Fluid Flow Models and Queues

8/11/2019 Fluid Flow Models and Queues

1/24

Fluid Flow Models and QueuesA Connection by Stochastic Coupling

Soohan Ahn* and V. Ramaswami*

AT&T Labs, Florham Park, New Jersey, USA

ABSTRACT

We establish in a direct manner that the steady state distribution of Markovian uid owmodels can be obtained from a quasi birth and death queue. This is accomplished throughthe construction of the processes on a common probability space and the demonstrationof a distributional coupling relation between them. The results here provide aninterpretation for the quasi-birth-and-death processes in the matrix-geometric approachof Ramaswami and subsequent results based on them obtained by Soares and Latouche.

Key Words: Fluid ows; Queues; Stochastic coupling; Matrix geometric method.

1. INTRODUCTION

Thesubject ofthispaper is the steady statedistributionofthe content ofa uidow buffermodulated by a nite state, continuous time, irreducible Markov chain of environmentalphases. Ina classicpaper, S. Asmussen [4] characterized that distribution to be of phase type.Later, V. Ramaswami [17] demonstrated that a representation of that phase type distributioncan be obtained from the G-matrix of a discrete time, discrete state Quasi Birth and DeathProcess [14,18] , thereby simplifying the analysis of the uid ow model. The connection to aQBD was a substantially new result that reduced the continuous time, continuous state space

325

DOI: 10.1081/STM-120023564 1532-6349 (Print); 1532-4214 (Online)Copyright q 2003 by Marcel Dekker, Inc. www.dekker.com

*Correspondence: Soohan Ahn and V. Ramaswami, AT&T Labs, Research, 180 Park Avenue,Florham Park, NJ 07932, USA; E-mail: [email protected]; [email protected].

STOCHASTIC MODELSVol. 19, No. 3, pp. 325348, 2003

M ARCEL D EKKER, I NC . 270 M ADISON A VENUE N EW YORK , NY 10016

2003 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.


2/24

problem of the uid model to the analysis of a discrete time, discrete state space QBD forwhich well tested stable algorithms that avoid the computational difculties arising in the

spectral methods (see Ref.[3]

and other references in Ref.[17]

) exist. The derivation in Ref.[17]

wasbased ona levelcrossing argument for the uid ow process, a duality resultingfromtimereversal, [16,5] and uniformization. Building on those results,Soares andLatouche [19] obtaineda representation in terms of a QBD which was arrived at without using time reversal in theanalysis, thereby simplifying the approach; a probabilistic interpretation was also attemptedby them of their QBD.

The QBDs arising in the matrix-geometric approach have not yet received asatisfactory interpretation. Our main purpose in this paper is to show that the approach viaQBDs can be interpreted as being rooted in a distributional coupling of the uid model to aqueue described by the QBD.

Our construction of the queue may have the avor of the approach in Ref. [6,13] , butunlike them. we take into account the detailed structure within the onoff periods, and thatis what leads to the consideration of a simple process, namely a discrete time QBD, thatprovides not only the distribution of the uid level but also the joint distribution of the uidlevel and the environment. While other embedded epochs may still yield a matrix-geometric structure, it is well-known that among matrix-geometric models, the QBD is thesimplest, both conceptually and algorithmically. (We hasten to note, however, that thepaper [13] covers very general models besides uid models driven by nite state Markovchains considered here.)

One may also consider other types of queueing models, one such being a queue withinterarrival dependent service times. However, the consideration of models withindependent service times is what leads us to a quasi birth and death process. Unlike this,the model with interarrival dependent service times does not possess a convenient Markov(renewal) structure at both arrival and departure epochs. The full power of our approachwill be seen in a subsequent paper [2] where we demonstrate some strong coupling resultsthat yield the transient analysis of the uid model as well. The steady state analysis requiresonly the weaker results which are much simpler, and are the only ones considered here.

In the sequel, we assume that the Markov chain of phases is irreducible and that itsstate space S 1 < S 2 < S 3 is such that: during sojourn of the Markov chain in state i of S 1 ,the uid level changes at rate ci . 0; during sojourn of the Markov chain in state j of S 2 ,the uid level changes at rate c j , 0; and, during sojourn in S 3, the uid level remainsconstant. We partition the states of the Markov chain in conformity with the three setsidentied above and denote the innitesimal generator of the underlying Markov chain by

Q

Q11 Q12 Q13

Q21 Q22 Q23

Q31 Q32 Q33

2664

3775

: 1

Here Q ij is a matrix of order jS i j jS jj:Let C denote a diagonal matrix formed by the absolute values jcij; i [ S 1 < S 2 and an

adequate number of 1s as needed to make C to be of the same order as Q. We deneT C 2 1Q and note that T is the innitesimal generator of a Markov chain. It is well-known that the distribution of the general uid ow model considered above can be

Ahn and Ramaswami326




3/24

determined easily from that of the uid ow governed by the Markov chain with generatorT in which the uid level increases at rate 1 in S 1 , decreases at rate 1 in S 2 , and remains

constant in S 3. In the sequel, we shall refer to the latter model as the homogenized owmodel. We shall derive the necessary conversion formulae at the end of the paper; thesecan also be found in Ref. [19] and are included here only for completeness.

