Discrete time tandem networks of queues

Performance Evaluation 47 (2002) 73–104

Discrete time tandem networks of queues Effects of different regulation schemes for simultaneous events Bernadette Desert, Hans Daduna∗ Universität Hamburg, Fachbereich Mathematik, Bundesstr. 55, D-20146 Hamburg, Germany Received 28 June 1999; received in revised form 11 June 2001

Abstract We consider tandem networks of discrete time Bernoulli servers with state dependent service rates and a state dependent Bernoulli arrival stream at the first node of the tandem. We investigate the effects of different regulation schemes for simultaneous events (e.g. joint arrivals and departures at some node, or joint departures at different nodes) on the performance behaviour of the network. The most serious effects occur with respect to arrival theorems which describe the distribution of the other customers present in the network and seen by an arriving customer, or observed by a customer during a jump inside the network. We prove necessary and sufficient conditions on the regulation scheme for a customer to observe always the time stationary behaviour of the network during his jumps. It turns out that we have to distinguish between local and global control for the regulation of simultaneous jumps. For different arrival schemes we compute the joint sojourn time distribution for a customer traversing the tandem. As a consequence we identify from this a regulation scheme which is known in the literature, where Little’s formula cannot be applied directly. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Early arrivals; Late arrivals; Departure first; Arrival first; State dependent arrivals; State dependent service; Arrival theorem; Sojourn time distribution; Little’s formula; Simultaneous events; Product form distribution

1. Introduction Networks of queues in discrete time have recently found much interest in the literature. The construction of discrete time queues in performance analysis of computer and communications systems dates back to the sixties, modelling time-shared computer systems (see e.g. [17]) and ALOHA network protocols. An overview of properties of single node systems is already presented in the book of Hunter [16]. Woodward [34] describes how satellite networks and local area networks can be modelled by using discrete time queueing networks, see [34, Chapters 6 and 7]. More recent results in discrete time system theory appeared in the books of Takagi [29] (on single node systems), and Bruneel and Kim [2]. Chapter 12 of the book of Chao et al. [4] and the lecture notes [8] are particularly devoted to discrete time network theory. A broad picture of the field with an emphasis on ∗

Corresponding author. Tel.: +49-40-428-384924; fax: +49-40-428-384930. E-mail address: [email protected] (H. Daduna). 0166-5316/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 5 3 1 6 ( 0 1 ) 0 0 0 5 7 - 8

74

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

single node systems can be found in the special issues of Performance Evaluation (Vol. 21 (1), 1994) and Queueing Systems and Their Applications (Vol. 18, 1994), and in the recent survey paper of Wittevrongel and Bruneel [32]. The revival of these models emerged from the introduction of ATM protocols as the multiplexing technique for Broadband Integrated Services Digital Networks (B-ISDN). ATM switches are often modelled as discrete time queueing systems because the switches are assumed to work on a common discrete time scale; for a discussion of the modelling principles, see in [2] the introduction to Chapter 4 and Section 4.1. Generic discrete time scales occur also in customer supply chain modelling: Tempelmeier [30] argues for using discrete time dynamic models because of the periodic review and order delivering policies. The most common time unit is 1 day. A specific feature that makes the analysis of discrete time networks technically more involved than its continuous time counterparts is the occurrence of simultaneous events: joint arrivals and departures at a specific node as well as multiple departures from different nodes at the same time instants, resulting possibly in concurrent arrivals at successor nodes, etc. The regulation of these simultaneous events in queueing systems (arrivals and departures) was already studied in depth with respect to steady-state behaviour by Hunter in the case of single node systems. The regulation schemes usually reflect physical properties of the described systems which are imposed on the running system by the protocols governing the system’s behaviour. In the present paper we study some of these regulation schemes and the effects which are consequences of working under different schemes. Most of these questions were already raised and investigated, e.g. by Hunter [16], Gravey and Hebuterne [13], and Pujolle et al. [25]. In Section 2, we reconsider three different regulation schemes and analyse their influence on single node systems: early arrivals (EA), late arrivals with departures first (LA-DF), late arrivals with arrivals first (LA-AF). Seemingly these are the most prominent regulation schemes for simultaneous events occurring in the literature. Studying the simple single node systems in detail enables us to clarify the special effects due to different regulation schemes before attacking the problem for networks of such nodes. We determine the steady-state behaviour and investigate the system’s behaviour from an arriving customer’s point of view by proving the appropriated arrival theorems. These arrival theorems determine the stationary and asymptotic distribution of the systems when the observation points are prescribed by an associated (embedded) point process. It turns out that the choice of specific regulation schemes is essential to obtain analogues of the properties found in continuous time systems. (For details, see the introductory remarks of Section 2.4.) This is similar to the findings of Gravey and Hebuterne [13]. Arrival distributions are the starting points to obtain customer-oriented performance measures such as passage time or waiting time distributions. Having these quantities at hand we find that in the EA system Little’s formula cannot be applied directly as it was done sometimes. (This is surprising because the systems under consideration seem to fit easily into the range of Little’s formula as proved, e.g. by Stidham [27] and do not show the structure of the usual counter-examples in connection with Little’s formulas.) It follows that comparisons of mean waiting times or mean sojourn times in systems with different regulation schemes cannot be performed directly by only computing mean queue lengths (see Pujolle et al. [25] for such a comparison). Having identified the problems connected with regulation schemes in Section 2, Section 3 is devoted to linear networks of single server nodes; we show to what extent the results from Section 2 generalise to an arbitrary number of queues in tandem. The delicate observation is: different schemes may be

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

75

applied in a network to the different nodes. Changing, e.g. from LA-DF to LA-AF inside the network and vice versa will not affect the steady-state behaviour, but the behaviour of the network seen by a jumping customer. Both classes of equilibrium distributions can be computed explicitly, even if we allow state dependent arrival streams and service probabilities. The control of the external arrival stream is a closed loop control where the decision variable is the total population size of the network. The service control is determined via the nodes’ local queue lengths. The steady states and the arrival probabilities factorise into terms concerning the arrivals and terms concerning the service behaviour. In the case of state independent systems we find classical product form probabilities similar to the networks’ continuous time counterparts. As far as the tandem’s overall behaviour is concerned, we have to distinguish between local and global control for the regulation of all the events occurring simultaneously in the tandem. Again we find important consequences of the chosen control with respect to arrival theorems and their generalisations to statements about what a jumping customer observes in the network. The question when an arriving or jumping customer will always observe the system in the time stationary equilibrium as well is answered in a sense that generalises the single node arrival theorem of Gravey and Hebuterne [13] to the case of vector-valued network processes. Loosely speaking, we need a global regulation scheme which ensures that a jumping customer finishes his jump before customers in front of him will move. Remarkably enough, this is even an if and only if condition. We close this section by computing stationary and asymptotic joint sojourn time distributions for customers traversing the tandem. A direct consequence is that we have at hand explicitly the end-to-end-delay for customers traversing the line. In Section 4 we reconsider a local access control to the tandem introduced by Pujolle et al. [25]. The computation of the steady state for such a tandem would enable us to compare different control schemes regulating the access to the network: either on the basis of global information as in Section 3 or on the basis of local information available at the entrance node. Unfortunately, at present we are not able to solve the equilibrium problem for such local control. The networks considered in this paper are operating on a discrete time scale Nθ = {0, θ, 2θ, . . . , } where θ > 0 is a prescribed fundamental time quantum, in ATM networks usually the time for transmission of one cell. For a simplified presentation we always assume θ = 1, i.e. the time scale is N = {0, 1, 2, . . . }. We use N+ = {1, 2, 3, . . . }, and for a, b ∈ R the complementary Kronecker-delta is
1 if a = b, η(a, b) = 0 if a = b.

2. Single server FCFS queues with infinite waiting room 2.1. Description of the system Let us first consider a single server node under FCFS queueing regime with unlimited waiting room. We assume that customers of a single type arrive at the node according to a state dependent Bernoulli process and have their service commenced in the order of their arrival. The time evolution of the system is described by a discrete time stochastic process X = (X(t) : t ∈ N) with state space N; X(t) indicates the number of customers in the queue at time t, including the customer being served. If at time t there are n customers present, n ≥ 0, an arrival occurs with probability

76

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

b(n) ∈ (0, 1); define c(n) = 1 − b(n). A service ends in the time segment [t, t + 1) with probability p(n) to be specified below; with probability q(n) = 1 − p(n) the customer in service will require at least one more unit of service time. It is assumed that a customer spends at least one time unit in the system. Suppose further that the decisions for customers’ service expiring or not and the arrival decisions depend on the total history of the system only through the actual state, and that at any fixed time instant, such simultaneous decisions are conditionally independent given the state of the system. From the statistical assumptions put on the system, the queue length process X is a discrete time, homogeneous Markov chain with state space N. Due to the single arrival property of the Bernoulli process and the single service channel structure of the system, X moves at most one step up or down in a time slot, i.e. X is a general random walk in the sense of Kemeny et al. [19, p. 84], or a birth and death chain in discrete time, see Hunter [16, Vol. I, p. 178]. For ease of notation, we fix henceforth a probability space (Ω, A, P ) and assume (unless explicitly specified otherwise) all the random variables which occur to be defined on this common space. (Sometimes we shall further specify the underlying probability law.) Before proceeding with the analysis of X, we must make further assumptions about the order in which arrivals and departures take place and specify whether these events occur at the beginning or at the end of a time slot. A description of specific regulation schemes is part of most research works on discrete time queues. For single server systems in isolation an investigation of different schemes was performed especially by Gravey and Hebuterne [13] and Hunter [16, Vol. II, Section 9.2]. We do not intend to give a complete review of the relevant literature on single node systems but to introduce for the reader’s convenience mainly those schemes that seem to be of interest to describe network movement protocols. Seemingly the first relevant paper in that direction was written by Pujolle et al. [25]; for tandem networks of single server systems they performed a comparison of the steady-state behaviour under different regulation schemes. We shall revisit this topic in Section 3. To describe the movements of the customers and the resulting state changes of the state process X, it is most convenient to think of the system as evolving in continuous time R+ = [0, ∞) and to embed the jump instants at points around the time points N = {0, 1, 2, . . . }. For the reader’s convenience, we define for t ∈ N : t −− < t − < t < t + , the difference between two consecutive points being “infinitesimal”. We always observe and record the state of the system at times t ∈ N.

2.2. Regulation schemes 2.2.1. “Early arrival” (EA) system An arrival occurs at the beginning of a time slot in (t, t + ]; a service is completed at the end of a time slot in [(t + 1)− , t + 1), t ∈ N. The situation is illustrated in Fig. 1. If an arriving customer finds the system empty, his service commences immediately and may be completed at the end of the same time slot with probability p(0) = 0. An arrival in (t, t + ] does not influence the service rate during the interval [t, t + 1).

