµi

)−2

. (4.4.7)

There are also statements which can be made about the currents of particles overevery link in the model. As we showed earlier, the distribution of particles leaving asite is a Poisson distributed variable with effective rate λij = µijρi and in this model

57



λij = ρij . So that we find for the current Jij between sites i and j:

Jij = −Jji = ρi − ρj . (4.4.8)

In this chapter we introduced the notion of a general zero-range process. We showed thatit is from a mathematical point of view equivalent to the Jackson network, i.e. a networkof M/M/1/∞ queues. We provided the necessary mathematical formalism and showedthat the steady-state solution is given by a universal product measure. The fugacitiesentering this product measure can be calculated by solving a linear system of equationsof size L, where L is the number of sites in the zero-range process. We also introduceda renormalization technique to solve n-dimensional zero-range processes in the case ofcondensation. In the next chapter we will provide numerical evidence for the validity ofthe introduced theory and will study some interesting effects of condensation.

58

5. Numerical and analytical results for the

n-dimensional ZRP

5.1. Introduction

In this chapter we will present numerical results for the n-dimensional zero-range pro-cess/ M/M/1/∞-queueing network to get a general impression of how such systemscan behave and eventually differ from the well-studied 1-d model. The dynamics andsteady-state solution of a n-dimensional ZRP will be highly dependent on the topologyof the network which we choose to study. Here it is not our aim to conclude any generalstatements, rather we choose to study the easiest possible n > 1−dimensional structure,which is the 2-dimensional grid as in figure 4.1. Our results include:

• a phase transition/condensation phenomenon

• the influence of boundary conditions

• symmetry with respect to in- and outflux in the model

• the influence of in- and outflux.

5.2. Description and general behavior of a global µi-model

We choose to study the general behavior of a zero-range process with constant andglobally defined µi = µ = const on a 5 × 5 homogeneous, finite grid (figure 5.1). Alsowe define a global rate of particles arriving to the grid λ0i = λ. Then, looking at thesteady-state probability distribution, we have for this system

P (N = k) ∼∏

i

(

ρiµ

)ki

(5.2.1)

withM−1λλλ = ρρρ (5.2.2)

or in a more compact form

P (N = k) ∼∏

i

(M−1eλ)kii

(

λ

µ

)ki

∼∏

i

(M−1eλ)kii κki (5.2.3)

59

Chapter 5: Numerical and analytical results for the n-dimensional ZRP

1 2 3 4

6 9 10

12 13 14 15

17 18 19 20

bbb

bcbcbc

22 23 24 25

5

11

16

21

7 8

Figure 5.1.: Illustration of a 5×5 grid with particle influx solely at node 1 and outflux solely atnode 3. We define the sets I = 1 and O = 3 as the sets of nodes with particleinflux and outflux, respectively. The transition probabilities in this model do onlydepend on the number of neighbors and out-links of a site. See table 5.1 for sometransition probabilities in this structure.

where we defined

κ =λ

µ(5.2.4)

and eλ as the unitary vector of all in-links

eλ = λλλ1

λ(5.2.5)

Hence the local utilization factor, controlling the convergence of the sum, for every sitei is give by

ρi = (M−1eλ)iκ, (5.2.6)

and the average steady-state occupation number

〈Ni〉 =1

1− ρi. (5.2.7)

The parameter κ is the ratio between arrival and hopping rate and is independent ofthe structure of the underlying network. If one fixes the underlying network structure,i.e. matrix M which defines the structure of the underlying process and the ”routing”of particles (the probabilities of a particle leaving site i to transfer to site j), hen κis the only adjustable parameter influencing the steady-state solution and whether ornot condensates will emerge. Here we choose to study a homogeneous model, withprobability pij of transferring from site i to j given as

pij =1

|∂i|, (5.2.8)

where |∂i| denotes the cardinality of the set of all graph-neighbors of site i, includingedges to leave the network. For example in figure 5.1 the probability for a particle leaving

60


a site in the bulk of the system1 to one of the 4 neighboring sites would be 1/4 = 0.25.For illustration purposes, some more transfer probabilities are shown in table 5.1.

departure site i arrival site j pij13 8, 12, 14, 18 1/411 6, 12, 16 1/325 20, 24 1/23 0, 2, 4, 8 1/41 2, 6 1/2

Table 5.1.: Some transition probabilities according to the ZRP on a 2-dimensional 5× 5 closedgrid as shown in figure 5.1. Site index 0 denotes the environment.

