Queueing Syst (2007) 56: 1–8DOI 10.1007/s11134-007-9022-0

On the stability of the multi-queue multi-server processorsharing with limited service

Andreas Brandt · Manfred Brandt

Received: 16 June 2006 / Revised: 27 March 2007 / Published online: 17 May 2007© Springer Science+Business Media, LLC 2007

Abstract We consider a multi-queue multi-server systemwith n servers (processors) and m queues. At the systemthere arrives a stationary and ergodic stream of m differ-ent types of requests with service requirements which areserved according to the following k-limited head of the lineprocessor sharing discipline: The first k requests at the headof the m queues are served in processor sharing by the n

processors, where each request may receive at most the ca-pacity of one processor. By means of sample path analysisand Loynes’ monotonicity method, a stationary and ergodicstate process is constructed, and a necessary as well as a suf-ficient condition for the stability of the m separate queuesare given, which are tight within the class of all stationaryergodic inputs. These conditions lead to tight necessary andsufficient conditions for the whole system, also in case ofpermanent customers, generalizing an earlier result by theauthors for the case of n = k = 1.

Keywords Head of the line processor sharing · Threadpools · Many queues · Many servers · General input · Batcharrivals · Permanent customers · Marked point process ·Stability condition · Loynes’ construction · Stationarity ·Ergodicity

This work was supported by a grant from the Siemens AG.

A. Brandt (�)Institut für Operations Research, Humboldt-Universität zu Berlin,Spandauer Str. 1, 10178 Berlin, Germanye-mail: brandt@wiwi.hu-berlin.de

M. BrandtKonrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB),Takustr. 7, 14195 Berlin, Germanye-mail: brandt@zib.de

Mathematics Subject Classification (2000) 60K25 ·68M20 · 60G10 · 60G17 · 60G55

1 Introduction

In telecommunication systems different processes consist-ing of requests have to be served. For improving the per-formance, in modern systems several processors are used inparallel for processing the requests. The system resources,in particular the processor capacities, have to be shared be-tween different types of requests in such a way that on theone hand certain performance characteristics for the dif-ferent processes can be guaranteed and on the other handa high system utilization is ensured. Furthermore, the re-sources should be allocated in a fair manner to the differ-ent processes. Various scheduling disciplines are known andimplemented for meeting these requirements. In the RoundRobin (RR)—also called time sharing—disciplines, cf. e.g.[8], a scheduler allocates a fixed quantum of service, i.e. ofprocessing time, to the requests present under service in aRR manner. RR disciplines ensure that requests with smallservice requirements have smaller sojourn times comparedto those with larger service requirements. For small servicequanta the RR disciplines are well approximated by cor-responding Processor Sharing (PS) disciplines, which aremore convenient for the analysis of time sharing systems, cf.e.g. [6]. PS systems have been studied by many researchers,cf. e.g. [1–4, 7, 11, 14, 18–24] and the references therein.

The fact that some requests may have to be served se-quentially implies for the system architecture that requestshave to be queued appropriately and that only the first re-quest in each queue is processed under the PS discipline.This discipline is known as Head of the Line ProcessorSharing (HOL-PS) discipline. In case of thread pools also

2 Queueing Syst (2007) 56: 1–8

Fig. 1 The multi-queue multi-server k-limited head of the line proces-sor sharing system with m queues and n processors in case of k = 2.

The boxes correspond to requests, the hatched boxes to requests inservice

more than one request of a queued thread can be processedsimultaneously by the processors. This leads to a general-ized HOL-PS discipline considered in this paper, which wecall k-limited HOL-PS discipline: the first k requests of allqueues are served by the processors in the PS mode.

More precisely, in this paper we consider a system con-sisting of m ≥ 1 queues and n ≥ 1 servers (processors), cf.Fig. 1. At the system there arrives a stream of m types ofrequests with service requirements. The input is describedby a marked point process � = {[T�, I�, S�]}∞�=−∞ on thereal line with the mark space K = {1, . . . ,m} × R+ and· · · ≤ T0 ≤ 0 < T1 ≤ · · · , where T� are the arrival instantsof the requests, I� ∈ {1, . . . ,m} indicates the type, i.e. thequeue where the request goes to, and S� ∈ R+ denotes therequired service time of the �-th request. We assume in thefollowing that � is a stationary and ergodic marked pointprocess, cf. e.g. [5]. Note that batch arrivals are includedand that there are no independence assumptions.