2. ASSOCIATED QUEUE

2.1. Motivation

We will construct a queueing model represented by a QBD and a process Y constructed from its work process such that: (a) the process Y is distributionallycoupled to the homogenized uid ow model at the epochs of a Poisson process; and(b) the steady state distribution of Y coincides with that of the uid ow model. Tothat end, it is convenient to assume that the phase process governed by T has beenuniformized by a Poisson process. That is, we choose a (xed) number l ^ max i2 T iiand view the Markov process of phases as a process that makes changes of states onlyat epochs of a Poisson process with rate l in such a way that successive states visitedform a discrete time Markov chain, independent of the Poisson process, with transitionmatrix P l 2 1T I : Here and throughout the rest of the paper, I denotes an identitymatrix of appropriate order. In the remainder of the paper, we shall consider thematrices T and P also as partitioned according as Q, with their submatrices denoted bythe symbols T ij and P ij respectively.

To effect the stochastic coupling arguments, rst assume the following as given on acommon probability space V ; A ; P : A pair of independent Poisson processes, say, Mand N , with common rate l ; a discrete time Markov chain { L n : n $ 0} of phases whichhas transition matrix P and is independent of the Poisson processes M and N . Withoutloss of generality, we shall assume that L 0 i for some i. With these as building blocks,we will construct for almost all sample points in V , (a) a phase process J { J t : t $ 0}which is a CTMC with generator T ; (b) a process F {F t : t $ 0} such that F (t )increases at rate 1 while J t [ S 1 ; decreases at rate 1 while J t [ S 2 and F (t ) is positive,and remains constant while J t [ S 3i :e:; F t ; J t is a version of the homogenizeduid ow process of interest; and (c) a queueing model Q {Qt : t $ 0} modulated bythe phase process, where Q(t ) is the queue length at time t . The queue Q is to beconstructed such that at the epochs { sk : k $ 0} of the process M % N ; where %denotes superposition of processes, (which will be such that the phase process J remainsconstant in each sk ; sk 1), the embedded sequence { F sk ; J sk : k $ 0} has thesame probability law as that of the embedded sequence { W sk ; J sk : k $ 0} ; where

W (sk ) is the amount of work in the queue at the epoch sk . (Throughout, unless otherwisestated, for any process under consideration state at a point t will always denote the stateat t .)

Associated with the work in the queue we also construct a process Y {Y t : t $ 0}as follows. First, we discard from the set { sk : k $ 0} of the points of M % N ; thepoints sk of the process N for which J sk 2S 1 < S 3 ; and call the resulting set of points as

Fluid Flow Models and Queues 327




4/24

{t k : k $ 0} : Now, for t [ t k ; t k 1; let

Y t W t k t 2 t

k ; if J t

k [ S

1

max 0; W t k 2 t 2 t k ; if J t k [ S 2

W t k ; if J t k [ S 3:

We will note that in each of the intervals t k ; t k 1the phase process remains constant andthe growth of Y mimics that of F :

So that we may motivate the details to follow, we state some facts we will establish inthe sequel: (a) The amount of work done for each customer in the queue Q will bedistributed as expl ; and {Y t k }k $ 0 can be considered to be a stochastic discretizationof the uid model with respect to the uniformization parameter l ; (b) if the uid model isstable, then the steady state distribution as t ! 1 of Y t ; J t will exist and equal thesteady state distribution of F t ; J t as t ! 1 ; (c) steady state results for Y can beobtained using the matrix-geometric method for a QBD.

2.2. The Construction

To avoid pedantry and to save notations, we shall suppress the sample point in theensuing discussion which is indeed a sample point by sample point construction.

Construction of the Phase Process : Let 0 a 0 and let 0 , a 1 , a 2 , denote thesuccessive epochs of the Poisson process M . We dene J t L n in the interval an %t , a n 1; where L n is the state visited by the discrete time Markov chain at step n. Clearly{ J t : t $ 0} is a continuous time Markov chain with innitesimal generator T , and theepochs { a n} form a set of l -uniformization epochs for the phase process.

Construction of the Homogeneous Fluid Flow : Without loss of generality, we willassume the initial condition F 0 0: We let F 0 0 and dene the process { F t } suchthat for t [ a n; an 1; F t F a n t 2 a n if J t [ S 1 ; F t max 0; F a n2 t 2a n if J t [ S 2; and nally F t F a nif J t [ S 3: Dened thus, clearly F increasesat rate 1 in S 1 , decreases at rate 1 in S 2 while it remains positive, and remains constant in S 3as required. Clearly, the joint process { F t ; J t } is stochastically equivalent to thehomogenized uid model that starts empty and in phase i, and is modulated by the Markovprocess with generator T .

Construction of the Queue Q : The queue Q will be dened in terms of thesuccessive embedded epochs t 0 0; and { t k : k $ 1} where there is an arrival, departure,or phase transition; we emphasize that some phase transitions may be from a phase toitself, as necessitated for instance in the uniformization process. It will be assumed that all

queues are FIFO, and service is rendered by the server (at unit rate per unit time) onlywhen the phase is in S 2; specically, no service is rendered when the phase is in S 1 < S 3Also, the queue size at time 0 will be dened to be 0 to match our initial conditionF 0 0 (other initial conditions can be accommodated with minor changes in theconstruction.) In the sequel we shall denote by Qk and J k the queue length (number of customers in the system Q ) and the phase J (t k ) at the epoch t k .





5/24

(a) Let t 0 0 and Q0 0; this initializes the queue size at time 0 to match ourinitial state specication F 0 0 for the uid model. Note that we have J 0 J 0 i from the construction of the phase process.