Dynamical modeling of n-dimensional ZRP

We now compare simulation results with our theory for the 2-d ZRP with global µi.For the simulations it is necessary to sample a valid trajectory of the ongoing stochasticprocesses in the ZRP. In general, in a model with total L sites and I sites with particlein-flux, we need to sample a consistent trajectory of L + I independent and parallelstochastic processes (L processes of particles leaving any of the L sites and I processesof particle influx). From a sequential programming point of view, it is not directly clearhow to achieve this task. However, sampling such a trajectory can be done via theGillespie-algorithm [17]. The idea of this algorithm is the following:

a) Sample a global time τ , which is the time span between the last stochastic event(at time t) and the next stochastic event in the complete system. In the ZRP asconsidered here, an event can either be the arrival of a new particle to the systemat site i (we denote this event by Ai) or the departure of one particle from sitei (which we denote by Di). Here the waiting times between any two consecutiveevents of the same type is exponentially distributed with rate λ0i for events Ai (i.e.particles arriving at site i) and with rate µi for event Di (i.e. particles leaving sitei). Then the global time τ for the next event will be exponentially distributed

τ ∼ µΣe−µΣt (5.2.9)

withµΣ =

∑

i

(µi + λ0i). (5.2.10)

b) Sample the kind of event, which will take place at t + τ according to the rates ofthe single event. Specifically, the probability pAi

that event Ai will take place is

pAi=

λ0i

µΣ(5.2.11)

1The bulk is defined as all sites which are not at the boundary. In the case of figure 5.1 this would bethe 3× 3 square at the center of the grid.

61


i ρi 〈Ni〉theo σtheo 〈Ni〉exp1 0.6467 1.83 2.28 2.205 0.3615 0.57 0.94 0.607 0.9125 10.42 10.91 11.1012 0.8905 8.13 8.61 9.7717 0.8761 7.07 7.55 7.08

Table 5.2.: Comparison of mean site occupations obtained a) via dynamical simulation (〈Ni〉exp)and b) as analytical result using equations (5.2.6) and (5.2.7).

0 1 2 3 4

0

1

2

3

4

0.0

1.5

3.0

4.5

6.0

7.5

9.0

10.5

Figure 5.2.: Average occupation numbers of a ZRP on a 5×5 grid as in figure 5.1 with κ = 0.15.Average particle numbers obtained using the Gillespie-algorithm over 10.000 timeunits after allowing to converge to the steady-state solution in 200.000 time units.

and equivalently for events of the D-type:

pDi=

µi

µΣ. (5.2.12)

Using this algorithm, one obtains a valid trajectory for the multiple parallel stochasticprocesses.

For illustration purposes, we do not show detailed long-term dynamics of the modelobtained using the described algorithm, but restrict ourselves to show the average num-ber of particles occupying every site for λ = 0.15, µ = 1, κ = 0.15 in Figure 5.2. Inthis example, sites 7, 12 and 17 show the highest average occupations. For this case itis possible to calculate the ρi’s using equation (5.2.6). Comparing the numerical andanalytical results (table 5.2) obtained using equation (4.4.6) one concludes that thetheoretical prediction holds (within the standard deviation).

62


We thus illustrate that the theoretical framework seems to describes the ZRP correctly.We will now study the influence of the parameter κ and the topology.

Influence of parameter κ

In this model the global parameter κ is certainly the most important and easy-to-studyparameter which influences the steady-state behavior of the system. To get an impres-sion of its impact, figures 5.3 (a)-(h) show the average number of particles occupyingthe sites, obtained analytically from results of equation (5.2.6) for different κ (aftercorrect renormalization). The following general statements can be made based on theobservation:

1. With increasing κ the total number of particles in the complete system seems toincrease.

2. With increasing κ the number of condensates seems to increase until the systemreaches a stationary state, where further increase in κ does not lead to more con-densates.