The input at queue i, i.e. the stream of type-i requestsincluding their required service times, is given by the sta-tionary ergodic marked point process

�i = {[Ti,�, Si,�]}∞�=−∞, (1.1)

where · · · ≤ Ti,0 ≤ 0 < Ti,1 ≤ · · · are the arrival instants oftype-i requests and Si,� their required service times. For theoffered load �i of the type-i requests it holds

�i = E

∞∑

�=−∞Si,�I{0 < Ti,� ≤ 1}

= E

∫

(0,1]×R+y�i(d(x, y)) = λimB0

i,

where λi = E�i((0,1] × R+) is the intensity of �i andmB0

i= ES0

i is the expectation of the service time S0i of a

typical type-i request, which is given by the Palm distribu-tion of P(�i ∈ (·)), cf. e.g. [10] formula (1.2.8).

The requests in the queues are served by the processorsaccording to the above mentioned k-limited HOL-PS disci-pline: The first k ≥ 1 requests of the m queues are served in

PS by the n processors. This means, if there are ki requestsin queue i, i = 1, . . . ,m, then there are altogether

b :=m∑

i=1

min(ki, k) (1.2)

requests in service, and, if b is positive, each of the firstmin(ki, k) requests from queue i receives the fraction

ϕ(b) := min

(n

b,1

)(1.3)

of the capacity of one processor.Note that the case of k = 1 corresponds to the ordinary

multi-queue multi-server HOL-PS system, which we simplycall multi-queue multi-server HOL-PS system. For k → ∞we obtain a G/G/n-PS system, where G/G correspondsto the cumulative arrivals and service times of all requests.If mk ≤ n then the m queues act as Gi/Gi/k/∞-FCFSqueues, where Gi/Gi stands for the arrival process and ser-vice times of the type-i requests.

The multi-queue single-server HOL-PS (k = 1) systemwith Poisson arrival processes and exponential service timeshas been considered and analyzed by several authors: In [13]the generating function of the occupancy distribution is de-rived in case of a completely symmetric system with twoqueues. In [12] a representation of the joint distribution ofthe queue length is derived by using power series expansionswith respect to the offered load; the established radius ofconvergence decreases rapidly in the number of queues. Forheavy traffic approximations we refer to [9, 13, 16]. HOL-PS systems with limited capacities are analyzed in [9, 17].In [14] approximations of the mean sojourn time for a PSsystem with Background Jobs are derived, which covers theHOL-PS model with exponential service times. The multi-queue single-server HOL-PS system with permanent cus-tomers is investigated in [2], where partially general servicetime distributions are considered. The single-queue multi-server system with k-limited HOL-PS discipline is a specialcase of the multi-programmed system given in [21]. For themulti-queue multi-server system with a k-limited HOL-PSdiscipline general results seem not to be known, even fork = 1, to our best knowledge.

The paper is organized as follows. In Sect. 2 by meansof Loynes’ monotonicity method, for the multi-queue multi-server k-limited HOL-PS system with a general stationaryand ergodic input, a stationary state process is constructed.In Sect. 3 a necessary as well as a sufficient condition forthe stability of the separate queues (Theorem 3.1, Corol-lary 3.1) are given. These stability conditions are tight, i.e.,they cannot be improved within the class of all stationaryand ergodic inputs (Example 3.1). They lead to tight neces-sary and sufficient conditions for the stability of the wholesystem (Corollary 3.2), also in case of permanent customers

Queueing Syst (2007) 56: 1–8 3

(Corollary 3.3), generalizing a corresponding result for themulti-queue single-server HOL-PS (k = 1) system with per-manent customers given in [3]. The gap between the nec-essary and sufficient stability conditions for the system isillustrated (Example 3.2).

2 Construction of a stationary state process

Let Ri,�(t) ∈ R+ the residual service time at time t of the�-th request arrived at queue i before t , � = 1,2, . . . , or-dered in the reversed order of the numbering of the arrivalinstants. Thus Ri,�(t) denotes the residual service time attime t of a �-th last arrived request at queue i before t whereall requests arrived before t are counted. Further let

Ri(t) = (Ri,1(t),Ri,2(t), . . .)—infinite vector

of the residual service times in queue i at time t,

R(t) := (R1(t), . . . ,Rm(t))—vector

of the residual service times at time t .