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

77

Fig. 1. Early arrival.

If X(t) = 0, the system remains in state 0 at time t + 1 if no arrival occurs or if a customer arrives in (t, t + ] and departs from the system in [(t + 1)− , t + 1). The state of the system at time t + 1 is 1 if a customer arrives and is not served immediately. Under the assumptions stated above, we obtain:  0 with probability c(0) + b(0)p(0), X(t + 1) = 1 with probability b(0)q(0). If X(t) = n ≥ 1, then    n − 1 with probability c(n)p(n), with probability b(n)p(n) + c(n)q(n), X(t + 1) = n   n + 1 with probability b(n)q(n). In the following cases, arriving customer cannot enter service until beginning of the next time slot. 2.2.2. Late arrival-departure first (LA-DF) system The events take place at the end of a time slot. An arrival occurs in [t − , t) and a service is completed in [t −− , t − ), t ∈ N; therefore a departure always takes place before an arrival. The situation is pictured in Fig. 2. In view of this regulation scheme, an arriving customer who finds the system empty cannot leave the node before the end of the next time slot; it follows that p(0) = 0. An arrival in [t − , t) does not influence the service rate until the beginning of the interval [t, t + 1). If X(t) = 0, the system remains in state 0 at time t + 1 if no arrival occurs. The state of the system at time t + 1 is 1 if a customer arrives in [(t + 1)− , t + 1). Therefore
0 with probability c(0), X(t + 1) = 1 with probability b(0).

Fig. 2. Late arrival-departure first (LA-DF).

78

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

Fig. 3. Late arrival-arrival first (LA-AF).

If X(t) = n ≥ 1, then the probabilistic behaviour of the system does not differ from that of an EA node any longer, and    n − 1 with probability c(n)p(n), with probability b(n)p(n) + c(n)q(n), X(t + 1) = n   n + 1 with probability b(n)q(n). 2.2.3. Late arrival-arrival first (LA-AF) system As in an LA-DF system, the events take place at the end of a time slot, but their order is interchanged. An arrival occurs in [t −− , t − ) and a service is completed in [t − , t), t ∈ N (see Fig. 3). An arriving customer who finds the node empty in [t −− , t − ) cannot depart from the system immediately in [t − , t), as his service is assumed to take at least one time unit. Again it follows that p(0) = 0. A customer who arrives to find n ≥ 1 people at the node will see with probability p(n) a customer leaving the system immediately after his arrival. Nevertheless, the order in which the events take place just before the beginning of a time slot does not influence the system states which are measured at times t ∈ N. Therefore the transition probabilities remain the same as in the LA-DF case. Remark. If we observe and record the state of the system at different time points, say t − instead of t, the regulation schemes described above may change. For example, the initial LA-DF system becomes an EA system, as a customer who arrives in [t − , t) and finds the system empty may be served within the time interval [t − , (t + 1)− ). The described three regulation schemes have particularly aroused our interest. There are several further definitions in the literature around the LA, EA, DF, AF properties combined with different specified observation points for the process. See, e.g. the model of Chaudhry and Gupta [3], who consider a batch arrival system under different regulation schemes. Their EA system coincides with our definition, the observation point t is described as “just before a potential” arrival, and the outside observer’s time instant is t + 21 . Apart from a slightly shifted modification, their LA system is similar to our LA-AF, with arrivals at t − , departures at t + , and observation point t −− “just before a potential” arrival, while the outside observer’s view at t + 21 is then the same as what we see at time t. Miyazawa and Takahashi [23] (Example 3.1) consider (in our terms) the LA-DF queue with batch arrivals, the queue length is counted at time t; additionally they distinguish observation points just after the departure and just after the arrival, if they occur. (See Section 2.4 below for our parallel investigation.) The scheduling of the inventory in [30] is performed by using an EA regime as described above.

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

79

Unless otherwise specified we henceforth assume p(n) ∈ (0, 1] for n > 0, and in the EA case p(0) ∈ (0, 1] as well. It follows that all the Markov chains considered in the following are irreducible and aperiodic. We shall use this without further mentioning. 2.3. Stationary distributions As pointed out at the end of Section 2.1 the state process X is a birth and death chain. We have seen that different regulation schemes result in different transition laws at the boundary of the state space. The results of the following theorem and corollary are therefore simple consequences of the limiting and stationary behaviour of birth and death chains (see Hunter [16, Vol. II, Example 7.2.2, p. 107]). Theorem 2.1. Let X = (X(t) : t ∈ N) with state space N denote the queue length process of the state dependent Bernoulli server under the EA system described above. X is ergodic if and only if N

EA

=

m ∞   b(i − 1)q(i − 1) m=0 i=1

c(i)p(i)

< ∞.

If X is ergodic, then its unique stationary and limiting distribution π EA (n : n ∈ N) is π EA (n) =

n  b(i − 1)q(i − 1) i=1

c(i)p(i)

[N EA ]−1 .

In the LA-DF and LA-AF cases we have q(0) = 1. Therefore n  q(i) LA −1 b(i − 1) n−1 LA LA-DF LA-AF i=1 (n) = π (n) = π (n) := π [N ] . n c(i) i=1 p(i) i=1 Remark. As the general form of arrival and service probabilities is allowed, it is in general not possible to give closed form solutions of the steady-state probabilities (even if these exist). This parallels the birth and death chain theory. Even the determination of recurrence and transience has to refer to the concrete given data of the system under consideration. For details, see Hunter [16, Vol. II, Example 7.2.2, p. 107]. An example with explicitly given norming constant is provided by systems with state independent arrival and service probabilities. This yields a (homogeneous) random walk with reflection at 0, the different regulation schemes take the form of different boundary behaviours of that random walk. Corollary 2.1. When the service and arrival probabilities p(n) = p, b(n) = b, n ∈ N, are state independent, then X is ergodic if and only if b/p < 1. If this condition is satisfied, we obtain

n

η(0,n) n

  bq bq b bq 1 EA LA π (n) = 1− 1− and π (n) = . cp cp q cp p Interarrival and service times are then geometrically distributed on N+ with parameters b and p, respectively.

80

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

For a better understanding of the difference between LA-DF and LA-AF, consider a loss system with a limited waiting room which can contain at most L customers, including the one in service. Then the probability that an arriving customer will be denied service is greater in the AF case than in the DF case, as a customer who sees L people in front of him is turned away, even if a service is completed immediately after his arrival and (L − 1) customers remain in the system at the beginning of the next time slot. Computing the mean number of rejected customers per time unit yields π(L)b(L) in the AF case and π(L)b(L)q(L) in the DF case. Although the stationary distributions obtained do not distinguish an LA-DF system from an LA-AF system, the order in which the events take place influences the perspective of an arriving customer and consequently, the arrival distribution. 2.4. Arrival theorems and PASTA analogue It is well known that continuous time systems with Poisson arrivals satisfy the PASTA property (Poisson arrivals see time averages) [33]: time stationary quantities are customer stationary quantities, as well. From a general point of view the PASTA theorem and its successors and relatives determine the stationary and asymptotic distribution of systems when the observation points are prescribed by an associated (embedded) point process. Comparing the stationary distribution of the embedded state process with the time stationary distribution of the systems (seen by an outside observer) is the topic of many research activities in queueing theory. Wolff’s PASTA theorem [33] opened the research into this field. Since then many generalisations of that property have been proved and popularised under names like ASTA, EPSTA, MUSTA. For a review see [1, Chapter 4, Section 3], or [18], and [11, Chapters 3 and 4]. Using point process terminology, EPSTA and its relatives are concerned with properties of special Palm measures of stationary point processes: Papangelou’s formula, which connects the point process intensity and time stationary expectations with Palm stationary expectations, allows the derivation of a variety of formulas summarised under the title job observer properties in single node systems as well as in networks of queues. For continuous time, see, e.g. [1, Example 3.2.2], [21, Example 3], [11, Section 4.3], and [10]. Starting from Palm theory in discrete time ([1, Chapter 1, Section 7.4]) a similar development is possible. Palm measures in discrete time are expressed as elementary conditional probabilities. This makes the theory more elementary although explicit computations are tedious, see Sections 3.4 and 3.5. In discrete time, however, a PASTA analogue usually does not hold, although exceptions can be found under certain conditions. An early result was proved by Halfin in [14]. Characterisation theorems of the PASTA type (thereby strengthening the BASTA — Bernoulli arrivals see time averages — results from [22]) were proved by El-Taha and Stidham [12] (see also [11, Section 2, Theorem 3.18 and Corollary 3.19]). Miyazawa and Takahashi [23] proved ASTA in a discrete time point process setting by using a rate conservation principle. They also observed that for some natural systems this property does not hold. We therefore present the “arrival theorems” for the different cases. These describe the individual customers’ behaviour when they enter the system and provide us with the necessary basic information to compute customer-oriented performance measures. The proofs can be performed either by applying the general theory or by directly evaluating elementary conditional probabilities, i.e. Palm probabilities in discrete time processes. Assume in each case that the Markov process which describes the time evolution of the system is stationary.

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

81

2.4.1. EA system Theorem 2.2. Let A(t) denote the event that at time t + an arrival occurs. Then in equilibrium we have ∞ m

−1 n−1 n   b(i) m−1 q(i) q(i) b(i) i=1 i=1 , π1EA (n) = P [X(t) = n|A(t)] = n m c(i) p(i) c(i) i=1 i=1 p(i) i=1 m=0 i=1 where π1EA (n : n ∈ N) is the probability that an arriving customer finds n other customers in front of him in service or waiting. For a state independent system, the following result holds:

n

 bq bq π1EA (n) = = π EA (n), n ∈ N. 1− cp cp 2.4.2. LA-DF system Theorem 2.3. Let B(t) = {At time t − a customer enters the node}. Then for n ∈ N we have ∞ m

−1 n  q(h) nh=0 b(h)  q(h) m h=0 b(h) LA-DF π1 (n) = P [X(t) = n + 1|B(t)] = . p(h) n+1 p(h) m+1 h=1 c(h) m=0h=1 h=1 c(h) h=1 If the system is state independent, we obtain for all n ∈ N

n

 bq bq LA-DF π1 (n) = 1− = π LA (n). cp cp 2.4.3. LA-AF system Theorem 2.4. Denote by C(t) the event that at time t −− an arrival occurs. Then in equilibrium we have for all n ∈ N ∞ m

−1 n   b(h) m−1 q(h) b(h) n−1 h=1 q(h) h=1 LA-AF −− n m π1 (n) = P [X(t ) = n + 1|C(t)] = . c(h) p(h) c(h) p(h) h=1 h=1 h=1 m=0h=1 For a state independent system, the following result holds: 

η(0,n) n

bq b 1 LA-AF π1 1− = π LA (n). (n) = q cp p 2.4.4. Order of events and PASTA analogue If the systems are state independent, any customer arriving at an EA or LA-AF node sees the other customers distributed according to the steady-state distribution π EA and π LA-AF , respectively. Thus, we obtain a PASTA analogue, which strongly suggests that the order of the events occurring during a time slot plays a crucial role. In both cases, an arrival always takes place before a departure.