The explanation for the first observation is directly given by equation (4.4.6) - withincreasing κ the average number of particles at every site increases. The first partof the second observation is somewhat intuitive: if we pump more particles into thesystem, more sites will get congested. However, the second part of this observation issomewhat counter intuitive. It seems that the system falls into a stable state, where nomore condensates emerge if κ is increased. A rigorous explanation for this observationwill be given later. If one takes a look at figure 5.3 again, it appears that in general(independent of κ) more particles gather in average in the bulk of the system than onthe rim. However, this is certainly more a topology-effect and will be discussed later.

Condensation and phase transition

In the description of a global µ and λ ZRP model, introduced previously, the convergencefactors ρi are given by (5.2.6). From this equation it is possible to calculate the criticalκc’s at which condensates (in the aforementioned sense) will emerge, given matrix M.Let us illustrate this. As said earlier, the first condensate will emerge at some site ξ, iffor the utilization factor it holds ρξ ≥ 1. This happens for the first time, if

ρi = (M−1eλ)iκ = 1 (5.2.13)

for any i. Hence we find the first critical value of κ after which at least one of the siteswill experience condensation to be

κc = mini

1

(M−1eλ)i

. (5.2.14)

63


0 1 2 3 4

0

1

2

3

4 0.45

0.60

0.75

0.90

1.05

1.20

1.35

1.50

(a) κ = 0.10

0 1 2 3 4

0

1

2

3

4

1

2

3

4

5

6

7

8

9

10

(b) κ = 0.15

0 1 2 3 4

0

1

2

3

4

.

15

30

45

60

75

90

105

120

(c) κ = 0.20

0 1 2 3 4

0

1

2

3

4

.

.

25

50

75

100

125

150

175

200

(d) κ = 0.25

0 1 2 3 4

0

1

2

3

4

.

.

.

40

80

120

160

200

240

280

320

360

(e) κ = 0.30

0 1 2 3 4

0

1

2

3

4

. .

.

.

6

12

18

24

30

36

42

48

54

(f) κ = 0.34

0 1 2 3 4

0

1

2

3

4

.

. .

.

.

3

6

9

12

15

18

21

24

27

(g) κ = 0.40

0 1 2 3 4

0

1

2

3

4

.

. .

.

.

3

6

9

12

15

18

21

24

27

(h) κ = 0.80

Figure 5.3.: Average occupation of sites for different κ on the setting shown in figure 5.1. Thedark red, dotted sites are condensates. The first condensate emerges at κ = 0.164.Note: the color maps are always normalized for every instance and not necessarilyinter-comparable.

64


One can indeed say more: if the values in the above set are not degenerated, exactly onecondensate will emerge when approaching κ = κc and the critical site ξ is obviously

ξ = argmini

1

(M−1eλ)i

. (5.2.15)

Here we defined the critical value κc as the one where the first of (eventually) multiplecondensates will emerge. In a sense, one can say that a phase transition takes place atκc, transforming the system from the uncongested phase to the 1-site-congested phase.However, if one looks at the steady-states obtained for different κ (figure 5.3) it seemsthat there are more than just one condensates in the large κ limit, hence multiple phasetransitions took place. Interestingly, the system seems to have a stable phase as well.Once the system reached that phase, increasing κ does not lead to the emergence offurther condensates.Calculating the critical κ values for every of those phase transitions is an extension of

the method presented above, utilizing the former described re-normalization procedure.However, (5.2.14) and (5.2.15) will not hold any longer in a renormalized model, sincea new in-link at one of the neighbor nodes k of ξ will not have the universal arrival rateλ, but arrival rates

λR0k = λ0k + µpξk. (5.2.16)

However, to account for this effect the scheme presented above can be generalized in thefollowing way. Let be λR = eRλ λ +wR

µ µ, where eRλ denotes the unit vector containingall in-links in the renormalized model with global arrival rate λ

(eRλ )k = λ0k1

λ(5.2.17)

and wRµ the weighted unit vector of new in-links due to the re-normalization

(wR

µ )k = pξk. (5.2.18)

Using this more general formulation, the utilization factors read

ρi = (M−1eRλ )iκ+ (M−1wR

µ )i (5.2.19)

and equation (5.2.14) generalizes to

κc = mini

1− (M−1wRµ )i

(M−1eRλ )i

. (5.2.20)