We want to construct a stationary state process based onLoynes’ monotonicity method, cf. e.g. [5, 15]. Let

MK ={ψ = {[t�, i�, s�]}∞�=−∞ : · · · ≤ t0 ≤ 0 < t1 ≤ · · · ,

lim�→±∞ t� = ±∞, i� ∈ {1, . . . ,m}, s� ∈ R+

}(2.1)

be the set of point process realizations where � is concen-trated on, i.e. P(MK) = 1. For τ > 0, let

r(τ)i (t,ψ) = (r

(τ)i,1 (t,ψ), r

(τ)i,2 (t,ψ), . . .),

i = 1, . . . ,m, (2.2)

r(τ)(t,ψ) = (r(τ)1 (t,ψ), . . . , r(τ)

m (t,ψ)) (2.3)

be the state of the system at t if it was started at timet − τ from the empty system with input realization ψ ∈ MK ,where the residual service times of the arrivals until t −τ are0 by definition. The workload in queue i at t is given by

v(τ)i (t,ψ) =

∞∑

�=1

r(τ)i,� (t,ψ), i = 1, . . . ,m, (2.4)

and the number of requests in queue i at t is given by

k(τ)i (t,ψ) =

∞∑

�=1

I{r(τ)i,� (t,ψ) > 0}, i = 1, . . . ,m. (2.5)

Note that requests arriving or departing at t are not countedin (2.5). As ϕ(b) is non-increasing with respect to b, cf.(1.3), it follows from the dynamics of the k-limited HOL-PS

discipline, in particular from the fact that all requests underservice receive an equal fraction of the processor capacity,that the vector of the residual service times r(τ)(t,ψ), i.e.each partial component r

(τ)i,� (t,ψ), is non-decreasing with

respect to τ , which is crucial in Loynes’ method. Becauseof this monotonicity, the limit as τ → ∞ exists:

ri,�(t,ψ) := limτ→∞ r

(τ)i,� (t,ψ),

i = 1, . . . ,m, � = 1,2, . . . , (2.6)

r(t,ψ) := limτ→∞ r(τ)(t,ψ). (2.7)

The state r(t,ψ) corresponds to the system state at t if thesystem was started at time −∞ from the initial state whereall residual service times are 0. Of course, the number ofrequests in queue i

ki(t,ψ) :=∞∑

�=1

I{ri,�(t,ψ) > 0}, i = 1, . . . ,m, (2.8)

may be infinite for some queues. However, r(t,ψ) satisfiesthe system dynamics due to continuity arguments. From nowon let

R(t) := r(t,�), t ∈ R, (2.9)

which is a stationary and ergodic process due to themonotonicity property and since the residual service timesof the arrivals until t − τ are 0 by definition. Note that R(t),t ∈ R, is the minimal stationary state process. Let

Ki(t) := ki(t,�), t ∈ R, (2.10)

the corresponding stationary number of requests in queue i

at t .

3 Stability conditions

A reasonable definition of stability for the model considered,cf. Theorem 3.1(i), is the following one:

Definition 3.1 Queue i ∈ {1, . . . ,m} is stable if P(Ki(0) <

k) > 0. The processor sharing system is stable if P(Ki(0) <

k) > 0 for i = 1, . . . ,m.

Let

pi := P(Ki(0) ≥ k), i = 1, . . . ,m, (3.1)

be the probability that at time 0 there are at least k type-irequests in the system. Note that in case of k = 1, i.e. in caseof the ordinary HOL-PS system, pi is the probability that

4 Queueing Syst (2007) 56: 1–8

queue i is non-empty at time 0. Because of the stationarityand ergodicity of R(t), it holds

pi = limt→∞

1

t

∫ t

0I{Ki(x) ≥ k}dx P-a.s.,

i = 1, . . . ,m. (3.2)

By Definition 3.1 queue i is stable if and only if pi < 1.