82

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

This confirms the results of Gravey and Hebuterne [13] who draw a comparison between an AF and a DF system and come to the conclusion that an arriving customer finds the system in equilibrium if an arrival always takes place before a departure, and if the sequence of arrivals after time t is independent of the system state up to t — a property which is referred to as the “independence condition”. To prove this result, they assume that an outside observer records the state of the system at time (i − 21 ), i ∈ N+ , and that an arrival occurs at time i. In the AF case, the arriving customer’s distribution coincides with the outside observer’s distribution. In the DF case, however, a customer who arrives at time i sees as many customers as the outside observer does at time (i − 21 ) if no departure occurs immediately before his arrival; otherwise he finds one less customer in the system. In view of the EA system, it is more convenient for us to observe the system at the beginning of a time slot than to rely on an outside observer’s perspective. The system state at the beginning of a time slot [t, t + 1) coincides with the formal state of the Markov process throughout the time segment [t, t + 1), as the process describing the evolution of the system does not record the events occurring after t until the beginning of the next time slot [t + 1, t + 2). In the state independent systems analysed in this paper, the “independence condition” is obviously satisfied. Steady-state probabilities correspond to the proportion of customers who arrive at the node when the process is in that state. In the EA and LA-AF cases, the state of the process coincides with the number of customers seen by an arrival, so that a PASTA analogue holds: π1 (k) = P [Xt is in state k|an arrival occurs] = P [Xt is in state k] = π(k)

∀k ∈ N.

In an LA-DF system, however, a departure may take place just prior to an arrival, in which case an arriving customer finds (k − 1) other customers although formally, the process is still in state k. Remark. 1. The result still holds for EA and LA-AF systems with state dependent service rates, provided the arrival rate remains state independent. 2. The result remains valid for a state independent system which can contain at most L customers. In the EA and LA-AF cases, the distributions of the customers seen by an arriving customer who either enters the node or departs immediately if the system already contains L customers equal the stationary distributions of the respective systems, which is due, once again, to the order of the events taking place in the systems under consideration. 2.5. Comparison of sojourn time distributions Individual customers’ sojourn time distributions are probably the most important customer-oriented performance measure. Unfortunately, almost nothing seems to be known about these distributions in systems with state dependent service probabilities, even in the single node systems described above. (See 3 in the Remarks below Corollary 2.2.) We therefore restrict our attention to systems with state independent service rates p ∈ (0, 1). Recall that we consider time quanta of unit length for determining the length of time a customer spends in the system. This implies, for instance, that the sojourn time of a customer arriving at time s − and departing at time t − is the same as the sojourn time of a customer arriving at s + and departing at t − .

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

83

As a consequence we observe, e.g. in an EA system a customer arriving to an empty system at least for a duration of one time segment in the server. This holds even if he leaves (with probability p(0)) the server at the end of the time slot in which he arrived and therefore does not occur in the state description of the system. The arrival theorems presented above are now used to compute the generating functions of the sojourn time distributions in the different systems. Theorem 2.5. Assume the Bernoulli server with state dependent arrival rates b(n) ∈ (0, 1), n ∈ N, and state independent service rates p ∈ (0, 1) is in equilibrium. Denote by T EA the sojourn time of a test customer C0 who arrives at time t + at the EA node and finds the other customers distributed according to π1EA . T LA-DF and T LA-AF are defined accordingly. Let α EA and α LA-DF denote the generating functions of the respective arrival distributions. From Theorems 2.2 and 2.3, we have

−1 ∞ m k  k ∞ ∞

   b(h) q m b(h) qx α EA (x) = x k π1EA (k) = , |x| ≤ 1. p c(h) m=0h=1 c(h) p k=0 k=0 h=1 α

LA-DF

∞ 

−1  ∞ ∞

  qx k kh=1 b(h)  q m m h=1 b(h) k LA-DF (x) = x π1 (k) = , k+1 m+1 p p h=1 c(h) m=0 h=1 c(h) k=0 k=0

|x| ≤ 1.

Then the generating functions of T EA , T LA-DF and T LA-AF are given by



  pθ pθ pθ pθ T EA EA T LA-DF LA-DF E[θ ] = α ]= , E[θ α , 1 − qθ 1 − qθ 1 − qθ 1 − qθ

η(0,n)

n+1 ∞  1 pθ T LA-AF LA-AF ]= π1 (n) E[θ . θ 1 − qθ n=0 Proof. Recall that A(t), B(t) and C(t) are events associated with arrivals in the different systems and defined in Theorems 2.2–2.4. Conditioning on these events and on the number of customers seen by C0 at his arrival instant, we obtain the formulas for the EA and LA-DF cases. In the LA-AF case, a customer may leave the system immediately after C0 ’s arrival, in which case n customers remain to be served during C0 ’s sojourn time. If no service is completed just after C0 ’s arrival, (n + 1) must be served. So E[θ

T LA-AF

∞  LA-AF ]= π1LA-AF (n)E[θ T |X(t −− ) = n + 1, C(t)] n=0

= π1LA-AF (0)



pθ 1 − qθ





n n+1 ∞  pθ pθ LA-AF + π1 (n) p +q 1 − qθ 1 − qθ n=1

η(0,n)

n+1 ∞  pθ 1 LA-AF = π1 (n) . θ 1 − qθ n=0




84

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

Corollary 2.2. If the arrival rates are state independent, the sojourn times are geometrically distributed on N+ with the same parameter ((p − b)/c) in all three cases. Remarks. EA LA-DF 1. The differences between E[θ T ] and E[θ T ], which reside in the arrival terms, reduce to an arriving customer’s perspective and can be attributed to the order of the events taking place during a time slot. As soon as C0 joins the queue or enters the service facility, the stochastic behaviour of the two systems does not differ any more. In the LA-AF case, the difference lies in the service rates. If there is at least one customer in the system when C0 arrives, an additional service can be completed during C0 ’s sojourn time at the node. LA-AF The additional factor (1/qθ)η(0,n) which appears in E[θ T ] may be regarded as accelerating C0 ’s passage at the node. A customer arriving to find n ≥ 1 people in the system at time t −− may have (n − 1) predecessors when the next slot begins at time t, which can reduce his sojourn time at the EA node. On the other hand, the arrival terms do not differ from those of E[θ T ] as in both cases, an arrival always takes place before a departure. 2. These differences disappear when the nodes are state independent, the three systems have the same stationary and asymptotic behaviour. During (t + , (t +1)−− ), t ∈ N, the queues contain exactly as many customers in all three cases. Their behaviour differs only during the infinitesimal intervals [t −− , t + ) which are negligible as compared with a time slot and can be viewed as short phases of reorganisation. This result no longer holds if the waiting room’s capacity is finite. In this case, the customers’ mean sojourn time in an LA-AF system becomes smaller, as a customer who sees n ≥ 1 people in the system at his arrival instant may have one less predecessor when his actual sojourn time begins. In other words, the acceleration of a customer’s passage time is connected with a greater loss probability. 3. A more explicit computation of sojourn time distributions in systems with state dependent service probabilities will depend strongly on the form of the service probabilities. An example for continuous time systems is provided by Whitt [31], where customers’ balking and reneging in a multiserver queue is expressed by adjusting the service rates suitably. The main property featured by the model as a special birth and death process is that service rates and waiting times of individual customers do not depend on the behaviour of the other customers who are behind them in the queue or parallel in service. Exploiting this, Whitt develops algorithms for computing (conditional) response time distributions. Applying similar restrictions to the discrete time queues will yield more explicit results as well. But even the standard multiserver queue poses additional difficulties, because the describing Markov queue length process is no longer a birth and death chain: downward jumps may be of a size greater than unity.

2.6. Little’s formula Little’s formula L = λW [20] is usually thought to hold in all queueing systems that occur in practice. The first counter-examples constructed (see, e.g., Stidham [27,28]) show the fundamental structure of the service protocols of most systems where the formula does not apply: the same customer coming and going several times from and to a queue. For in-depth discussion of Little’s Theorem, see, e.g. the books of Baccelli and Bremaud [1] and El-Taha and Stidham [11]. Miyazawa and Takahashi [23, Example 4.1], applied a discrete time rate conservation principle to prove a general form of Little’s formula for batch arrival systems in discrete time.

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

85

The systems considered here at a first glance seem to fit into the range of Little’s Theorem as proved, e.g. in the references cited. Thus using Little’s formula, Pujolle et al. [25] conclude that a customer spends on average less time in an EA system than in an LA system, which contradicts the result of Corollary 2.2 stating that in both cases, the sojourn times are geometrically distributed on N+ with the same parameter ((p − b)/c), which implies that E[T ] = c/(p − b). The resolution of this contradiction: for EA systems the application of Little’s formula requires a careful interpretation of the quantities involved. (Similar subtleties were found when starting from a general Little’s formula for a black-box system and considering special internal service regimes other than FCFS; see [11, Sections 6.2 and 6.3.2].) We observe in general that if the service rates are state dependent, the customers’ mean residence time cannot be computed directly from the mean queue length. The question remains whether or not such a computation is possible in the case of a state independent queueing system with infinite waiting room under the different regulation schemes considered in this paper. In detail: let L denote the mean queue length and W be the mean sojourn time. Self-explanatory superscripts are assigned to these expected values in the following. In the EA case we have LEA =

bq . p−b

It follows: q LEA c = E[T EA ] = W EA = = , p−b b p−b which contradicts the result of Corollary 2.2. Computing the expected queue length in the LA case yields LLA =

bc , p−b

which implies LLA c = E[T LA ] = W LA = . p−b b Therefore Little’s formula can be directly applied to a discrete time queueing system in the LA case only. At the beginning of a time slot, the state of the process is k if the queue really contains k customers. The customers arrive at time t − or t −− and are recorded at time t. This guarantees the correct coincidence of physical states and describing process at time t. In the EA case, however, the state of the process can be k in [t, t + 1) although the physical queue length of the system is k + 1 in (t + , t + 1). Thus, the first time slot a customer spends in the system is not counted. An arriving customer who finds the system empty and is served within one time slot is not counted at all; the process remains in state 0. The difference between the two expected values W LA and W EA determined above corresponds to the time slot which is not counted in the EA case: W LA − W EA = 1. In the EA case, the physical queue length L˜ EA can be observed at time t + 21 , which leads to L˜ EA :=

∞  k[cπ EA (k) + bπ EA (k − 1)] = k=1

bc = LLA . p−b

86

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

Remark. Similar problems arise in networks in discrete time including nodes with EA regulation of simultaneous events.