(b) Having dened t k and Qk ; J k ; we rst specify the next time point t k 1 and thenthe value of the queue size and phase immediately after that epoch. The queuesize is assumed to remain constant over intervals of the form t k ; t k 1; that is,we shall set Qt Qk for all t [ t k ; t k 1: There are several cases to consider.

Case 1 : If J k [ S 1; then t k 1 is the rst epoch in M to come after t k , and the next queuelength value Qk 1 is set to 1 Qk that is, in this case, the epoch t k 1 isdened to be an arrival epoch to the queue Q . The phase at J k 1 is set to J t k 1; note that a phase change occurs at the newly dened epoch iff at theepoch t k 1 a different phase is entered in the uniformization scheme; otherwise,that epoch will constitute a self-transition for the phase in the queue Q .

Case 2 : If J k [ S 3 ; then the next epoch t k 1 is once again the rst epoch in M to comeafter t k , but the queue length value Qk 1 is set to the same value as Qk that is,a construct is made that makes the queue length remain constant just as theuid level would remain constant over the interval under consideration (notethat we are assuming that no work is being done in S 3.) The phase J k 1 is set to J t k 1; note that a phase transition to a different phase occurs at the newlydened epoch iff the new phase entered is indeed different; otherwise, theepoch is to be treated as a self-transition epoch.

Case 3 : If J k [ S 2; then the next epoch t k 1 is the rst epoch in the superposition M % N to come after t k . The queue length at that epoch is set depending on whether thatepoch comes from M or from N . Specically, the next queue length value Qk 1is set to the same value as Qk if t k 1 [ M ; it is changed to max0; Qk 2 1if thenew epoch t k 1 [ N : Thus, the next epoch is just a phase transitionepoch (withno effect on queue size) if it is an epoch of M , and a departure epoch (with nophase change) if the epoch is in N and a departure is indeed possible; note thatexcept when the epoch is in M and the new phase entered is different, the newepoch is a dummy phase change transitionepoch(i.e., with a phase selftransition).

Remark 1. If one considers the uid model at the end of sojourn in each phase of S 1 wherethe trajectory is upward, onecould replace the continuous upward increments by a jump andpretend as thougha customer with an exponentially distributed service time hasbeen added.Butwe need to also generate thedepartureepochs of those customers ifwe areto analyze thequeue. In our construction, the process N is used to insert the departure epochs.

3. DISTRIBUTIONAL COUPLING

The following result is obvious from the way the construction has been carried out.

Theorem 1. a The queue Q is modulated by the same continuous time phase process J that modulates the uid model.





6/24

b The queue {Qt ; J t : t $ 0} is represented by a QBD in continuous time ,and the embedded sequence {Qsk ; J sk : k $ 0} is a discrete time QBD.

Note that in trying to match certain behavior in the constructed queue to that of theuid model, we could specify the queue only in terms of its queue length process. We nowshow formally that each arrival in the queue brings in a random amount of work distributedas expl independently of the history of the process up to the arrival epoch. This is the lastassertion of the next theorem whose other assertions are essentially obvious consequencesof our construction.

Theorem 2. The queueing model Q satises the following conditions:a. Arrivals to the queue occur only at those epochs t k for which J t k 2 [ S 1; that is,

the epoch is a phase transition epoch in M from S 1 which may very well be a phase self transition.

b. Departures to the queue can occur at t k only if J t k 2 [ S 2; Qk 2 1 . 0 and t k [ N : Also, the phase immediately after each departure epoch is the same as that immediately prior to that epoch.

c. Assume that work gets depleted at unit rate while J t [ S 2 ; and that no service isrendered while J t [ S 1 < S 3 : Then the amounts of work done between successivedeparture epochs are iid random variables distributed as exp l :

Proof. We only need to prove (c). The service of a customer begins at an epoch sk with J k [ S 2 : The service completion epoch is the rst epoch sk r ; r ^ 1 to come after t k forwhich J sk r [ S 2 and sk r [ N : We shall evaluate now the distribution of the work done for the customer, given the phase at the beginning of the service.

First of all, note that if we denote by L 12 and L 32 the matrices of rst passageprobabilities for the phase process into S 2 from S 1 and S 3 respectively, then the matrix L P 22 P 21 L 12 P 13 L 32 is the matrix of return probabilities into the set S 2 , and thatunder the assumption that P is irreducible (and hence recurrent non-null), L is stochastic;i.e., L 1 1; where 1 denotes a column vector of 1s.

Let f isdenote the transform of the amount of work done by the server until the nextdeparture epoch, given that service starts in phase i [ S 2: Let f s denote the vector withelements f is; i [ S 2 : A simple conditioning argument gives us the following equation:

f s 2ls 2l

12

1 2ls 2l

12

L f s: 2

In the above equation, the rst term covers the case when the rst step of M % N itself

marks the service completion; this happens if the next epoch is an epoch of N , and thathas probability 1/2. The second term corresponds to the case when the rst stepcorresponds to a point from M (this marks a phase transition without a departure) and theprobability of this is again 1/2. In each case, the amount of service rendered in the rst stephas an exponential distribution with parameter 2 l , and to this random variable must beadded the remaining amounts of service, if any, rendered to the customer. In the second





7/24

case, the phase could change, and we need to wait until a return into S 2 is made beforeservice commences again; all that is reected in the term L we have used there.