Theorem 3.1 Let

�1 ≥ �2 ≥ · · · ≥ �m > 0. (3.3)

For the minimal stationary state process R(t) given by (2.9),(2.7) it holds:

(i) If queue i is stable then queue j is stable if �j ≤ �i .If queue i is unstable then queue j is unstable if �j ≥�i .

(ii) Queue i ∈ {1, . . . ,m} is stable if there exists a h ∈{ik, ik + 1, . . . ,mk} such that

(h/k)�i + (�h/k� − h/k)��h/k�

+m∑

j=�h/k�+1

�j < min(n,h). (3.4)

(iii) If queue i ∈ {1, . . . ,m} is stable then

�i < k, i�i +m∑

j=i+1

�j ≤ n (3.5)

with equality only in case of �i = �1.

Proof If the system is non-empty at t then any served re-quest receives the random fraction

C∗(t) = min

(n∑m

j=1 min(Kj (t), k),1

)(3.6)

of the capacity of one processor at t , cf. (1.2), (1.3). Fortechnical convenience, let C∗(t) := 1 if the system is emptyat t . Thus queue i receives the random multiple

Ci(t) = min(Ki(t), k)C∗(t), i = 1, . . . ,m, (3.7)

of the capacity of one processor at time t . Because of thestationarity and ergodicity of R(t), for the mean multiple ofthe processor capacity received by queue i it holds

ECi(0) = limt→∞

1

t

∫ t

0Ci(x)dx P-a.s., i = 1, . . . ,m.

(3.8)

Let

�∗ := kEC∗(0). (3.9)

Then from (3.1), (3.6), (3.7), (3.9) it follows

�∗ − k(1 − pi) ≤ ECi(0) ≤ �∗ − min

(n

mk,1

)(1 − pi),

i = 1, . . . ,m. (3.10)

We use Loynes’ construction (2.1–2.7). Let τ > 0 befixed. Starting the dynamics of the processor sharing systemat time −τ from the empty system then

c(τ)i (�) := v

(τ)i (0 + 0,�) − v

(τ+1)i (1 + 0,�)

+∞∑

�=−∞Si,�I{0 < Ti,� ≤ 1}, i = 1, . . . ,m,

(3.11)

is just the amount of service that receive the type-i requestsduring the interval (0,1] by the processors. Taking expecta-tions we find

Ec(τ)i (�) = Ev

(τ)i (0 + 0,�) − Ev

(τ+1)i (1 + 0,�) + �i.

(3.12)

By the stationarity of � and since v(τ)i (0 + 0,�) is non-

decreasing with respect to τ , it holds

Ev(τ+1)i (1 + 0,�) = Ev

(τ+1)i (0 + 0,�)

≥ Ev(τ)i (0 + 0,�).

Thus (3.12) yields

Ec(τ)i (�) ≤ �i, i = 1, . . . ,m.

By the stationarity of � , taking the limit as τ → ∞ we ob-tain

ECi(0) ≤ �i, i = 1, . . . ,m. (3.13)

Let pi < 1. Because of (3.2), then we conclude that in anyneighborhood of infinity queue i possesses periods whereless than k requests are in queue i. Thus, in view of thestationarity of � and of the system dynamics, all requestsarriving at queue i will be served. Since

∫ t

0 Ci(x)dx is theamount of service received by queue i during the interval(0, t], we find

�i = limt→∞

1

t

∫

(0,t]×R+y�i(d(x, y))

= limt→∞

1

t

∫ t

0Ci(x)dx = ECi(0), (3.14)

i.e., �i is just the mean multiple of the processor capacityreceived by queue i provided pi < 1.

Queueing Syst (2007) 56: 1–8 5

If �i < �∗ then from (3.13) it follows ECi(0) < �∗,and hence from (3.10) we find pi < 1, i.e., queue i is sta-ble. Let �i ≥ �∗. Assuming pi < 1, from (3.14) it followsECi(0) = �i ≥ �∗, and hence (3.10) provides a contradic-tion. Therefore pi = 1, and queue i is unstable if �i ≥ �∗.Summarizing, it holds the following necessary and sufficientstability condition for queue i: queue i is stable if and onlyif

�i < �∗. (3.15)

This stability condition implies (i).From (3.15), (3.14), (3.10), (3.7) and (3.6) we find

m∑

j=1

min(�j , �∗) =

m∑

j=1

(I{pj < 1}�j + I{pj = 1}�∗)

=m∑

j=1

ECj(0)

= E

[min

(n,

m∑

j=1

min(Kj (0), k)

)].