3. Tandems of single server nodes 3.1. Description of the system We now consider an open tandem of J single server nodes, J ≥ 2, with infinite waiting rooms. We further assume that the customers arrive at the first node according to a state dependent Bernoulli process and are served there and at each subsequent node on a FCFS basis. A customer leaving node j , j = 1, . . . , J − 1, immediately enters node j + 1 and departs from the system when he has passed through all queues. The evolution of the network is described by a discrete time stochastic process X = (X(t) : t ∈ N) with state space NJ ; the joint queue length vector X(t) = (X1 (t), . . . , XJ (t)) = (x1 , . . . , xJ ) ∈ NJ indicates the number of customers present at each node at time t, t ∈ N. The arrival probabilities depend on the total number of customers in the network, while the service rates at a given node depend on the number of customers present at that node at the beginning of the corresponding time slot. Thus, if at node j there are xj customers present, j = 1, . . . , J , an arrival occurs with probability b(x1 + · · · + xJ ) ∈ (0, 1); at node j a service ends with probability pj (xj ). Define c(x1 + · · · + xJ ) = 1 − b(x1 + · · · + xJ ) and qj (xj ) = 1 − pj (xj ). The set of all decisions for customers’ service expiring or not and the arrival decisions of the Bernoulli process depend on the total history of the network only through the actual state, and for a fixed time instant such simultaneous decisions are conditionally independent, given the actual state of the system. It follows from these assumptions that the process X is a discrete time, homogeneous, irreducible and aperiodic Markov chain with state space NJ . Therefore X can be considered as a multivariate birth and death process (or compound birth–death and migration process, in continuous time such processes are investigated, e.g. by Serfozo [26] and Daduna and Szekli [9]). With respect to network processes large classes of multidimensional birth and death processes — even with concurrent movements of customers — have been investigated during the last years; Chao et al.’s book [4] surveys this research mainly for the continuous time case. The methods and results carry over to the discrete time case as well (see Section 12 of [4]). This is due to the structure of the transition laws for the systems. But because of our overall dependence of the arrival probability on the state of the network it seems to be difficult to derive, e.g. the stationary distribution (see Theorem 3.1) from the general results given there. In general terms we shall compute in this section the steady-state distribution for multivariate random walks on the positive orthant NJ with drift towards the origin. While in the interior of the orthant there are similarities common to the process transition behaviour originating from the different regulation schemes, these schemes generate different types of behaviour on the boundary. Note that in general the steady states of compound birth and death processes and migration processes are not known. Theorem 3.2 contributes to this problem by providing an explicit product form solution for a system with practical relevance. See Section 4, e.g., where we did not succeed in computing the steady state of a multidimensional random walk. There the transition probabilities at the boundaries are not compatible with the transition behaviour inside the state space in a way to allow product form expressions.

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

87

Fig. 4. EA at the first node, LA at the subsequent nodes.

The node types EA, LA-DF and LA-AF defined in Section 1 can be arbitrarily combined to form an open tandem of queues. In this section we shall essentially investigate the following three cases: 1. LA-DF at each node. An arrival at the first node occurs at time t − , t ∈ N, a service at an arbitrary node is completed at time t −− . As a customer departing from a node at time t −− joins the next queue at time t − , a departure takes place before an (internal) arrival at each node. 2. EA at the first node, LA at the subsequent nodes (see Fig. 4). An arrival at the first node occurs at the beginning of a slot at time t + , t ∈ N; a service at an arbitrary node is completed at the end of a time slot. At the first node, therefore, an arrival always takes place before a departure. An arriving customer who finds the first node empty can be served within one time slot with probability p1 (0) = 0. If the rest of the network is composed of LA-DF nodes, a customer leaving node j ≤ J − 1 at time t −− , arrives at node (j + 1) at time t − , just after a possible departure from node j + 1. Nevertheless, we can also construct a tandem consisting of an EA node and (J − 1) LA-AF nodes. To this end, let us define infinitesimal time points t J − < t (J −1)− < · · · < t 2− < t − as follows: 2.1. A service at the first node may end in (t J − , t (J −1)− ). 2.2. For all j = 1, . . . , J − 2, a customer may enter node j + 1 at time t (J −j )− ; another customer may depart from that node in (t (J −j )− , t (J −j −1)− ). 2.3. An arrival may occur at node J at time t − ; a customer may leave the system in (t − , t). 2.4. Thus, a customer entering node j , j = 2, . . . , J , can see another customer departing from that node immediately after his arrival, provided node j is not empty.

88

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

3. LA-AF at each node. The events take place at the end of a time slot. An arrival at the first node occurs at time t J − ; the infinitesimal time points t J − < t (J −1)− < · · · < t 2− < t − are defined as in the preceding case to regulate the subsequent customer movements. At each node, therefore, an arrival occurs before a departure. For notational ease, the above described networks are referred to in the following as “LA-DF systems”, “EA systems” and “LA-AF systems”. Note that in the case of LA-DF and EA with LA-DF the synchronisation of joint arrivals and departures is performed on the basis of local information only, while in the other two cases a global synchronisation of the customers’ movements is necessary. This is formally dictated by the successive time instants t j − which constitute an artificial device to model the global synchronisation. We shall investigate later on the structural implications of local and global synchronisations for the individual customers’ behaviour. 3.2. Stationary distributions As the order of the events taking place immediately before the beginning of a time slot [t, t + 1) does not influence the system states which are always measured at time t, t ∈ N, the following results hold for networks consisting either of LA-DF or of LA-AF nodes with an infinite number of waiting places. As far as steady states are concerned, these networks will therefore be referred to as LA systems or LA networks. Theorem 3.1 (Daduna [6]). Let X = (X(t) : t ∈ N) denote the Markov chain describing the time evolution of an LA tandem. X is ergodic if and only if x1 +···+xJ −1 J xj −1  b(h)  h=1 qj (h) h=0 LA N (J ) := < ∞. x1 +···+xJ xj c(h) j =1 h=1 pj (h) h=1 (x ,...,x )∈NJ 1

J

If X is ergodic, its unique stationary and limiting distribution is given by x1 +···+xJ −1 J xj −1 b(h)  h=1 qj (h) LA h=0 LA π (x1 , . . . , xJ ) = x1 +···+xJ [N (J )]−1 ∀(x1 , . . . , xJ ) ∈ NJ . xj c(h) p (h) h=1 h=1 j j =1 Let us now examine the EA system described above. It turns out that the equilibrium distribution of such a network can be derived from the result given in Theorem 3.1. Theorem 3.2. Let X = (X(t) : t ∈ N) denote the Markov chain describing the time evolution of the EA network. For all (y1 , . . . , yJ ) ∈ NJ , the system of steady-state equations for X is solved by x1 +···+xJ −1 1 −1 J xj −1 b(h) xh=0 qj (h) EA −1 q1 (h)  h=1 h=0 EA x1 π (x1 , . . . , xJ ) = x1 +···+xJ N (J ) xj p (h) c(h) p (h) j h=1 1 h=1 h=1 j =2 = q1 (0)π LA (x1 , . . . , xJ )

[N EA (J )]−1 , [N LA (J )]−1

and π EA (0, x2 , . . . , xJ ) = π LA (0, x2 , . . . , xJ )

[N EA (J )]−1 , [N LA (J )]−1

if x1 > 0

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

89

where N EA (J ) and N LA (J ) denote the normalising constants in the EA case and LA case, respectively. Proof. It is the behaviour of the first node that distinguishes the EA system under consideration from the LA network. In the LA case, a customer who arrives to find the first node free at time t − cannot be served before the end of the next time slot [t, t + 1). In the EA case, however, an arriving customer who finds the first node empty at time t + may depart from that node at the end of the same time slot with probability p1 (0) = 0. It is the probability of such a transition which differentiates the EA system from the LA network considered in Theorem 3.1 and which necessitates the condition p1 (0) = 0. This difference does not affect the rest of the network which consists of LA nodes in both cases. If there is at least one customer at the first node, the stochastic behaviour of the two systems is identical, which may be explained as follows: For X(t) = x = (x1 , . . . , xJ ) ∈ NJ , let A(x) denote the set of nodes which are busy at time t + 21 . (Note that 1 ∈ A(x) does not necessarily imply X1 (t) = x1 > 0.) Let Aa (x) = A(x) ∪ {a}, where a symbolises an arrival at the first node. In both cases a transition out of state x = (x1 , . . . , xJ ) ∈ NJ into state y = (y1 , . . . , yJ ) ∈ NJ is for all x ∈ NJ with x1 ≥ 1 and x = y completely determined by the set B(x) ⊆ Aa (x) containing all nonempty nodes at which a service is completed and taking account of a possible arrival at the first node. For all x = (x1 , . . . , xJ ) ∈ NJ with x1 ≥ 1 and all B(x) ⊆ Aa (x), there exists a unique successor state y = HB(x) (x) resulting from the following transitions: 1. 2. 3. 4.

If a ∈ B(x), a customer enters the first node and joins the end of the queue. For all j ∈ B(x), j ≤ J − 1, a customer departs from node j and immediately goes to node j + 1. If J ∈ B(x), a customer leaves the system. At the nodes j ∈ A(x) − B(x), the ongoing services continue.

Conversely, given any pair of states x, y ∈ NJ with x1 ≥ 1 and x = y such that y can be reached from x in a single-step transition, there is exactly one set B(x) ⊆ Aa (x) such that y = HB(x) (x). If pEA (x, y) and p LA (x, y) denote the transition probabilities from state x to state y in the EA case and LA case, respectively, the following equality holds: p LA (x, y) = pEA (x, y) for all x, y ∈ NJ with x1 ≥ 1. This condition fails if x1 = 0. We shall therefore focus our attention on the behaviour of the first node and compare the steady-state equations for the respective networks in the following cases: 1. The first node is empty. 2. There is exactly one customer at the first node. 3. At least two customers are staying at the first node. In these three cases, the number of customers present at the subsequent nodes is arbitrary.