Again, the site at which the condensate emerges is obtained by replacing min by argminin the equation above.To show that the method works, we calculate the critical values κc and the correspond-

ing sites at which the condensates emerge for the model studied earlier. The results areshown in table 5.3. Comparing with the sequence earlier (figures 5.3 (a)-(h)), our pre-diction seems to hold. After the threshold κc = 0.355 there are no more condensates,

65


# condensates κc site

1 0.164 72 0.218 123 0.279 164 0.303 65 0.355 1

Table 5.3.: Order of emerging condensates, with critical values κc, for the zero-range processon a 5× 5 structure as in figure 5.1.

since in this case the denominator of equation (5.2.20) becomes zero, meaning that thenext such phase transition will occur for κ → ∞. To explain this behavior, let us lookat the denominator

(M−1eλ)i (5.2.21)

and determine when this quantity is zero for all i. This will obviously be the case, whenthe unit vector eλ is the null vector. This however is exactly the case, when there are nomore in-links into the system with global rate λ, which can happen when all sites withsuch an in-link in the base (not renormalized) scenario of the system form condensates.This explanation fits exactly the data obtained in table 5.3 and figure 5.3 since here thelast site which experiences condensation is site 1, the only site with an λ-in-link. Hencewe have the following general statement.

Stable configuration In a global µi, n-dimensional zero-range process, the stable con-figuration is exactly then achieved, when all sites with λ-in-links formed condensates.

Mathematically, this results is clear. From a high-level point of view it is rather re-markable for the following reason. To increase parameter κ we have two possibilities:a) either choose to increase λ or b) choose to decrease the global parameter µ. Thatincreasing λ has no further effect if all λ-in-link sites already formed islands (and henceprovide the system in the large time limit with infinite supply of particles) is easilyseen. That decreasing µ has no effect is more subtle: since there are still out-links ofthe system, one would suggest that decreasing µ and hence decreasing the effective rateof particles leaving the network will lead to more congestion in the system. This seemsnumerically and theoretically not to be the case. One has to keep in mind that in thestable configuration µ is not only the rate of particles leaving the network but is also therate of particles entering the uncongested part of the system and the rate of particlesdeparting at a site and transferring to one of its neighboring sites. Hence the local uti-lization factors ρi, which basically are the ratio between local in-rate and local effectivedeparture-rate will stay the same. This is true, since all effective departure-rates andin-rates are multiples of the global rate µ. Thus, if one scales (decrease/increase) µ bya factor of α, the utilization factors are not changed:

ρi =µin

µout→

α · µin

α · µout=

µin

µout= ρi (5.2.22)

66


1 2 3 4

6 9 10

12 13 14 15

17 18 19 20

bb

b

bcbcbc

22 23 24 25

5

11

16

21

7 8

Figure 5.4.: Illustration of a 5 × 5 torus with possible particle influx at site 1 and outfluxat site 3. On a torus periodic boundary conditions are imposed, i.e. every sitehas 4 nearest neighbors (plus eventual in- or out-links). As an illustration of thisperiodicity, we hint to the fact that in such a setting e.g. site 25 has not only sites20 and 24 as graph neighbors, but also sites 5 and 21.

Hence we directly conclude, that in the stable configuration not only the number ofcondensates will be stable, but also the utilization factors ρi and hence the steady-statedistribution of the system (including the mean number of particles present at a not-congested site) will stay the same. This is a very interesting observation since it impliesrobustness of the system after a critical value of κ is passed. Once in the stable phase,local fluctuations in λ and global fluctuations in µ will not perturbate the system out ofthe stable solution.

Study of some topological influence

After we have shown the influence of parameter κ, we will now take a look at the secondfactor which influences the non-equilibrium steady state of such a global µi system: thetopology.

The torusIn the former studied realization we chose to consider a finite grid (figure 5.1). Here wedeviate from this setting and introduce the ZRP on a torus, i.e. a periodic grid (figure5.4). The torus is more homogeneous than the closed grid since there is no rim and bulk- every site has 4 nearest neighbors and hence particle routing is equivalent. The onlydeviation from the transfer probabilities pij = 0.25 is seen at nodes with out-links, wherepi0 = pij = 0.2. Figures 5.5 (a)-(c) show the average particle occupations of the ZRPon a torus. The first striking difference is that, compared to the finite grid, the stablesolution is achieved at a much lower value of κ and with only one condensate emergingat the site which hosts the in-link. This is a very interesting result for which we unfor-tunately lack intuitive understanding at this point. What is also very apparent is the

67


0 1 2 3 4

0

1

2

3

41.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

(a) κ = 0.10

0 1 2 3 4

0

1

2

3

4

.