(3.16)

Let us assume pi = 1. Because of (3.3) and (i), then queue j

is unstable for j ∈ {1, . . . , i}, i.e.

1 = pj = P(Kj (0) ≥ k), j = 1, . . . , i. (3.17)

Hence from (3.16) it follows

m∑

j=1

min(�j , �∗) ≥ E

[min

(n,

i∑

j=1

min(Kj (0), k)

)]

= min(n, ik). (3.18)

Taking into account (3.3) and (3.15), thus we find

i�i +m∑

j=i+1

�j =m∑

j=1

min(�j , �i)

≥m∑

j=1

min(�j , �∗) ≥ min(n, ik). (3.19)

This estimate can be tightened as follows.Firstly, we consider the modified system with mk queues

and nk processors where the input at queue j is given by

�(1)j := ��j/k�, j = 1, . . . ,mk. (3.20)

Thus for the offered load at queue j in the modified systemit holds

�(1)j = ��j/k�, j = 1, . . . ,mk. (3.21)

Note that the input at the modified system consists of k

copies of the input at the original system and that moreoverthe dynamics of the modified system consist of k copies ofthe dynamics of the original system. Hence queue j in themodified system is stable if and only if queue �j/k� in theoriginal system is stable, in particular, in the modified sys-tem queue ik is unstable.

Let h ∈ {ik, ik + 1, . . . ,mk}. Secondly, we scale the ser-vice times per queue in the modified system such that forthe offered load �

(2)j of type-j requests in the newly modi-

fied system it holds

�(2)j :=

⎧⎪⎪⎨

⎪⎪⎩

�(1)j , j = 1, . . . , ik,

�(1)ik , j = ik + 1, . . . , h,

�(1)j , j = h + 1, . . . ,mk.

(3.22)

In view of �(2)j ≥ �

(1)j for j = 1, . . . ,mk, from the monoton-

icity property of Loynes’ construction with respect to theservice times it follows that also in the newly modified sys-tem queue ik is unstable. Because of (3.22), (3.21), (3.3)and (i), thus queue j is unstable for j ∈ {1, . . . , h} in thenewly modified system. Hence (3.19) applied to the newlymodified system provides

h�(2)h +

mk∑

j=h+1

�(2)j ≥ min(nk,hk).

In view of (3.22), (3.21), thus we obtain

h�i +mk∑

j=h+1

��j/k� ≥ min(nk,hk),

which is equivalent to the following tightening of (3.19):

(h/k)�i + (�h/k� − h/k)��h/k�

+m∑

j=�h/k�+1

�j ≥ min(n,h). (3.23)

Thus queue i in the original system is stable if there exists ah ∈ {ik, . . . ,mk} such that

(h/k)�i + (�h/k� − h/k)��h/k�

+m∑

j=�h/k�+1

�j < min(n,h), (3.24)

which is part (ii).Let pi < 1. From (3.15), (3.9) and (3.6) it follows the first

part of (iii):

�i < �∗ ≤ k. (3.25)

6 Queueing Syst (2007) 56: 1–8

Because of (3.16), it holds

m∑

j=1

min(�j , �∗) ≤ n, (3.26)

and in view of (3.3) and (3.15), we obtain the second part of(iii):

i�i +m∑

j=i+1

�j =m∑

j=1

min(�j , �i)

≤m∑

j=1

min(�j , �∗) ≤ n (3.27)

with equality of the last both sums only if �i = �1 as the firstsummands are equal only if �i = �1. �

Choosing h = ik for n < ik, h = n for ik ≤ n < mk andh = mk for mk ≤ n in Theorem 3.1, respectively, we obtainthe following corollary:

Corollary 3.1 Let �1 ≥ �2 ≥ · · · ≥ �m > 0.