Case 1. The first node is empty. Consider the ways in which the system can reach state (0, y2 , . . . , yJ ) ∈ NJ . In both cases, it turns out that transitions out of states (0, x˜2 , . . . , x˜J ) into state (0, y2 , . . . , yJ ) arise when nothing happens at the first node, and are completely determined by a set B(x) ˜ ⊆ Aa (x) ˜ with a ∈ / B(x). ˜

90

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

In both cases transitions out of states (1, x2 , . . . , xJ ) into state (0, y2 , . . . , yJ ) are triggered by a service completion at the first node and are uniquely determined by a set B(x) ⊆ Aa (x) with 1 ∈ B(x) and a∈ / B(x). In the EA case, we must additionally consider the transitions out of states (0, x2 , . . . , xJ ) into state (0, y2 , . . . , yJ ) which arise when a customer arrives at the first node and is served within one time slot. These transitions are completely determined by the set B(x) ∪ {a}, where B(x) denotes the same set as in the preceding case. Using the notation introduced above, we obtain for all y = (0, y2 , . . . , yJ ) ∈ NJ 

π LA (y) =

π LA (0, x˜2 , . . . , x˜J )c(x˜2 + · · · + x˜J )

(0,x˜2 ,...,x˜J )∈NJ ∃B(x)⊆A ˜ ˜ ˜ a (x),H B(x) ˜ (x)=y a ∈B( / x) ˜

×





pj (x˜j )

j ∈B(x) ˜ j =1

qj (x˜j )

j ∈A(x)−B( ˜ x) ˜ j =1



+

π LA (1, x2 , . . . , xJ )c(1 + x2 + · · · + xJ )p1 (1)

(1,x2 ,...,xJ )∈NJ ∃B(x)⊆Aa (x),HB(x) (x)=y a ∈B(x),1∈B(x) /

×





pj (xj )

j ∈B(x) j =1

qj (xj )

j ∈A(x)−B(x) j =1

and 

π EA (y) =

π EA (0, x˜2 , . . . , x˜J )c(x˜2 + · · · + x˜J )

(0,x˜2 ,...,x˜J )∈N ∃B(x)⊆A ˜ ˜ ˜ a (x),H B(x) ˜ (x)=y a ∈B( / x) ˜



×

 j ∈B(x) j =1



qj (xj ) +

j ∈A(x)−B(x) j =1

pj (xj )

π EA (0, x2 , . . . , xJ )b(x2 + · · · +xJ )p1 (0)

(0,x2 ,...,xJ )∈NJ ∃B(x)∪{a}⊆Aa (x),HB(x)∪{a} (x)=y 1∈B(x)

pj (xj )



j ∈B(x) j =1

(1,x2 ,...,xJ )∈N ∃B(x)⊆Aa (x),HB(x) (x)=y a ∈B(x),1∈B(x) /



qj (x˜j )

j ∈A(x)−B( ˜ x) ˜ j =1

π EA (1, x2 , . . . , xJ )c(1 + x2 + · · · + xJ )p1 (1) J

×



pj (x˜j )

j ∈B(x) ˜ j =1

J

+





qj (xj ).

j ∈A(x)−B(x) j =1

Substituting the proposed forms for π EA (x1 , . . . , xJ ) and π EA (0, x2 , . . . , xJ ) yields

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104



π EA (y) =

(0,x˜2 ,...,x˜J )∈N

×

 [N EA (J )]−1 LA π (0, x ˜ , . . . , x ˜ )c( x ˜ + · · · + x ˜ ) pj (x˜j ) 2 J 2 J [N LA (J )]−1 J j ∈B(x) ˜ j =1





qj (x˜j ) +

j ∈A(x)−B( ˜ x) ˜ j =1

q1 (0)

(1,x2 ,...,xJ )∈NJ

[N EA (J )]−1 [N LA (J )]−1

× π LA (1, x2 , . . . , xJ )c(1 + x2 + · · · + xJ )p1 (1)





(0,x2 ,...,xJ )∈N



×

qj (xj )

j ∈A(x)−B(x) j =1

[N EA (J )]−1 LA π (0, x2 , . . . , xJ )b(x2 + · · · + xJ )p1 (0) [N LA (J )]−1 J

pj (xj )

j ∈B(x) j =1



pj (xj )

j ∈B(x) j =1

+

91



qj (xj ).

j ∈A(x)−B(x) j =1

Here the second and the third summands can be collected, since they describe transitions associated with a service completion at the first node. Moreover, it follows from Theorem 3.1 that π LA (1, x2 , . . . , xJ )c(1 + x2 + · · · + xJ )p1 (1) = π LA (0, x2 , . . . , xJ )b(x2 + · · · + xJ ). Therefore 

π EA (y) =

(0,x˜2 ,...,x˜J )∈N

×

 [N EA (J )]−1 LA π (0, x ˜ , . . . , x ˜ )c( x ˜ + · · · + x ˜ ) pj (x˜j ) 2 J 2 J [N LA (J )]−1 J

 j ∈A(x)−B( ˜ x) ˜ j =1

j ∈B(x) ˜ j =1

qj (x˜j ) +



[q1 (0) + p1 (0)]

(1,x2 ,...,xJ )∈NJ

× π LA (1, x2 , . . . , xJ )c(x2 + · · · + xJ )p1 (1)



j ∈B(x) j =1

[N EA (J )]−1 [N LA (J )]−1

pj (xj )



qj (xj ).

j ∈A(x)−B(x) j =1

Comparing this result with the steady-state equation obtained in the LA case yields π EA (0, y2 , . . . , yJ ) = π LA (0, y2 , . . . , yJ )

[N EA (J )]−1 . [N LA (J )]−1

Case 2. There is exactly one customer at the first node. Consider the transitions corresponding to an arrival at the first node, i.e. the probability of going to state (1, y2 , . . . , yJ ) ∈ NJ when the process is in state (0, x2 , . . . , xJ ). In the LA case, such a transition is completely determined by a set B(x) ⊆ Aa (x) with a ∈ B(x).

92

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

In the EA case, however, we must specify that 1 ∈ / B(x), as an arriving customer may be served within one time slot. Therefore the transition probabilities obtained in the LA case must be multiplied by q1 (0) in the EA case. It is not necessary to examine more closely the other transitions which can carry the process to state (1, y2 , . . . , yJ ), as in these cases, no arriving customer can find the system empty, so that the transition probabilities obtained for both systems are equal. Using the notation introduced above, we obtain for all y = (1, y2 , . . . , yJ ) ∈ NJ 

π LA (y) =

π LA (0, x2 , . . . , xJ )b(x2 + · · · + xJ )





pj (xj )

j ∈B(x) j =1

(0,x2 ,...,xJ )∈NJ ∃B(x)⊆Aa (x),HB(x) (x)=y a∈B(x)

+



qj (xj )

j ∈A(x)−B(x) j =1

π LA (x˜1 , . . . , xJ )pLA [(x˜1 , . . . , x˜J ), y]

(x˜1 ,...,x˜J )∈NJ 1≤x˜1 ≤2

and 

π EA (y) =

π EA (0, x2 , . . . , xJ )b(x2 + · · · + xJ )q1 (0)

×



qj (xj ) +

j ∈A(x)−B(x) j =1

pj (xj )

j ∈B(x) j =1

J

(0,x2 ,...,xJ )∈N ∃B(x)⊆Aa (x),HB(x) (x)=y a∈B(x),1∈B(x) /





π EA (x˜1 , . . . , xJ )pEA [(x˜1 , . . . , x˜J ), y].

(x˜1 ,...,x˜J )∈NJ 1≤x˜1 ≤2

Inserting the suggested forms for π EA (x1 , . . . , xJ ) into the second equation and keeping in mind the fact that pLA (x, y) = pEA (x, y) if x1 ≥ 1 yields 

π EA (y) =

π LA (0, . . . , xJ )

(0,x2 ,...,xJ )∈NJ

×

 ¯ =1 j ∈B,j

qj (xj ) +

 [N EA (J )]−1 b(x + · · · + x )q (0) pj (xj ) 2 J 1 [N LA (J )]−1 j ∈B,j =1



q1 (0) π LA (x˜1 , . . . , x˜J )

(x˜1 ,...,x˜J )∈NJ 1≤x˜1 ≤2

× pLA [(x˜1 , . . . , x˜J ); y)] = q1 (0)π LA (y)

[N EA (J )]−1 [N LA (J )]−1

[N EA (J )]−1 . [N LA (J )]−1

Case 3. At least two customers are staying at the first node. In this case, no arriving customer can find the system empty, which implies that pLA (x, y) = pEA (x, y) for all y ∈ NJ with y1 ≥ 2. Setting up the system of steady-state equations in the EA case and substituting accordingly establishes the desired result for all y = (y1 , . . . , yJ ) ∈ NJ with y1 ≥ 2.

π EA (y) =



B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

93

π EA (x1 , . . . , xJ )pEA [(x1 , . . . , xJ ); y]

(x1 ,...,xJ )∈NJ x1 ≥1

=



q1 (0)π LA (x1 , . . . , xJ )

(x1 ,...,xJ )∈NJ x1 ≥1

= q1 (0)π LA (y)

[N EA (J )]−1 LA p [(x1 , . . . , xJ ); y] [N LA (J )]−1

[N EA (J )]−1 . [N LA (J )]−1

Hence Theorem 3.1 allows us to deduce that π EA ((x1 , . . . , xJ ), (x1 , . . . , xJ ) ∈ NJ , is the equilibrium distribution for the process X, which completes the proof of Theorem 3.2.
Corollary 3.1. When the systems are state independent, the Markov chains are ergodic if and only if (b/pj ) < 1, j = 1, . . . , J . If the ergodicity condition is fulfilled, the equilibrium distribution of the above described systems is given by the products 

 J 





bq1  1 η(0,xj ) bqj xj b bq1 x1 π EA (x1 , . . . , xJ ) = 1− 1− cp1 cp1 j =2 qj cpj pj and π

LA





 J

 1 η(0,xj ) bqj xj b 1− , (x1 , . . . , xJ ) = qj pj cpj j =1

(x1 , . . . , xJ ) ∈ NJ .

Remark. The results established in the case of state independent systems are highlighted by their product form, a structure exhibited by the equilibrium distribution of their continuous time exponential analogues. Compare the result of Hsu and Burke [15]. At least so far as steady states are concerned, the state independent networks behave as if their nodes were independent single node systems as are discussed in Section 1. Of course, the independence established in the corollary is that of the random variables X1 , . . . , XJ observed at a fixed point in time. The asymmetric term which appears in the EA case reflects the different behaviour of the first node. 3.3. Arrival theorems Consider a tandem network composed of J nodes, J ≥ 2, in equilibrium. We now wish to compute the stationary distribution of the joint queue length process seen by arrivals in the three different systems described at the beginning of this section. The proofs of the theorems are possible by either elaborating on formulas from the general Palm theory in discrete time (ASTA, etc., see Sections 1–2.4) or by evaluation of conditional probabilities. For Theorem 3.3, see [7]; the proofs of Theorems 3.4 and 3.5 follow similar lines. It is of special interest to investigate under which conditions an arriving customer finds a network in equilibrium and to what extent the results obtained by Gravey and Hebuterne [13] for single node systems generalise to linear networks.