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

(b) κ = 0.20

0 1 2 3 4

0

1

2

3

4

.

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

(c) κ = 0.80

Figure 5.5.: Average occupation of sites for different κ on the 5×5 torus setting as in figure 5.4.The dark red, dotted site is the condensate. The condensate emerges at κ = 0.152

.

symmetry in the solution, due to the perfect symmetry of the underlying grid. Basicallyin this easy setting of one in- and one out-link, the local solution is solely dependenton the distance and relative position of the local site to the in-link site and the out-linksite. For example sites 7 and 22 are equivalent in this regards and hence show the samesteady-state. There are no boundary effects in such a setting.

Position of in- and out-links:Since the base models studied so far are highly symmetric (torus) or somewhat symmet-ric (closed grid), changing the in- and out-link from figure 5.1 and 5.4 in a ”correlation”preserving way, should not change the quantitative behavior of the system. To illustratethis principle, figures 5.6 (a) and (b) show the influence of mirroring the setting of in- andout-links for the ZRP on a finite 5× 5 grid (figure 5.1) along the central vertical axis aswell as rotating by π

2 around site 13. As expected, the steady-states stay quantitativelythe same and are only a mirrored/rotated version of the former solution. However, in

68


0 1 2 3 4

0

1

2

3

4

.

.

.

.

.

3

6

9

12

15

18

21

24

27

(a) mirrored

0 1 2 3 4

0

1

2

3

4

. .

.

.

.

3

6

9

12

15

18

21

24

27

(b) rotated

0 1 2 3 4

0

1

2

3

4

.

.

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

22.5

(c) translated

Figure 5.6.: Stable configuration with mirrored, rotated and translated in- and out-links of theZRP on a finite 5× 5 grid with boundaries, as in figure 5.1.

this special case with boundary there is no general rotational or translative symmetry,as can be seen in figure 5.6 (c), where the in- and out-links have been translated to theright by a distance of one. That this symmetry does not hold is because in this modelthe routing dynamics are different between bulk and rim sites. In general, a change of in-and out-links will only give an equivalent solution if their positions have been changedusing such a transformation, which is a symmetry-preserving transformation for the un-derlying routing-network (as is clearly the case with the previous mirror and π

2 -rotationfor the boundary case). The torus is completely symmetric and so is a transformedsolution, given that the relative distances between all in- and out-links stay the same.Figure 5.7 (a)-(c) show again the steady-states of the torus setting with mirrored (a),rotated (b) and translated (c) in- and out-links.

Number of in- and out-linksCertainly, the number of in- and out-links will have an influence on the steady-statesand the stable configuration as well. Just to illustrate this influence, figure 5.8 shows

69


0 1 2 3 4

0

1

2

3

4

.

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

(a) mirrored

0 1 2 3 4

0

1

2

3

4

.

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

(b) rotated

0 1 2 3 4

0

1

2

3

4

.

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

(c) translated

Figure 5.7.: Stable configuration with mirrored, rotated and translated in- and out-links of theZRP on a periodic 5× 5 grid (torus), as in figure 5.4.

the stable configurations for one random assignment of 3 in- and 4 out-links in the finitegrid and the torus. The steady-states in the stable configurations are very different fromthe so far studied model of one in- and one out-link. A common observation for thetorus is however that in this model with 3 sites having in-links, the stable configurationagain consists of exactly three condensates. Nevertheless, due to the vast possible com-binations of assignments of in- and out-links, we are not able to conclude any generalstatements from this example. Hence we choose to perform Monte-Carlo sampling tostudy several relations:

a) Phase diagram: no condensate → one condensateAn interesting transition in the studied models is certainly the transition of the phasewithout any condensates to the phase with one condensate. This transition occurs ata (for every model) typical value κc. Hence we choose to show in figure 5.9 the phasediagram of this transition in a plot spanning the space of parameters in our model: thenumber of in-links I, the number of out-links O and the parameter κ. Since the number

70


0 1 2 3 4

0

1

2

3

4

.