(i) Let n < ik. Queue i is stable if

i�i +m∑

j=i+1

�j < n, (3.28)

and if queue i is stable then

i�i +m∑

j=i+1

�j ≤ n (3.29)

with equality only in case of �i = �1.(ii) Let ik ≤ n < mk. Queue i is stable if

(n/k)�i + (�n/k� − n/k)��n/k�

+m∑

j=�n/k�+1

�j < n, (3.30)

and if queue i is stable then

�i < k, i�i +m∑

j=i+1

�j ≤ n (3.31)

with equality only in case of �i = �1.(iii) Let mk ≤ n. Queue i is stable if and only if

�i < k. (3.32)

The following example tells us that the sufficient as wellas the necessary stability conditions given in Corollary 3.1are tight for any choice of the offered loads �i , i.e., theycannot be improved within the class of all stationary andergodic inputs.

Example 3.1 Let n < mk and �1 ≥ �2 ≥ · · · ≥ �m > 0 begiven and U be uniformly distributed on [0,1].

(i) Consider the case of batch arrivals of size mk where atthe time instants � + U , � ∈ Z, k requests with service time�j/k arrive at queue j , j ∈ {1, . . . ,m}.

In this case the service of an arriving batch at queue i

takes at least the time∑m

h=i (�h − �h+1)max(h/n,1/k)

where �m+1 := 0. The stability of queue i implies that thisduration has to be less than 1, i.e. (3.28) if n < ik and (3.30)if ik ≤ n < mk. Thus the sufficient stability conditions aretight.

(ii) Consider the case where at the time instants � +∑j−1h=1 �h + U , � ∈ Z, j ∈ {1, . . . ,m}, a request with service

time �j arrives at queue j .Let n < ik, �i = �1 and (3.29) be fulfilled or let ik ≤ n <

mk, �i = �1 and (3.31) be fulfilled.Then it holds �j ≤ �1 = �i < k for j ∈ {1, . . . ,m} and∑mh=1 �h ≤ n.Starting from the empty system, each request receives the

capacity of one processor as long as there are at most k re-quests in each queue and at most n requests in the system.However, as long as each request receives the capacity ofone processor, due to �j < k there are at most k requests ineach queue, and due to

∑mh=1 �h ≤ n there are at most n re-

quests in the system. Hence queue i is stable as �i < k. Thusthe necessary stability condition (3.29) is tight, and the nec-essary stability conditions (3.31) are tight in case of �i = �1.

Let ik ≤ n < mk, �i < �1 and let (3.31) be fulfilled.Then it holds �j ≤ �i < k for j ∈ {i, . . . ,m} and i�i +∑mh=i+1 �h < n.Let

∑mh=i �h ≤ n− (i −1)k. Starting from the empty sys-

tem, each request of the queues j ≥ i receives the capacityof one processor as long as there are at most k requests ineach of these queues and altogether at most n − (i − 1)k re-quests in these queues. However, as long as each request ofthese queues receives the capacity of one processor, due to�j < k for j ≥ i there are at most k requests in each of thesequeues, and due to

∑mh=i �h ≤ n − (i − 1)k there are alto-

gether at most n − (i − 1)k requests in these queues. Hencequeue i is stable as �i < k.

Let∑m

h=i �h > n − (i − 1)k. Assume that queue i is un-stable. In view of Theorem 3.1(i), then the first i queues areunstable. Due to the monotonicity property of Loynes’ con-struction with respect to the service times, the first i queuesare unstable also in the modified system where the servicetime �j of the requests of queue j is raised to max(�j , k) forj < i. Let us analyze the modified system by starting it fromthe empty system at time t − τ where τ ≥ ∑m

h=i �h(> k).Let T ∗ the arrival instant of the request which arrived atqueue i during [t − ∑m

h=i �h, t − ∑mh=i �h + 1). Then for

fixed � ∈ {0, . . . , ∑mh=i �h� − 1} the last of the requests

which arrived before time t at the time instants T ∗ + � +∑j−1h=i �h at queue j ≥ i is a.s. in the system at time t as

Queueing Syst (2007) 56: 1–8 7

the request arrived at queue j during [t − �j , t) and as therequest has service time �j and receives at most the capac-ity of one processor. Thus there are a.s. altogether at least∑m

h=i �h� ≥ n − (i − 1)k requests in the queues j ≥ i attime t . Note that at most k of the requests under considera-tion are in the same queue j as �j < k for j ≥ i and the inter-arrival times for each fixed queue are equal to 1. Moreover,there are a.s. at least k requests in each of the queues j < i attime t as the service times of these requests are at least k inthe modified system. Thus there are a.s. altogether at least n

served requests in the modified system at time t , and henceall processors are busy a.s. at time t . In the minimal steadystate given by Loynes’ construction thus all processors arebusy a.s., and hence each of the unstable queues j ≤ i re-ceives at least the mean multiple (n − ∑m

h=i+1 �h)/i > �i

of the capacity of one processor in contradiction to (3.13)applied to queue i. Therefore queue i is stable in the modi-fied and thus in the original model, too.