94

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

3.3.1. LA-DF system Theorem 3.3. Let X = ((X1 (t), . . . , XJ (t)) : t ∈ N) be the joint queue length process of the state dependent LA-DF network. Define for this system the event Bi (t) = {At time t − an arrival occurs at node i } for all i ∈ {1, . . . , J }. Then in equilibrium we have for all x = (x1 , . . . , xJ ) ∈ NJ πiLA-DF (x) := P (X(t) = (x1 , . . . , xi−1 , xi + 1, xi+1 , . . . , xJ )|Bi (t)) xj −1 x1 +···+xJ xi J   b(h) qi (h) h=1 qj (h) h=0 [HiLA-DF (J )]−1 , = xj x1 +···+xJ +1 p (h) p (h) i c(h) h=1 j h=0 j =1,j =i h=1 where the norming constant HiLA-DF (J ) is chosen so that the distribution πiLA-DF sums to unity with the sum taken over the state space. Corollary 3.2. If i = 1 and if the system is state independent π1LA-DF (x)

=

bq1 cp1

x1

bq1 1− cp1

 J

j =2

1 qj

η(0,xj )

bqj cpj

xj

b 1− pj

 = π LA (x),

x ∈ NJ .

3.3.2. EA system Theorem 3.4. The network under consideration now consists of an EA node and (J − 1) LA nodes, J ≥ 2. For all x = (x1 , . . . , xJ ) ∈ NJ , an arriving customer sees at time t + the system in state x with probability π1EA (x)

=

x1 +···+x  J h=1

1 −1 J xj −1 qj (h) EA q1 (h)  h=1 b(h) xh=0 x1 [H (J )]−1 , xj c(h) h=1 p1 (h) j =2 h=1 pj (h) 1

where H1EA (J ) denotes the norming constant. For a state independent system

π1EA (x) =

bq1 cp1

x1

1−

bq1 cp1

 J

j =2

1 qj

η(0,xj )

bqj cpj

xj

1−

b pj

 = π EA (x),

x ∈ NJ .

3.3.3. LA-AF at the first node Theorem 3.5. The network is now composed of J LA-AF nodes. Then for all x = (x1 , . . . , xJ ) ∈ NJ the arriving customer will see at time t −− the system in state x with probability π1LA-AF (x)

=

x1 +···+x  J h=1

J xj −1 qj (h) LA-AF −1 b(h)  h=1 ] , [H1 j c(h) j =1 xh=1 pj (h)

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

95

where H1LA-AF (J ) denotes the norming constant. For a state independent system π1LA-AF (x)





 J

 1 η(0,xj ) bqj xj b = 1− = π LA (x), qj cpj pj j =1

x ∈ NJ .

3.4. External arrival instants and PASTA analogue From the single node results in Sections 2.3 and 2.4 it follows that to prove PASTA analogues, we have to restrict our considerations to systems with state independent arrival rates. In this section we therefore confine our attention to a network composed of J state independent nodes, J ≥ 2. A customer arriving from the exterior sees the system in equilibrium if his arrival occurs before all the other events which can take place in the network during a time interval [t, t + 1), t ∈ N. This condition is always satisfied when the first station of the network is an EA node. An LA-AF system with global synchronisation of simultaneous events also fulfills this condition, as an arrival occurs at time t J − , immediately before all the other events which take place in (t J − , t). A short reflection shows that this result can be extended to other types of tandems, provided an arrival from the exterior occurs before all the service completions which can take place at the different nodes in an arbitrary order. In this respect, recall that the stationary queue length distribution of an LA-AF system does not differ from that of an LA-DF system or of a network composed of both LA-AF and LA-DF nodes. Theorem 3.6. Consider a network consisting of J state independent LA nodes, J ≥ 2. 1. If during a time slot, a service completion at an arbitrary node takes place before an arrival from the exterior, the arriving customer cannot find the system in equilibrium. 2. Conversely, if a customer arriving from the exterior sees the system in equilibrium, his arrival occurs before all the other events which can take place in the network during a time slot. Proof. 1. Without loss of generality, it suffices to consider a network consisting of J nodes, J ≥ 2, in which a service at an arbitrary node i, i ≥ 2, is completed before an arrival at the first node, and to check that the distribution of the customers seen by an arrival does not coincide with the stationary distribution. 2. Conversely, suppose a customer enters the first node at time (t + 1)−− , t ∈ N, and sees the system in equilibrium. Let A denote the event {at time (t + 1)−− an arrival occurs at the first node} and π1 be the conditional probability that an arriving customer finds the system in state x for all x = (x1 , . . . , xJ ) ∈ NJ . From the assumption it follows for all x ∈ NJ : π1 (x) = P [X[(t + 1)3− ] = (x1 , . . . , xJ )|A] = π LA (x). On the other hand, the state independent arrival rates imply bπ LA (x) = P [X(t) = (x1 , . . . , xJ )|an arrival occurs] π LA (x) =  b y∈NJ π LA (y)

(*)

96

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

as the system states are measured at time t, t ∈ N. From the equality (*), it follows for all x ∈ NJ P [X[(t + 1)3− ] = (x1 , . . . , xJ )|A] = P [X(t) = (x1 , . . . , xJ )|an arrival occurs]. Therefore, a customer arriving at the first node at time (t + 1)−− sees the system in the state in which he would have found it at time t. The system state at the arrival instant has remained unchanged since the beginning of the time slot; additional service completions before t + 1 must take place in ((t + 1)−− , t + 1).
Remark. In a state independent system, we obtain a PASTA analogue when the state of the network found by a customer arriving in [t, t + 1) coincides with the state of the Markov process X(t) at the beginning of the corresponding time slot, i.e., when a customer’s arrival at the first node is the first event which can take place in the network within a time slot. Theorem 3.6 can therefore be regarded as a generalisation of the Gravey and Hebuterne theorem [13] to vector-valued system processes. Note that the proof just given parallels the argument of Section 1 involving the order of the events within a time slot and the “independence condition”. Furthermore, as in the case of a single node system, the result remains valid without the restriction that the service rates be independent. 3.5. A customer’s jump between two arbitrary nodes We are now interested in the system state observed by a customer jumping between two arbitrary nodes. From Theorem 3.3 we know that in a state dependent LA-DF network, a customer going from one node to the next never sees the system in equilibrium. Nevertheless, the results of Section 3.5 strongly suggest that a customer jumping between two nodes in a state independent network will see the system in equilibrium if an internal arrival always takes place before a departure at that node. The following theorem confirms this suggestion. Theorem 3.7. Consider a network consisting of J LA-AF nodes, J ≥ 2 with global synchronisation of simultaneous events. For all x = (x1 , . . . , xJ ) ∈ NJ let πiLA-AF (x) denote the conditional probability that a customer going from node (i − 1) to node i sees exactly xj customers at node j , j = 1, . . . , J . Here node 0 denotes the outside. When the system is state dependent, we obtain for all i = 1, . . . , J  −1 y1 +···+yJ x1 +···+x J xj −1 J yj −1  J b(h)     qj (h) qj (h) b(h)   b(0) πiLA-AF (x) = b(0) h=1 h=1 xj yj c(h) c(h) p (h) p (h) J h=1 j h=1 j h=1 h=1 j =1 j =1 (y1 ,...,yJ )∈N

=

π1LA-AF (x1 , . . . , xJ ).

In the case of a state independent system, the following equality holds: πiLA-AF (x)





 J

 1 η(0,xj ) bqj xj b 1− = π LA (x1 , . . . , xJ ) = cp q p j j j j =1

Proof. By induction on i: For i = 1 the result holds by Theorem 3.5.

∀(x1 , . . . , xJ ) ∈ NJ .

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

97

Assume as induction hypothesis that the result holds for each integer smaller than or equal to i, 1 ≤ i < J . A service at node i is completed after all the events which can take place at the previous nodes 1, . . . , i−1 in the corresponding order. Suppose that a customer Ci going from node i to node i + 1 sees the network in state (x1 , . . . , xi , xi+1 , . . . , xJ ). It is difficult to reconstruct the state of the system at the beginning of the time slot, as all possible combinations of events which can occur until Ci ’s jump must be taken into account. It suffices, however, to consider the state of the system observed by the last customer who jumped between two arbitrary nodes (j − 1) and j before Ci ’s departure from node i, j = 1, . . . , i. The corresponding conditional distributions πjLA-AF (x), x ∈ NJ , which are known by the induction hypothesis, already contain the necessary information about the events which have taken place until Ci ’s jump. If there is at least one customer present at each of the first i nodes, it suffices therefore to consider (i + 1) possibilities: • If a service is completed at node (i − 1) immediately before Ci ’s departure from node i, the customer jumping between node (i − 1) and node i sees the system in state (x1 , . . . , xi−1 , xi , xi+1 , . . . , xJ ). • If the last service before Ci ’s jump is completed at node j , j = 1, . . . , i − 2, the customer going from node j to node (j + 1) will observe the system in state (x1 , . . . , xj , xj +1 − 1, xj +2 , . . . , xi−1 , xi + 1, xi+1 , . . . , xJ ). • If an arrival occurs and no service is completed until Ci ’s jump, the arriving customer finds the system in state (x1 − 1, x2 , . . . , xi−1 , xi + 1, xi+1 , . . . , xJ ). • If Ci ’s departure from node i is the first event taking place after the beginning of the time interval, the state of the system at the beginning of the slot is (x1 , . . . , xi−1 , xi + 1, xi+1 , . . . , xJ ). In each case, we must determine the transition probabilities describing the evolution of the system between the last service completion and Ci ’s jump. The infinitesimal time points t J − < t (J −1)− < · · · < t (J −i−1)− < t (J −i)− are defined at the beginning of Section 2. A customer jumping between node (j − 1) and node j , j = 1, . . . , J , sees the subsequent nodes j, j + 1, . . . , J in the state in which they have remained since the beginning of the time slot, which influences the state dependent service rates describing the evolution of the system between the service completion at node (j − 1) and Ci ’s jump. Denote by Ai+1 the event that at time t (J −i)− Ci is about to arrive at node i + 1. In the following LA-AF expressions the arrows indicate that customers are about to reach the corresponding nodes. If πi+1 (x) denotes the conditional probability that Ci sees the system in state x when jumping between node i and node i + 1, we obtain for all x = (x1 , . . . , xJ ) ∈ NJ with xj > 0 for j = 1, . . . , i LA-AF πi+1 (x) = P [X(t (J −i−1)− ) = (x1 , . . . , xi−1 ,  xi , xi+1 , . . . , xJ ),

X(t (J −i)− ) = (x1 , . . . , xi−1 , xi ,  xi+1 , . . . , xJ )|Ai+1 ] + P [X(t (J −i−2)− ) = (x1 , . . . , xi−2 ,  xi−1 − 1, xi + 1, xi+1 , . . . , xJ ), X(t (J −i−1)− ) = (x1 , . . . , xi−1 , xi + 1, xi+1 , . . . , xJ ), X(t (J −i)− ) = (x1 , . . . , xi−1 , xi ,  xi+1 , . . . , xJ )|Ai+1 ] + P [X(t (J −i−3)− ) = (x1 , . . . , xi−3 ,  xi−2 − 1, xi−1 , xi + 1, xi+1 , . . . , xJ ), X(t (J −i−2)− ) = X(t (J −i−1)− ) = (x1 , . . . , xi−1 , xi + 1, xi+1 , . . . , xJ ),