.

.

.

.

.

8

16

24

32

40

48

56

64

72

(a) grid

0 1 2 3 4

0

1

2

3

4

.

.

.

2.4

3.0

3.6

4.2

4.8

5.4

6.0

6.6

7.2

(b) torus

Figure 5.8.: Stable configuration for one randomly chosen combinations of 3 in-links (at sites1, 4 and 17) and 5 out-linkes (at sites 4, 10, 19 and 25) for the closed 5 grid (a)and the 5× 5 torus.

of combinations of possible in- and out-link assignments is huge2 we choose to sample forevery pair (O, I) over 100 random samples. Although the sampling has been random,there appears to be a smooth phase-dividing surface, with the condensate-phase lyingabove it. Hence κc(O, I) seems to be a self-averaging quantity in this kind of system.Also, somehow surprisingly, the qualitative behavior of the phase diagram seems to beequivalent for the grid and the much more symmetric torus. However, looking at thestandard deviations for both diagrams, the statistics in the torus case is more smooththan for the closed grid (table 5.4), as can be seen by the smaller maximum relativestandard deviation. In general the relative standard deviation is biggest for those sam-

case I O 〈κc〉σ

〈κc〉

grid (1,25) (1,25) (0.005,0.75) (0.04,0.24)torus (1,25) (1,25) (0.005,0.75) (0.005,0.09)

Table 5.4.: Range of means for the first critical value of κ and their relative standard deviationsfor the data shown in figure 5.9.

ples involving a lot of disorder, i.e. systems with small I and O. One observed differencethough is that whereas in the torus case the relative standard deviation decreases mostsignificantly in the direction of increasing I, in the closed grid case this quantity de-creases most in direction of increasing O.

b) Maximum number of condensates/stable configurationAs we have illustrated earlier, the number of maximum condensates (i.e. the numberof condensates in the stable configuration) seems to be much closer to the number of

2To be precise, in a 5 × 5 model the number of combinations for I in-links and O out-links is exactly(

25

I

)

·(

25

O

)

, which is maximal at O = I = 12, 13 with sim1013 combinations.

71


in nodes

510

1520

out n

odes

5

10

15

20

kappa_c

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(a) grid

in nodes

510

1520

out n

odes

5

10

15

20

kappa_c

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(b) torus

Figure 5.9.: 2-Phase diagram over the space of free parameters O, I and κ for the closed 5× 5grid (a) and the 5 × 5 torus (b). Each data point was obtained via Monte-Carlosampling over 100 instances.

in-links in the case of a totally symmetric torus, than in the case of a finite grid withboundary. To study this effect further, we performed again Monte-Carlo sampling inthe 5 × 5 grid and torus and show the results in figure 5.10. Figure 5.10 (b) clearlyreinforces the observation that in the completely symmetric case of a torus, the numberof condensates, C, in the stable configuration equals the number of sites with in-links:

C ≈ I. (5.2.23)

The model only deviates from this behavior for the case of relatively large numbersof in-links and small number of out-links. The closed grid on the contrary shows theopposite behavior (see also figure 5.10 (c)). In the vast majority of cases the numberof condensates in the stable solution is larger than the number of sites having in-links:C > I. It is only in the case of large amounts of in-links that the number of condensatesis somewhat equal. However, this might only be due to the fact that in those casesthe maximum number of condensates is anyways bounded from above by the number ofsites in the system, hence no large deviations are possible. Also, in both cases (torusand finite grid), with increasing number of out-links the maximal number of condensatesseems in average to be closer to the number of sites with in-links

C ≈ I if O ≈ 25, (5.2.24)

hence providing more evidence for above statement positively.Before we close this chapter, we want to stress that although most effects in this

chapter were studied on a relatively small toy-example with two different boundary con-ditions, we expect a lot of the statements to hold in more general settings. For grids insize much larger than 5×5, their behavior should be very close to the here studied torusmodel, since in those cases the boundary is virtually only present at (spatial) infinity.Hence boundary effects will not have a big influence on the dynamics and statistics in

72


in nodes

510

1520

out n

odes

5

10

15

20

max. co

ndensa

tes

5

10

15

20

(a) grid

in nodes

510 15

20

out

nod

es

5

10

15

20

max. co

ndensa

tes

5

10

15

20

(b) torus

0 5 10 15 20 25in nodes

0

5

10

15

20

25

max.

condensa

tes

(c) grid - projection

Figure 5.10.: Number of maximal condensates C as a function of O and I for the closed 5× 5grid (a) and the 5× 5 torus (b). Each data point was obtained via Monte-Carlosampling over 100 instances. (c) shows a projection of (a) on the C-I subspace.

the bulk of the system.