Solving Eq. (2) for f s and simplifying the result shows that

f s I 2l

s 2l L

2 1 ls 2l

1

X1

n0

ls 2l

L n ls 2l

1

ls l

1; 3

and we have used the fact that L 1 1: Thus, the marginal distribution of the amount of

service rendered (irrespective of the starting phase) is exponential with parameter l , andthe proof is complete. A

The next result provides a distributional coupling that plays a key role in our analysis.

Theorem 3. The process {Y sk ; J sk : k $ 0} has the same probability law as the process {F sk ; J sk : k $ 0} :

Proof. We will show by mathematical induction that for all m $ 0; the subsequences upto k m have the same joint distribution. First of all, note that the phase process isidentical for both the queue and the uid. Also, since F 0 Y 0 0; the distributional

equality holds trivially for m 0:To complete the proof by induction, s ince { Y sk ; J sk : k $ 0} ; and

{F sk ; J sk : k $ 0} ; are both Markov, it sufces to show that for all x; j; theconditional distribution of Y sm 1; J sm 1 given Y sm; J sm x; jis the same asthat of F sm 1; J sm 1 given F sm; J sm x; j:

We now need to consider several cases: (a) If J sm [ S 1; then both F sm and Y smare each increased by independent random variables that have exp2l distributions toyield F sm 1and Y sm 1; (b) If J sm [ S 3 ; by our construction, the values of neither thework in the queue nor the uid level changes at the next step; (c) In the case where J sm[ S 2; in the interval sm; sm 1 both the uid level and the work in the queue aredepleted at a unit rate as long as they remain positive and left alone once they hit zero;furthermore, no increment occurs to either values at sm 1 : In all the cases, therefore, we getequality for the conditional distributions of the values at the next step given they have thesame values at the current step. Hence the proof is complete by mathematical induction.

A

Remark 2. We emphasize that what the theorem above gives is only an equality of distributions and not the equality of values of the random variables F sk and Y sk at





8/24

the embedded points. In other words, what we have is a distributional coupling only.The main value of the theorem lies in the fact that the work in the queue we have

constructed increases, whenever it does, by an amount that is independent of the pasthistory of the queue. Unlike this, the uid level increases by exactly the length of thesojourn interval preceding the epoch of increase. Note that the paths of the uidmodel considered directly as paths of the process of work in some queue does notlead to a Markov renewal structure at the embedded epochs; in such a construction,each service time is precisely equal to some previous interarrival time, anddependencies are not just of one step.

Remark 3. The successive epochs { t k } mark epochs at which, compared to theprevious epoch, the uid level is up or down by exp l distributed amounts orremains constant. Hence, the term stochastic discretization. Such a procedure has

been considered by Adan and Resing in Ref.[1]

in an operational manner; we thank Adan for bringing [1] to our attention. Our work here and in Ref. [2] makes theapproach rigorous. More importantly, the work here sets the stage for transient resultsin Ref. [2] which are not obtainable by the approach of [1] base on long run averagerewards only.

4. ANALYSIS OF Q

The queue Q can be analyzed through the embedded QBD at the epochs { sk : k $ 0} :Here, however, we will analyze it through the embedded QBD at the epochs { t k : k . 0}since that will help us to reconcile results obtained here with those in previous papers moreeasily.

4.1. Embedded QBD

Let us dene Qk Qt k : It is trivial to see that the process { Qk ; J t k : k $ 0} is aMarkov chain. In the following, the possible values of the queue length Qn will be called aslevels.

The probabilities of the chain Qn; J t nmaking a transition to the next higher level isclearly governed by the substochastic matrix

P 0

P 11 P12 P13

0 0 0

0 0 0

26643775

: 4





9/24

Similarly, the probabilities of going downward in levels is governed bythe substochastic matrix

P 2

0 0 0

0 12 I 0

0 0 0

26643775

: 5

Finally, in the steps where the process Qn does not change level, the probabilities aregoverned by the substochastic matrix

P 1

0 0 012 P 21

12 P 22

12 P 23

P 31 P32 P33

2664

3775

: 6

Indeed, if one considers the successive epochs of arrivals, service completions andphase changes as dened above, one gets a discrete time QBD with transition matrix

M P0 0 0

P 2 P1 P0 0

0 P2 P1 P0

26666643777775

; 7

where M P 2 P 1 governs the movements within level 0.Also, note that the process { Qk ; J t k ; t k : k $ 0} is a Semi Markov Process (SMP)

with exponential sojourn times; furthermore, the mean sojourn time in state n; j is given

by 1/ l if j [ S 1 < S 3 ; and by 1=2l if j [ S 2 :Clearly, the evolution of the continuous time process dened by Qt can be describedby the continuous time QBD

B A0 0 0

A2 A1 A0 0

0 A2 A1 A0

26666643777775

8

where

Ai diag l I ; 2l I ; l I P i 2 d i1 I

with d i1 being 1 or 0 depending on whether i 1 or not, and the three blocks of thediagonal matrix on the right side of the above equation are respectively of sizes jS iji 1; 2; 3: Finally, B A2 A1 : This is clear by considering the instantaneous rates of transitions along with the probabilities of successive states.

We summarize the above discussion as a theorem.





10/24

Theorem 4. The process {Qk ; J t k ; t k : k $ 0} is a semi-Markov process withexponential sojourn times. The mean sojourn time in state n; jis 1=l if j [ S 1 < S 3; and it is 1=2l if j [ S 2: The embedded Markov chain {Qk : k $ 0} is a discrete time QBDwith transition matrix given in Eq. 7. The continuous time process {Qt ; J t : t $ 0}is a continuous time QBD with innitesimal generator given in Eq. 8.