98

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

X(t (J −i)− ) = (x1 , . . . , xi−1 , xi ,  xi+1 , . . . , xJ )|Ai+1 ] + · · · + P [X(t (J −1)− ) = (x1 ,  x2 − 1, . . . , xi−1 , xi + 1, xi+1 , . . . , xJ ), X(t (J −2)− ) = · · · = X(t (J −i−1)− ) = (x1 , . . . , xi−1 , xi + 1, xi+1 , . . . , xJ ), X(t (J −i)− ) = (x1 , . . . , xi−1 , xi ,  xi+1 , . . . , xJ )|Ai+1 ] + P [X(t J − ) = ( x1 − 1, x2 , . . . , xi−1 , xi + 1, xi+1 , . . . , xJ ), X(t (J −1)− ) = · · · = X(t (J −i−1)− ) = (x1 , . . . , xi−1 , xi + 1, xi+1 , . . . , xJ ), X(t (J −i)− ) = (x1 , . . . , xi−1 , xi ,  xi+1 , . . . , xJ )|Ai+1 ] + P [X(t − 1) = · · · = X(t (J −i−1)− ) = (x1 , . . . , xi−1 , xi + 1, xi+1 , . . . , xJ ), X(t (J −i)− ) = (x1 , . . . , xi−1 , xi ,  xi+1 , . . . , xJ )|Ai+1 ]. Inserting the corresponding conditional probabilities yields  LA-AF (x) = πiLA-AF (x1 , . . . , xi , xi+1 , . . . , xJ )pi (xi ) πi+1 LA-AF + πi−1 (x1 , . . . , xi−1 − 1, xi + 1, xi+1 , . . . , xJ )qi−1 (xi−1 − 1)pi (xi + 1) LA-AF (x , . . . , x − 1, x , x + 1, x , . . . , x )q (x − 1) +π i−2

1

i−2

i−1

i i+1 J i−2 i−2 LA-AF · · · +π2 (x1 , x2 −1, x3 , . . . , xi−1 , xi

× qi−1 (xi−1 )pi (xi + 1) + + 1, xi+1 , . . . , xJ ) × q2 (x2 − 1)q3 (x3 ) . . . qi−1 (xi−1 )pi (xi + 1) + π1LA-AF (x1 − 1, x2 , x3 , . . . , xi−1 , xi + 1, xi+1 , . . . , xJ )q1 (x1 − 1)q2 (x2 ) · · · × qi−1 (xi−1 )pi (xi + 1) + c(x1 + · · · + xj + 1)  × π LA (x1 , x2 , . . . , xi−1 , xi +1, xi+1 , . . . , xJ )q1 (x1 )q2 (x2 ) · · · qi−1 (xi−1 )pi (xi +1) N −1 , LA-AF where the norming constant N is chosen so that the distribution πi+1 sums to unity. Applying the induction hypothesis to the above result, we obtain for all x ∈ NJ with xj > 0 for j = 1, . . . , i   x1 +x J xj −1 2 +···+xJ  q (h) b(h) j LA-AF  (x) = b(0) πi+1 h=1 xj c(h) p h=1 j (h) h=1 j =1

× [pi (xi ) + pi−1 (xi−1 )qi (xi ) + pi−2 (xi−2 )qi−1 (xi−1 )qi (xi ) + pi−3 (xi−3 )qi−2 (xi−2 )qi−1 (xi−1 )qi (xi ) + · · · + p3 (x3 )q4 (x4 ) · · · qi−2 (xi−2 ) × qi−1 (xi−1 )qi (xi ) + p2 (x2 )q3 (x3 ) · · · qi−2 (xi−2 )qi−1 (xi−1 )qi (xi ) + p1 (x1 )q2 (x2 ) · · · × q (x )q (x )q (x ) + q1 (x1 )q2 (x2 ) · · · qi−2 (xi−2 )qi−1 (xi−1 )qi (xi )]N −1  i−2 i−2 i−1 i−1 i i  x1 +x J xj −1 2 +···+xJ  q (h) b(h) j  N −1 . = b(0) h=1 xj c(h) h=1 pj (h) h=1 j =1 The desired equalities follow from Theorem 3.5. If l arbitrary nodes j1 , . . . , jl are empty, l = 1, . . . , i and jm ∈ {1, . . . , i} for m = 1, . . . , l, when Ci departs from node i, (i − l + 1) disjoint possibilities must be taken into account. If jm ∈ {2, . . . , i}, m = 1, . . . , l, no service has been completed at node jm − 1; if jm = 1, no arrival has occurred. The summands πjm are then left out, as the corresponding events could not occur, and the

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

99

factors qjm (0) which appear in the other summands vanish, since qjm (0) = 1. The remaining terms sum to unity. 䊐 Remark. The above result also holds for a network which differs from the LA-AF system under consideration in that the first station is an EA node. The difference reduces to the external arrival which takes place at the beginning of a time slot. The same method can be applied to prove the result as the order of the events within a time slot remains unchanged. 3.6. Sojourn time distributions As in the case of a single node system, the arrival theorems are used to determine the sojourn time distribution of a typical customer in the respective networks. In other words, we study the networks from the point of view of a test customer C0 who arrives at time t and finds the other customers distributed according to the arrival distributions computed in Section 3.6. We confine our attention to systems with state independent service rates and compare C0 ’s sojourn time in two different tandems of queues. We first present the results obtained for an LA-DF system [5–7] and then show that C0 ’s sojourn time distribution in an EA network can be derived from this result. The latter result solves the sojourn time problem for the networks considered by Pujolle et al. [25] who only computed mean sojourn times. 3.6.1. LA-DF system We assume that C0 arrives at time t − at the first node of the system and finds the other customers distributed according to π1LA-DF . We denote by B1 (t) the event that at time t − an arrival occurs at the first node. Theorem 3.8. Let T LA-DF = (T1 , . . . , TJ ) denote the vector of C0 ’s successive sojourn times at the LA-DF nodes during his passage starting at time t − . The generating function of T LA-DF is given by     J J    T T π1LA-DF (x1 , . . . , xJ )E  θj j |X(t) = (x1 + 1, . . . , xJ ), B1 (t) E  θj j  = j =1

j =1

(x1 ,...,xJ )∈NJ

x1 +···+xJ b(h) LA-DF −1 [H1 (J )] x1h=0 = +···+xJ +1 c(h) h=1 (x1 ,...,xJ )∈NJ 

×

qj pj

xj

p1 θ1 1 − q1 θ1

x1 +1  J

j =2

1 θj



q1 p1

η(0,xj )

x1  J

j =2

pj θj 1 − qj θj

1 qj

η(0,xj )

xj +1

,

|θj | ≤ 1, j = 1, . . . , J. In an open tandem of queues, the joint sojourn time vector of a customer who finds the system in state x = (x1 , . . . , xJ ) ∈ NJ and sees x1 + · · · + xJ = M customers when he arrives at the first node is distributed as the joint sojourn time vector of a customer in a closed network with the same nodes and M + 1 people cycling, who observes the other M customers in state (x1 , . . . , xJ ). This is due to the strong Markov property of the system processes, the state independent service rates and the FCFS queue

100

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

discipline. Using this observation, the result of Theorem 3.8 is proved by reduction to the sojourn time theorem for closed networks [5], which uses the result in [24]. Remark. The difficulty of the proof is due to the relation   x1 +1  

xj +1

J

J  pj θj 1 η(0,xj ) p1 θ1 Tj   E θj |X(t) = (x1 + 1, . . . , xJ ), B1 (t) = . 1 − q1 θ1 θj 1 − qj θj j =1 j =2 3.6.2. EA system In this case, C0 arrives at time t + at the first node of the tandem and finds the other customers distributed according to π1EA . Let A1 (t) denote the event that at time t + an arrival occurs at the first node. Theorem 3.9. Let T EA = (T1 , . . . , TJ ) denote the vector of C0 ’s successive sojourn times at the different nodes of the EA system during his passage starting at time t + . Then the generating function of T EA is given by     J J    T T E  θj j  = π1EA (x1 , . . . , xJ )E  θj j |X(t) = (x1 , . . . , xJ ), A1 (t) j =1

j =1

(x1 ,...,xJ )∈NJ

=



[H1EA (J )]−1

x1 +···+x  J h=1

(x1 ,...,xJ )∈NJ

b(h) c(h)



p1 θ1 1 − q1 θ1



q1 θ1 1 − q1 θ1





xj J

 pj θj qj θj 1 η(0,xj ) × , 1 − qj θj 1 − qj θj qj θj j =2

x1

|θj | ≤ 1, j = 1, . . . , J.

Proof. The FCFS queue discipline, the strong Markov property of X and the state independent service rates ensure that C0 ’s conditional joint sojourn time distribution in the network is completely determined by the number of customers he finds at the successive nodes of the system at his arrival instant. Conditioning on the event A1 (t) and on the vector of initial population sizes, we obtain     J J    T T E  θj j  = π1EA (x1 , . . . , xJ )E  θj j |X(t) = (x1 , . . . , xJ ), A1 (t) j =1

j =1

(x1 ,...,xJ )∈NJ

=

∞ 

[H1EA (J )]−1

M=0

M  b(h) h=1

 ×E

J 

j =1

T

c(h)

 (x1 ,...,xJ )∈NJ x1 +···+xJ =M



q1 p1

x1  J

j =2



θj j |X(t) = (x1 , . . . , xJ ), A1 (t) .

1 qj

η(0,xj )

qj pj

xj

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

101

Comparing the conditional joint sojourn time distribution in an EA system (left-hand side) and in an LA-DF system (right-hand side) yields     J J   T T E  θj j |X(t) = (x1 , . . . , xJ ), A1 (t) = E  θj j |X(t) = (x1 + 1, . . . , xJ ), B1 (t) j =1

j =1

as the difference reduces to C0 ’s exact arrival instant. A customer’s treatment after his arrival in one of the networks is the same. Hence  

x1  η(0,xj ) xj M J ∞ J

    qj b(h) q 1 Tj 1 EA −1 E  θj  = G1 (M + 1, J ) [H1 (J )] c(h) p1 qj pj J j =1 M=0 h=1 j =2  ×E

(x1 ,...,xJ )∈N x1 +···+xJ =M

J 



T θj j |X(t)

= (x1 + 1, . . . , xJ ), B1 (t) [G1 (M + 1, J )]−1 .

j =1

The second sum corresponds to the generating function of the cycle time distribution of a customer who arrives at the first node of a closed network with M + 1 people cycling and sees the other M customers distributed according to

x1  

 J

q1 1 η(0,xj ) qj xj M+1,J π1 = [G1 (M + 1, J )]−1 , p1 q p j j j =2 where G1 (M + 1, J ) denotes the norming constant [5]. It follows by Theorem 3.8 that   

x1

J M ∞    b(h) q1 θ1 p1 θ1 Tj EA −1   E θj = [H1 (J )] c(h) 1 − q1 θ1 1 − q1 θ1 j =1 M=0 h=1 



xj J

 pj θj qj θj 1 η(0,xj ) × qj θj 1 − qj θj 1 − qj θj j =2 

=

[H1EA (J )]−1

(x1 ,...,xJ )∈NJ

×

x1 +···+x  J h=1

b(h) c(h)



p1 θ1 1 − q1 θ1



q1 θ1 1 − q1 θ1

x1





xj J

 pj θj qj θj 1 η(0,xj ) , 1 − q 1 − q q θ θ θ j j j j j j j =2

|θj | ≤ 1, j = 1, . . . , J.