Disordered n-dimensional ZRPAlthough here we studied a global µi = µ and λ0i = λ model, the analysis can be easilyextended to a case with quenched disorder in the rates. If one defines the locally differentrates µi with respect to some base µ as

µi = αiµ (5.2.25)

and the same with λ0i

λ0i = βiλ (5.2.26)

equation (5.2.6) will be transformed into

ρi =(M−1M−1M−1eλ)i

αiκ. (5.2.27)

73


with the definition

eλ = λλλ1

λ

unchanged, but now due to λλλi = βiλ, will be evaluated as eλi = βi. Hence our modelprovides a natural way of treating disordered n-dimensional zero-range processes. Itshould also be clear that the study of topological irregularities is easily possible withinthe presented framework via routing matrix M. In general, the framework offers a widevariety of possible future studies, may it be theoretical or application-close.

In this chapter we gave some numerical evidence for the correctness of our proposedn-dimensional ZRP framework. We also studied the influence of a global parameter κand established that the system eventually falls into a stable configuration, where anyincrease of κ will not have an influence on the steady-state of the system. We alsoshowed that there are eventually multiple phase transitions with the appearance of mul-tiple condensates before the stable configuration is achieved. We also showed that thebehavior of the finite and periodic 5× 5 grid is rather different, due to boundary effects.

74

6. Conclusion and Further Studies

In this work we established a strong connection between the concept of queueing networksand the zero-range process. We showed that it is possible to extent the 1-dimensionaltheory of zero-range processes to a more general n-dimensional theory and that one isstill able to solve this non-equilibrium problem for the steady-state solution. Based onthis theory we studied the ZRP on a 2-dimensional grid, especially focusing on the influ-ence of an external parameter κ as well as several topological properties. We furthermoreobserved an interesting condensation phenomenon, which is similar but in detail distinctfrom the 1-dimensional case.

The here presented treatment and study of the n-dimensional ZRP allows for a numberof natural extensions and further studies:

• Percolation. As we have shown, the n-dimensional ZRP may exhibit condensationat eventually a large number of sites. We have also shown, that such sites canbasically be considered as ”excluded” from the model. Hence starting from a ZRPon a connected graph, this effect may lead to separation of the graph (breakdown ofthe giant connected component). This can be considered an ”inverse” percolationproblem[18], starting in the supercritical phase and via increasing κ leading toundergo the percolation threshold. If one denotes the (percolation) critical valueof κ with κpc, the following expression for the percolation threshold should hold:

pc =N −Nc(κpc)

N. (6.0.1)

Here N is the total number of sites and Nc(κ) the number of congested sites forgiven κ. To prove percolation, one would have to study the sub-critical phase(which here is the phase with κ > κpc) and whether or not the cluster size decaysexponentially with increasing κ. Based on preliminary results, we suspect that the2-dimensional ZRP, as presented here, undergoes a percolation transition. Thishowever work in progress and provides an interesting research question for furtherstudies.

• Disorder. As mentioned in the last chapter, the treatment of a disordered n-dimensional ZRP model is easily possible within the proposed framework and pro-vides an interesting direction for further research.

• Different topologies. In this thesis we focused in the last chapter on an exemplarytreatment of the 5×5 grid to illustrate the power of the proposed framework. Thiscan however be naturally extended to the study of ZRP’s on other topologies, for

75

Chapter 6: Conclusion and Further Studies

example also including the study of zero-range processes on (3-dimensional) cubesor classes of famous network types like random graphs.

• Applications. It would be interesting to see if the here presented theory of n-dimensional ZRP’s can be applied e.g. as a straight forward extension of an 1-dZRP application. In the field of porous materials this could be of interest. Alsoone could ask if the recent 1-dimensional EP/ZRP based treatment of traffic flowscan be enhanced[20].

76

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