Remark 4. Obviously, there are many dierent QBD s that one can associate with thehomogeneous uid model: (i) For instance, in the construction, instead of considering thequeue size and phase to the right of the transition epochs, one may consider those to the leftof such epochs and consider the resuting QBD ; one such leads to results in the formobtained by Soares and Latouche [19] using other techniques. See Section 6. (ii) One mayalso consider the time reversed uid process and work in terms of the QBD s that arise fromit. The QBD in Ramaswami [17] corresponds to one such construction; we omit the details.

4.2. Steady State Queue Length

Here, we characterize (when it exists) the steady state distribution as k ! 1 of theembedded queue length and phase processes Qt k ; J t k : This distribution will be usedlater to derive the steady state distribution of the uid ow model.

Dene p to be the steady state probability vector of the CTMC of phasesi.e., theunique vector satisfying p T 0; p 1 1; where 1 is a column vector of ls, and partition itas p p 1 ; p 2 ; p 3 corresponding to the sets S i; i 1; 2; 3: The necessary and sufcientcondition for stability of the uid ow is well-knownnamely,

p 11 , p 21: 10

In the sequel, we assume this to be the case.

Theorem 5. The uid model is stable iff the queue Q is stable.

Proof. Let

h h 1 ; h 2; h 3 p 1; 2p 2 ; p 3:

One can verify easily that p T 0 is equivalent to the condition that

h h P 0 P 1 P 2:

Now, the inequality (10) is equivalent to h P 21 . h P 01 which is the necessary andsufcient condition for the stability of the QBD given by Eq. (7). Thus, Q is stable iff F isstable. A





11/24


12/24

Remark 5. The above theorem provides a way to compute R via the G-matrix of the timereversed dual which is denoted here by the symbol G . The computation of G can be done

efciently using the logarithmic reduction algorithm. [12] This approach is the analog of theone developed in Ref. [17] , but using the QBD considered in this paper. The matrix G can berelated to the busy period of the time reversed uid ow corresponding to thehomogenized uid model under consideration. The details are routine and omitted; seeRef. [17] for similar results.

Another approach for computing R is given by the following result which leads to anapproach similar to that adopted in Ref. [19] , but using the QBD constructed in this paper. Itexploits the connection between the R matrix and the G matrix of a QBD and avoids timereversal arguments entirely. One can interpret the matrix G below as related to the busyperiod of the given homogenized uid model itself; we omit the details which are routine;see Ref. [19] for similar results.

Theorem 8. The matrix R which is the minimal solution of Eq. 11 is given by

R P 0 I 2 P 1 2 P 0 G2 1 ; 15

where the stochastic matrix G is the minimal nonnegative solution of the equation

G P 2 P 1 G P 0 G 2 16

and has the structure

G

0 G12 0

0 G22 0

0 ^

G32 0

2664

3775

: 17

Proof. The relationship between R and G is a familiar relation obtained byG. Latouche. [l0] The stochasticity of G is due to the fact that the discrete time QBD isstable, and its special structure is a result of the structure of the matrix P 2. A

Remark 6. The relationship between R and G exploited here is special to the QBD anddoes not extend to more general models. However, the duality results presented earlier doextend in a natural manner. [16] Thus, the approach using the dual allows one to analyzemuch more general uid ow models (for instance those with jumps) than the onesconsidered here. We omit the details as it would constitute a major digression.

Before we conclude this section, we present one more result which is needed in thenext section. It is obtained by invoking a well-known result [7] that expresses the meanreturn time of a state in a Markov renewal process in terms of the steady state probabilitiesof the embedded Markov chain and the fundamental mean c of the process. The quantityc is dened as the steady state expected length of a sojourn interval. One can obtain c as





13/24

the weighted average of the mean sojourn times of individual states weighted by theirsteady state probabilities in x, but we will have no need for its explicit value.

Theorem 9. Let mn; j denote the mean return time of the state n; j in the SMPconsidered in Theorem 4. We have m n; j c= xn; j; where c is the fundamental mean of the semi-Markov process.

5. STEADY STATE RESULTS

We rst prove the following important result which explains our interest in thequeueing model constructed by us.

Theorem 10. The process {F t ; J t : t $ 0} has a steady state distribution as t ! 1iff the process {Y t ; J t : t $ 0} has a steady state distribution as t ! 1 ; and when theyexist both distributions are the same.

Proof. That the existence or otherwise of the stationary distributions is simultaneous forboth processes is immediate from Theorem 5. It is obvious from the construction that thePoisson process M % N is such that { M t s N t s 2 M t N t : s $ 0} isindependent of { F u2 ; J u2 : 0 # u # t } as well as the process { Y u2 ; J u2 :0 # u # t }; the latter depend only on the history of the superposition in the interval 0; t :Thus, one can use the PASTA (Poisson Arrivals see Time Averages) theorem [9,20] to assert

the following two equations:lim t !1 P F t # x; J t j limk !1 PF sk 2 # x; J sk 2 j ;

lim t !1 P Y t # x; J t j limk !1 P Y sk 2 # x; J sk 2 j :

The equality of the stationary distribution now follows from the equidistribution of thepairs F sk 2 ; J sk 2 and Y sk 2 ; J sk 2 for each k. A

Now, using the steady state queue length distribution obtained in the previous section,we can obtain the steady state joint distribution of Y t ; J t as t ! 1 ; thereby obtainingthe steady state distribution of the homogenized uid model.