Hence the necessary stability conditions (3.31) are alsotight in case of �i < �1.

Choosing i := 1 in Corollary 3.1 provides tight stabilityconditions for the whole system:

Corollary 3.2 Let �1 ≥ �2 ≥ · · · ≥ �m > 0.

(i) Let n < mk. The system is stable if

(n/k)�1 + (�n/k� − n/k)��n/k�

+m∑

j=�n/k�+1

�j < n, (3.33)

and if the system is stable then

�1 < k,

m∑

j=1

�j ≤ n. (3.34)

(ii) Let mk ≤ n. The system is stable if and only if

�1 < k. (3.35)

Remark 3.1 Corollary 3.2 covers well-known stability re-sults:

(i) For mk ≤ n, which corresponds to a system of m

Gi/Gi/k/∞ queues, Corollary 3.2(ii) yields that the m

queues are simultaneously stable if and only if �i < k

for i ∈ {1, . . . ,m}.In particular, for m = 1 it follows that the G/G/k/∞system is stable if and only if �1 < k.

(ii) For k ≥ n the n processors are all busy if at least n

requests are in the system, and from Corollary 3.2 weconclude that

∑mj=1 �j < n implies the system stabil-

ity and that for a stable system it holds �j < k forj ∈ {1, . . . ,m} and

∑mj=1 �j ≤ n.

In particular, in case of k = n = 1 and m > 1 we have asingle-server HOL-PS system, where

∑mj=1 �j < 1 im-

plies the system stability and where for a stable systemit holds

∑mj=1 �j ≤ 1.

Example 3.2 Let m = 3, n = 2, k = 1 and �1 ≥ �2 = 0.7,�3 = 0.5. Then the sufficient stability condition (3.33) reads�1 < 0.75, and the necessary stability conditions (3.34) read�1 ≤ 0.8. Therefore the system is stable if �1 < 0.75 andunstable if �1 > 0.8.

Simulations of the HOL-PS system with Poisson arrivalstreams of intensity �i , i = 1,2,3, and exponential servicetimes with mean 1 provide the approximate necessary andsufficient stability condition �1 < 0.76 in the special case ofthe Markov model with equal mean service times.

Consider the modified k-limited HOL-PS system wherein each of the first l < m queues there are k permanent cus-tomers, i.e., where in each of the first l queues there are k

requests with infinite service requirement. In this case a rea-sonable definition of stability for the whole system would bethe stability of the remaining m − l queues, cf. [3].

We model the case of permanent customers as describedabove by choosing the offered loads for the first l queuessufficiently large. Thus we may assume that �1 ≥ · · · ≥ �l >

�l+1 ≥ · · · ≥ �m > 0. Choosing now i := l + 1 in Corol-lary 3.1 provides tight stability conditions for the wholemulti-queue multi-server k-limited HOL-PS system withpermanent customers, generalizing a corresponding resultfor the ordinary multi-queue single-server HOL-PS systemwith permanent customers given in [3], Theorem 3.5(iv):

Corollary 3.3 Consider the modified processor sharing sys-tem where in each of the first l queues there are k permanentcustomers and 0 < l < m. Let �l+1 ≥ �l+2 ≥ · · · ≥ �m > 0.

(i) Let (l + 1)k < n < mk. The system is stable if

(n/k)�l+1 + (�n/k� − n/k)��n/k�

+m∑

j=�n/k�+1

�j < n, (3.36)

and if the system is stable then

�l+1 < k, l�l+1 +m∑

j=l+1

�j < n. (3.37)

(ii) Let n ≤ (l + 1)k or mk ≤ n. The system is stable if andonly if (3.37) holds.