3.6.3. State independent systems If the EA and LA-DF networks are state independent, it follows from Corollary 3.2 and Theorem 3.4 that the arrival distributions are equal. We obtain therefore the following corollary:

102

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

Corollary 3.3. Let C0 denote a test customer who arrives at time t + at the first node of a state independent EA network. The generating function of C0 ’s sojourn time in the system is given by   J J   ((pj − b)/c)θj Tj   E θj , |θj | ≤ 1, j = 1, . . . , J. = 1 − (1 − (pj − b)/c)θj j =1 j =1 This result does not differ from the generating function of the sojourn time of a customer who arrives at time t − at the first node of an LA-DF system and sees the other customers distributed according to π1LA-DF . Remarks. 1. The differences between the generating functions determined above reduce to an arriving customer’s perspective and disappear in the case of state independent systems; both networks show the same stationary and asymptotic behaviour with respect to sojourn time distributions, a property already observed in the case of a single node system. The behaviour of the first node differs only during infinitesimal time intervals [t − , t + ), t ∈ N, which are negligible as compared with a time slot. 2. In a state independent system, a customer’s sojourn time at node j , j = 1, . . . , J , has the same distribution as if node j were an isolated queue with individuals arriving there in a Bernoulli process of rate b. 4. Tandems of queues with local access control The entrance control applied in our networks to regulate the arrivals is a global closed loop control, where the total population size is the decision variable for the arrival control. It would be most interesting to compare such control rules with the effects of local closed loop control regimes where the decision variable for the admission is the queue length of the first node only. Such systems were investigated by Pujolle et al. [25]; they assert that an LA system consisting of J nodes, J ≥ 2, and satisfying the conditions mentioned above possesses the following equilibrium distribution: π(x1 , . . . , xJ ) =

J  j =1

Cj

b(0)b(1)qj (1) · · · b(xj − 1)qj (xj − 1) , c(1)p1 (1) · · · c(xj )pj (xj )

(x1 , . . . , xJ ) ∈ NJ ,

where b(k) denotes the probability of an arrival when k customers are staying at the first node at the beginning of the corresponding time slot, k ∈ N, pj (i) is the probability that a service is completed at node j , j = 1, . . . , J , if that node is in state i, i ≥ 1, and C1 , . . . , CJ denote norming constants. Yet, the model presented by the authors has a system of steady-state equations which is not solved by π under the stated assumptions. Theorem 4.1. Consider a network composed of two LA-DF nodes with state dependent service rates and arrival rates depending only on the number of customers present at the first node. In general the system of stationary equations is not solved by π(x1 , x2 ) =

2  j =1

Cj

b(0)b(1)qj (1) · · · b(xj − 1)qj (xj − 1) c(1)p1 (1) · · · c(xj )pj (xj )

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

103

Proof. Inserting the corresponding values of π(x1 , x2 ) into the steady-state equation, π(0, 0) = c(0)π(0, 0) + c(0)p2 (1)π(0, 1) yields



 b(0) π(0, 0) = π(0, 0) c(0) + c(0)p2 (1)  π(0, 0) = c(1)p2 (1)

if c(0) = c(1),

which contradicts the general assumption that the arrival rates depend on the number of customers present
at the first node. Remark. Pujolle et al. [25] considered additionally a tandem system where an EA regime holds at all nodes in the networks. It follows, e.g. that a customer arriving at an empty system at time t + leaves node J at time (t + 1)− and is never counted in the state process. For these tandems a similar observation can be made as stated in the preceding theorem. These counter-examples, along with Theorem 3.1, suggest that to obtain explicit steady states of product form the arrival rates of a tandem of queues must depend on the total number of customers present in the system; seemingly dependence of the rates cannot be restricted to the number of customers staying at the first node. Acknowledgements We are grateful to two anonymous referees for their careful reading of the first version of this paper. Their comments and suggestions were helpful to enhance the paper’s contents as well as its presentation. References [1] F. Baccelli, P. Bremaud, Elements of Queueing Theory, Springer, New York, 1994. [2] H. Bruneel, B.G. Kim, Discrete-time Models for Communication Systems Including ATM, Kluwer Academic Publishers, Boston, 1993. [3] M.L. Chaudhry, U.L. Gupta, Queue-length and waiting-time distributions of discrete-time GIX /Geom/1 queueing systems with early and late arrivals, Queueing Syst. Appl. 25 (1997) 307–324. [4] X. Chao, M. Miyazawa, M. Pinedo, Queueing Networks: Customers, Signals, and Product Form Solutions, Wiley, Chichester, UK, 1999. [5] H. Daduna, The joint distribution of sojourn times for a customer traversing a series of queues: the discrete time case, Queueing Syst. Appl. 27 (1997) 297–323. [6] H. Daduna, Some results for steady-state and sojourn time distributions in open and closed linear networks of Bernoulli servers with state dependent service and arrival rates, Perform. Eval. 30 (1997) 3–18. [7] H. Daduna, The cycle time distribution in a cycle of Bernoulli servers in discrete time, Math. Meth. Oper. Res. 44 (1996) 295–332. [8] H. Daduna, Discrete Time Queueing Networks: Recent Developments, Tutorial Lecture Notes, Performance’96, Lausanne, 1996. [9] H. Daduna, R. Szekli, Dependencies in Markovian networks, Adv. Appl. Prob. 25 (1995) 226–254. [10] H. Daduna, R. Szekli, Conditional job observer properties in multitype closed queueing networks, Technical Report No. 108, Wroclaw University, Wroclaw, 2000. [11] M. El-Taha, S. Stidham Jr., Sample-path Analysis of Queueing Systems, Kluwer Academic Publishers, Boston, 1999. [12] M. El-Taha, S. Stidham Jr., A filtered ASTA property, Queueing Syst. Appl. 11 (1992) 211–222.

104

B. Desert, H. Daduna / Performance Evaluation 47 (2002) 73–104

[13] A. Gravey, G. Hebuterne, Simultaneity in discrete-time single server queues with Bernoulli inputs, Perform. Eval. 14 (1992) 123–131. [14] S. Halfin, Batch delays versus customer delays, The Bell Syst. Tech. J. 62 (1983) 2011–2015. [15] J. Hsu, P.J. Burke, Behaviour of tandem buffers with geometric input and Markovian output, IEEE Trans. Commun. 24 (1976) 358–361. [16] J. Hunter, Mathematical Techniques of Applied Probability, Vols. 1 and 2, Academic Press, New York, 1983. [17] L. Kleinrock, Analysis of a time-shared processor, Naval Res. Logist. Quart. 10 (1964) 59–73. [18] D. Koenig, V. Schmidt, EPSTA: the coincidence of time-stationary and customer-stationary distributions, Queueing Syst. Appl. 5 (1989) 247–264. [19] J.G. Kemeny, J.L. Snell, A.W. Knapp, Denumerable Markov Chains, 2nd Edition, Springer, New York, 1976. [20] J.D.C. Little, A proof of the queueing formula L = λW , Oper. Res. 9 (1961) 383–387. [21] B. Melamed, W. Whitt, On arrivals that see time averages, Oper. Res. 38 (1990) 156–172. [22] A. Makowsky, B. Melamed, W. Whitt, On averages seen by arrivals in discrete time, in: Proceedings of the IEEE Conference on Decision and Control, Vol. 28, Tampa, FL, 1989, pp. 1084–1086. [23] M. Miyazawa, Y. Takahashi, Rate conservation principle for discrete-time queues, Queueing Syst. Appl. 12 (1992) 215–230. [24] V. Pestien, S. Ramakrishnan, Features of some discrete-time cycling queueing networks, Queueing Syst. Appl. 18 (1994) 117–132. [25] G. Pujolle, J.P. Claude, D. Seret, A discrete-time queueing system with a product form solution, in: T. Hasegawa, H. Tagaki, V. Takahashi (Eds.), Computer Networking and Performance Evaluation, Elsevier, Amsterdam, 1986, pp. 139–147. [26] R. Serfozo, Reversibility and compound birth–death and migration processes, in: Queueing and Related Models, Oxford University Press, Oxford, 1992, pp. 65–90. [27] S. Stidham Jr., L = λW : a discounted analogue and a new proof, Oper. Res. 20 (1972) 1115–1126. [28] S. Stidham Jr., A last word on L = λW , Oper. Res. 22 (1974) 417–421. [29] H. Takagi, Queueing Analysis: A Foundation of Performance Analysis, Vol. 3, Discrete-time Systems, North-Holland, New York, 1993. [30] H. Tempelmeier, Inventory service-levels in the customer supply chain, OR Spektrum 22 (2000) 361–380. [31] W. Whitt, Improving service by informing customers about anticipated delays, Mgmt. Sci. 45 (1999) 192–207. [32] S. Wittevrongel, H. Bruneel, Discrete-time ATM queues with independent and correlated arrival streams, in: D.D. Kouvatsos (Ed.), Performance Evaluation and Applications of ATM Networks, Kluwer Academic Publishers, Boston, 2000, pp. 387–412. [33] R.W. Wolff, Poisson arrivals see time averages, Oper. Res. 30 (1982) 223–231. [34] M.E. Woodward, Communication and Computer Networks: Modelling with Discrete-time Queues, IEEE Computer Society Press, Los Alamitos, CA, 1994. Bernadette Desert was born and raised in France and studied English and German literature in Paris, Middlebury (Vermont, USA) and Hamburg. She worked as a technical translator in Karlsruhe and in Kiel for five years before studying mathematics at the University of Hamburg where she specialized in stochastic processes. She is now writing a thesis about discrete-time queueing networks. She has been working with an insurance company in Hamburg for two years; her responsibilities include planning, designing and maintaining data warehouse solutions to support data mining and predictive modelling efforts.

Hans Daduna studied Mathematics at the University of Hamburg, Germany. He was an assistant at the Department of Mathematics of the Technical University of Berlin. Since 1984 he is a Professor of Mathematics at the Department of Mathematics at Hamburg University. His research interests are in the field of stochastic processes with applications in computer sciences and operations research.