5.1. Steady State Results for Y t; J tLet V j x denote the steady state probability as t ! 1 that Y 1t # x and the phase

J t is j. For x . 0; v j x d dx V j x is the steady state joint density; its existence isestablished inter alia in our derivation. In the following, we let V x and v x denoterespectively the vectors with elements V j x; j [ S and v j x; j [ S : Also, assumed is that





14/24

in accordance with the partition of the state space, these have also been partitioned; that is,V x V 1 x; V2 x; V3 x; and v x v1 x; v2 x; v3 x: Likewise, we also

partition the vector of steady state probabilities xn , corresponding to queue length n asx n1;x n2;x n3;.

5.1.1. Steady State Density in the Set S 1

Let us dene r n; j0;idu to be the element in the Markov renewal kernel (see Ref.[7] )

associated with the semi-Markov process Qt ; J t that is usually interpreted as theelementary probability of a transition by the SMP into the state ( n, j) at u, given that itstarts in (0, i) at time 0. Conditioning on the last epoch of jump of the SMP t k before time t ,we can write the joint transform given by

E 0;ie2 sY t

I J t j 18in terms of the Markov renewal kernel. Indeed, we note the following facts:

(a) For the phase at t to be j, the phase entered at t k must be j, and no additionaltransition from j should occur before time t .

(b) If j [ S 1 ; then Y t Y t k t 2 t k :(c) If Qk n; i.e., the queue size at t k is n, then Y t k W t k is distributed as the

sum of n iid expl random variables.

Piecing together the facts noted above, we can write for j [ S 1 the transform inEq. (18) as given by

X1

n0 Z t

0 r n; j0;idu

ls l

n

e2 st 2 u

Pt k 12

t k .

t 2

u 19

We note that the probability appearing in the above integral is given by exp2 l t 2 u:Now, for j [ S 1; taking the limit as t ! 1 in Eq. (19) using the Key Renewal

Theorem (see Ref. [7] ), we get the limit of that transform as

X1

n0c 2 1 xn; jZ

1

0

ls l

n

e 2 st e 2 l t dt X1

n0c 2 1 xn; j

ls l

n 1s l

20

Using the matrix-geometric result for xn and the structure of R given in Theorem 6 andEq. (12), which imply that xn1 x 01 Rn11 ; we can write the quantities in Eq. (20) for j [ S 1together in vector form as

c 2 1x01X1

n0 Rn11 ls l

n

1s l

: 21

The above expression can be simplied to

c 2 1x01 sI 2 l R11 2 I 2 1 ; 22





15/24


16/24

Proof. We invoke Theorem (10) and evaluate the density for the process Y . By a routineargument conditioning on the last epoch t k before time t , we can write for i [ S 1; j [ S 2;

v2 x j t !1limX1

n1 Z t

0r n; j0; idue

2 2l t 2 ue 2 l x l xn2 1

n 2 1!l : 31

The above is obtained by noting that for Y (t ) to be in x; x dx and the phase at t to be j [ S 2 ; at the last epoch of transition u before time t , there should be some n . 0customers in the system, there should be no further transitions in u; t ; and the totalremaining work at t of those n customers (which is distributed as a sum of n exponentialrandom variables) should be in x; x dx:

Taking the limit as t ! 1 in the above and writing in vector form, we get

v 2 x c2 1X

1

n1xn2Z

1

0e 2 2l t dt e 2 l x

l xn2 1

n 2 1!l

c 2 1X1

n1x01 Rn

2 111 R12

12

e 2 l x l xn2 1

n 2 1! c 2 1x01 e Kx

12

R12 v 1 xC : 32

In the above, the second equality is obtained using the matrix-geometric structure of xnand the structure of R by which xn2 x 01 Rn

2 111 R12 ; and the last equality is obtained by

using Eqs. (24) and (23). A

Corollary 1. The probability that the uid buffer is empty and the phase is in S 2 is givenby the vector

V 20 p 2 2 p 1C : 33

Proof. Note that p 2 is the steady state probability for the set S 2. Subtracting from this theintegral of v2 x in 0; 1 obviously should give the required vector of emptinessprobabilities for S 2. Hence the result. A

5.1.3. Steady State Density in the Set S 3

Now, for j [ S 3; we have

P 0;i0 , Y t % x; J t j

Z t

0 X1

n1r n; j0;idue

2 l t 2 uZ x

0

l n

n 2 1!e 2 l y yn2 1dy:

34

This follows from the fact that for the said event to occur, at the time of the last epoch u of transition of the SMP, the phase should be j, no transition should occur in ( u, t ] and





17/24


18/24

We also have

V30 p

32 p

1Q

:

43

6. COMPUTATIONAL SIMPLIFICATION

The key quantities needed in the computation of the steady state uid ow distributionare the matrices K , C and Q . We have already noted in Eq. (42) that Q can be expressed interms of C and the matrix T . The following simple result shows that K can be written interms of C and Q so much so that the only non-trivial computation needed is that of thematrix C . After noting this, we also provide some results that further simplify thecomputation of C .