References

1. Boxma, O.J., Groenendijk, W.P.: Waiting times in discrete-time cyclic-service systems. IEEE Trans. Commun. 36, 164–170(1988)

8 Queueing Syst (2007) 56: 1–8

2. Brandt, A., Brandt, M.: On the sojourn times for many-queuehead-of-the-line processor-sharing systems with permanent cus-tomers. Math. Methods Oper. Res. 47, 181–220 (1998)

3. Brandt, A., Brandt, M.: A note on the stability of the many-queuehead-of-the-line processor-sharing system with permanent cus-tomers. Queueing Syst. 32, 363–381 (1999)

4. Brandt, A., Brandt, M.: A sample path relation for the sojourntimes in G/G/1-PS systems and its applications. Queueing Syst.52, 281–286 (2006)

5. Brandt, A., Franken, P., Lisek, B.: Stationary Stochastic Models.Akademie-Verlag/Wiley, Berlin/Chichester (1990)

6. Coffman, E.G., Muntz, R.R., Trotter, H.: Waiting time distribu-tions for processor-sharing systems. J. Assoc. Comput. Mach. 17,123–130 (1970)

7. Cohen, J.W.: The multiple phase service network with generalizedprocessor sharing. Acta Inf. 12, 245–284 (1979)

8. Doshi, B.T., Rege, K.M.: Analysis of a multistage queue. AT&TBell Lab. Tech. J. 64, 369–390 (1985)

9. Fendick, K.W., Rodrigues, M.A.: A heavy-traffic comparison ofshared and segregated buffer schemes for queues with the head-of-line processor-sharing discipline. Queueing Syst. 9, 163–190(1991)

10. Franken, P., König, D., Arndt, U., Schmidt, V.: Queues and PointProcesses. Akademie-Verlag/Wiley, Berlin/Chichester (1982)

11. Guillemin, F., Robert, P., Zwart, B.: Tail asymptotics forprocessor-sharing queues. Adv. Appl. Probab. 36, 525–543 (2004)

12. Hooghiemstra, G., Keane, M., van de Ree, S.: Power series for sta-tionary distributions of coupled processor models. SIAM J. Appl.Math. 48, 1159–1166 (1988)

13. Konheim, A.G., Meilijson, I., Melkman, A.: Processor-sharing oftwo parallel lines. J. Appl. Probab. 18, 952–956 (1981)

14. Leung, K.K.: Performance analysis of a processor sharing policywith interactive and background jobs. IFIP Trans. C-5, 189–207(1991)

15. Loynes, R.M.: The stability of a queue with nonindependent inter-arrival and service times. Proc. Camb. Philos. Soc. 58, 497–520(1962)

16. Morrison, J.A.: Diffusion approximation for head-of-the-lineprocessor sharing for two parallel queues. SIAM J. Appl. Math.53, 471–490 (1993)

17. Morrison, J.A.: Head of the line processor sharing for many sym-metric queues with finite capacity. Queueing Syst. 14, 215–237(1993)

18. Núñez-Queija, R.: Sojourn times in a processor sharing queue withservice interruptions. Queueing Syst. 34, 351–386 (2000)

19. Ott, T.J.: The sojourn-time distribution in the M/G/1 queue withprocessor sharing. J. Appl. Probab. 21, 360–378 (1984)

20. Ramaswami, V.: The sojourn time in the GI/M/1 queue withprocessor sharing. J. Appl. Probab. 21, 437–442 (1984)

21. Rege, K.M., Sengupta, B.: Sojourn time distribution in a multipro-grammed computer system. AT&T Bell Lab. Tech. J. 64, 1077–1090 (1985)

22. Sericola, B., Guillemin, F., Boyer, J.: Sojourn times in theM/PH/1 processor sharing queue. Queueing Syst. 50, 109–130(2005)

23. van Uitert, M., Borst, S.C.: A reduced-load equivalence for gen-eralised processor sharing networks with long-tailed input flows.Queueing Syst. 41, 123–163 (2002)

24. Yashkov, S.F.: Mathematical problems in the theory of shared-processor systems. Itogi Nauki Tekh., Ser. Teor. Veroyatn., Mat.Stat., Teor. Kibern. 29, 3–82 (1990)