Theorem 14. The matrix K is given byK T 11 C T 21 Q T 31 : 44

Proof. Using the partitioned structures of R and P i in Eq. (11) and (4)(6), one canobtain the following:

R11 P 11 C P 21 R13 P 31 :

Substituting K l R11 2 I and Q R13 and substituting for P ij their values in terms of the corresponding T ij , the result is immediate. A

In constructing the embedded Markov chain at epochs t k , we noted that we could have

also used the states to the left of t k . Considering the states to the left of the epochs t k , k $

1leads to the consideration of the discrete time QBD in which the up, within level, anddownward in level transitions are governed respectively by the matrices:

B0

P 11 0 012 P 21 0 0

P 31 0 0

26643775

; B1

0 P12 P13

0 12 P 2212 P 23

0 P32 P33

26643775

; B2

0 0 0

0 12 I 0

0 0 0

26643775

: 45

For that QBD, the matrix G representing the downward passage probabilities from level 1to level 0 is the minimal nonnegative solution of the equation

G B2 B1 G B0 G 2 : 46

Because of the structure of B 2, we also have the partitioned structure

G

0 G12 0

0 G22 0

0 G32 0

26643775

: 47





19/24

We can now deduce the following result which helps us derive a simpler procedure tocompute C .

Theorem 15. The matrix C is also given by C G12 ; where G 12 is the submatrix of Gcorresponding to row indices in S 1 and column indices in S 2.

Proof. By the results in Ref. [15] , there exists a kernel ^ Rdu such that R ^ R1 ; andfurthermore Rdu ij is the elementary probability that the SMP Qt k ; J t k ; t k : k $ 0}visits the state 1; j in u; u du avoiding level 0 given that it starts in state 0; i: Now,the matrix

C 12

R12

Z 1

0

^ R12 du

Z 1

0

2l e 2 2l xdx12

I :

The integral on the right clearly records the probability of the queue emptying out fromvarious phases in S 2 given that the semi-Markov process starts a busy cycle in state 0; i;i [ S 1; the queue must reach the state 1; jin its rst busy period, and then empty out fromthat phase in one step. Thus, C records the probability that the process { Qt k 2 ; J t k 2 :k $ 2} visits 0; j when it visits level 0 for the rst time given that Qt 1 2 ; J t 12 0; i: Therefore, C G12 and hence the result. A

The next result allows one to work with matrices of smaller orders when iterativelycomputing C . The resulting formula for C is similar to that obtained by Soares andLatouche. [19]

Theorem 16. The matrix

G 0 G12

0 G22" #is the minimal nonnegative solution to the equation

G B2 B1G B0G 2; 48

where

B2 0 0

0 12 I 24

35

; B1 0 P12 P 13 I 2 P 33

2 1P 32

0 12 P 22 P 23 I 2 P 33 2 1P 322

435

;

B0 P 11 P 13 I 2 P 33

2 1P 31 012 P 21 P 23 I 2 P 33

2 1P 32 024 35:





20/24

Proof. Considering the QBD of Theorem 15 restricted to the set S 1 < S 2; we get theQBD governed by Bi, i 0; 1; 2: That the G-matrix of the censored QBD is given by thematrix G dened above is trivial to see as well. A

7. PHASE TYPE REPRESENTATION

So far, we have worked with the homogeneous uid ow and we need to translate ourresults to the original model given by Q and C ; we shall briey refer to that model as theQ-model. In the sequel, we shall assume that the diagonal matrix C is also partitionedaccording to the state space as C diag C 11 ; C 22 ; I where each block is also diagonal.

The following result concerning steady state probabilities is a trivial consequence of the relation T C 2 lQ:

Theorem 17. Let j denote the steady state probability vector corresponding to theinnitesimal generator Q. Then

j 1

p C 2 11p C 2 1; 49

where, p is the steady state probability vector of T .

Now, let W i x and w i x denote the vectors in the Q-model that are analogous toV i x and v i x:

A standard argument considering the epochs t and t D t leads to a set of partialdifferential equations for the time dependent state probabilities of the uid model, fromwhich one gets by letting t ! 1 the following equations for the steady state densities wi:

w 1 xQ11 w 2 xQ21 w 3 xQ31 w 01 xC 11

w 1 xQ12 w 2 xQ22 w 3 xQ32 2 w 02 xC 22

w 1 xQ13 w 2 xQ23 w 3 xQ33 0

Comparing the above with the analogous equations for the homogeneous uid ow, wecan immediately deduce that

w i x h v i xC 2 1ii for i 1; 2; 3; 50

for some constant h .Thus, we can write in partitioned form

w x h v 1 xC 2 111

..

.C

C 2 122

..

.Q

C 2 133

2hp 1Ke

Kx

C 2 111

..

.C

C 2 122

..

.Q

C 2 133

2 hp 1Ke KxC 2 111 I

..

.C 11 C C

2 122

..

.C 11 Q C

2 133 2 hp 1C

2 111

~Ke ~Kx I ..

.~C ..

.~Q ;

2 h p C 2 11j 1 ~Ke ~Kx I ..

.~C ..

.~Q ;





21/24


22/24

The actual numerical computation of the steady state probability density or cdf can beaccomplished by using the available techniques for phase type distributions (see Ref. [11] ,

Chapter 2) since we can show that the distribution of the uid ow is indeed aPH-distribution.

Theorem 19. For the uid model given by Q; C , the steady state distribution of the uid level is a phase type distribution. One representation for that phase type distributionis given by the pair b ; U , where

b 1 ~C 1 ~Q 10 ~D 1;

U ~D2 11

~K 0 ~D 1; the matrix ~D 1 , is a diagonal matrix with j 1 on its diagonal, and 1 is avector of 10s of appropriate order determined by the context in which it appears.

Proof. The proof proceeds by using the duality arguments similar to those in Ref. [16] .We write the steady state uid density

f x X3

j1w j x1

as also given by

f x 2 j 1 ~Ke ~Kx1 ~C 1 ~Q 1 10 10 ~Q 0 10 ~C 0e

~Kx0~K 0j 1 0 b e Ux2 U 1:

The second equality above is obtained by noting that the transpose of a scalar is itself, andthe last is obtained by supplying the necessary products ~D 1 ~D

2 11

~D2 11

~D 1 I in betweenthe terms of the previous equation. The fact that U satises the condition for a generatorcan be veried by considering the duality formula for K ; we omit the details which areroutine; like in Ref. [17] , it is easy to show that for the time reversed dual ow of the Q-ow,e Ux indeed records the probability of the busy period ending in various phases given thephase at the beginning of that busy period. A

Remark 7. There is a discrepancy between our results here and those reported in Ref. [19]

due to a minor error which crept into that paper. Specically in Ref. [19] , a term j 1 appearssuperuously in the Eq. (4) of that paper, and that error then gets propagated to Theorem5.1 of that paper. We are grateful to Soares and Latouche for conrming the correctness of our ndings.

ACKNOWLEDGMENT

The authors thank Ward Whitt for many helpful discussions, and Benny VanHoudt foridentifying several typos in an earlier version. Soohan Ahn was supported (partially) by





23/24

the Post-Doctoral Fellowship Program of Korea Science & Engineering Foundation(KOSEF).

REFERENCES

1. Adan, I.J.B.F.; Resing, J.A.C. A two-level trafc shaper for an on-off source.Perform. Eval. 2000 , 42 , 279298.

2. Ahn, S.; Ramaswami, V. Transient analysis of uid ow models via stochasticcoupling to a queue, 2003, to be submitted.

3. Anick, D.; Mitra, D.; Sondhi, M.M. Stochastic theory of a data handling system withmultiple sources. Bell Syst. Tech. J. 1982 , 61 , 1871 1894.

4. Asmussen, S. Stationary distributions for uid ow models with or withoutBrownian noise. Stochast. Models 1995 , 11 , 1 20.

5. Asmussen, S.; Ramaswami, V. Probabilistic interpretations of some duality resultsfor the matrix paradigms in queueing theory. Stochast. Models 1990 , 6 , 715734.

6. Choudhury, G.L.; Mandelbaum, A.; Reiman, M.I.; Whitt, W. Fluid and diffusionlimits for queues in slowly changing environments. Stochast. Models 1997 , 13,121146.

7. Cinlar, E. Markov renewal theory. Adv. Appl. Probab. 1969 , 1 , 123187.8. Grassmann, W.K.; Taksar, M.J.; Heyman, D.P. Regenerative analysis and steady

state distributions for Markov chains. Oper. Res. 1985 , 33 , 1107 1116.9. Ko nig, D.; Schmidt, V. Imbedded and non-imbedded stationary characteristics of

queueing systems with varying service rate and point processes. J. Appl. Probab.1980 , 17 , 1048 1061.

10. Latouche, G. A note on two matrices occurring in the solution of quasibirth-and-death processes. Stochast. Models 1987 , 3 , 251257.

11. Latouche, G.; Ramaswami, V. Introduction to matrix analytic methods in Stochastic Modeling ; SIAM & ASA: Philadelphia, 1999.

12. Latouche, G.; Ramaswami, V. A logarithmic reduction algorithm for Quasi-Birth-and-Death Processes. J. Appl. Probab. 1993 , 30 , 650674.

13. Kella, O.; Whitt, W. A storage model with a two state random environment. Oper.Res. 1992 , 40 (2), S257S262.

14. Neuts, M. F. Matrix-Geometric Solutions in Stochastic Models, An Algorithmic Approach ; The Johns Hopkins University Press: Baltimore, MD, 1981.

15. Ramaswami, V. The busy period of queues which have a matrix-geometric steadystate probability vector. Opsearch 1982 , 19 , 238261.

16. Ramaswami, V. A duality theorem for the matrix paradigms in queueing theory.Stochast. Models 1990 , 6 , 151161.

17. Ramaswami, V. Matrix analytic methods for stochastic uid ows. In Teletrac

Engineering in a Competitive World ; Smith, D., Key, P., Eds.; Proc. of the 15thInternational Teletrafc Congress; Elsevier, 1999; 10191030.18. Ramaswami, V. Algorithmic analysis of stochastic modelsThe changing face of

mathematics. In Mathematics for the New Millennium ; Vijayakumar, A.,Krishnakumar, S., Eds.; Ramanujan Memorial Lecture; Anna University: Chennai,India, 2000.





24/24

19. da Silva Soares, A.; Latouche, G. Further results on the similarities between uidqueues and QBDs. In Matrix-Analytic Methods, Theory and Applications ;

Latouche, G., Taylor, P.G., Eds.; Proc. of the Fourth International Conference onMatrix Analytic Methods; World Scientic Publishing Co.: Singapore, 2002.

20. Wolff, R.W. Poisson arrivals see time averages. Oper. Res. 1982 , 30 , 223231.

Received May 10, 2002Revised October 18, 2002Accepted February 7, 2003




Documents

Fluid Flow Models and Queues