Electronic Notes in Theoretical Computer Science 65 No. 6 (2002) URL: http://www.elsevier.nl/locate/entcs/volume65.html 15 pages

Performance Analysis of Retrial Queueing Systems Using Generalized Stochastic

Petri Nets

Nawel Gharbi 1 and Malika Ioualalen

Institut Génie Electrique, Département Informatique Université des Sciences et de la Technologie Houari Boumediene - USTHB

Alger, Algérie

Abstract

We consider retrial queueing systems, in which an arriving customer finding the server busy, may repeat his call after a random duration. The consideration of repeated calls introduces great analytical difficulties. In fact, detailed analytical results exist for some special retrial queueing systems, while for many others, the performance evaluation is limited to numerical algorithms, approximation methods and simulation. Retrial queues have been widely used to model many problems in telephone switching systems, telecommunication and computer networks. In this paper, we show a method for modeling and analyzing retrial queueing systems, using the Generalized Stochastic Petri nets (GSPNs).

Key words: Retrial Queueing systems, Generalized Stochastic

Petri nets, Modeling, Performance evaluation.

1 Introduction

Queueing systems are often used in modeling manufacturing [1,9], telecommunication [29] and computer systems. The classical theory of queues gives two principal ways to solve the conflict which will occur when the arriving customer in the system finds the server busy :

• The customer may leave the system forever without receiving service : this is the well known "Erlang loss system",

• The customer may wait in line, to receive service after some departure : this is the classical "System with queue".

1 Email: nawel_gharbi@hotmail.com

©2002 Published by Elsevier Science B. V.

Fig. 1. Schematic diagram of a Retrial Queueing System

An alternative way authorizes the customer to repeat his call after some random time. Between trials, the customer is said to be in "orbit". Such a system is called : Retrial queueing system. For a review of the main results and bibliography about this topic see these survey papers : [5,17,18,21,33].

A retrial queueing system consists of a service station and an orbit as shown schematically in Figure 1. Customers can arrive at the service station either from outside the system or from the orbit. If an arriving customer finds the server idle, he receives service immediately and leaves the system after service completion; otherwise, if the customer is blocked from entering the service station because the server is not available, he may join the orbit and attempts service again after some random delay, and continues to do so until the server is found idle. At this time, the retrying customer is served and leaves the system. On retrials, each orbiting customer is treated the same as a new arriving one.

Since the work of Kosten (1947) [20], Wilkinson (1956) [32] and Cohen (1957) [16], retrial queues are considered as an interesting problem in tele-traffic theory and telephone switching systems, where subscribers redial after receiving a busy signal. The study of retrial queueing systems is actually motivated also by new developments and new technologies, particularly in computer and telecommunication networks, where for example, several terminals are making retrials to receive service from a central processor, or in the case of local area networks (LAN) under the non-persistent CSMA (carrier-sense multiple access) protocol, where terminals are making retrials in order to access the communication bus and transmit messages.

The consideration of repeated calls introduces great analytical difficulties. In fact, detailed analytical results exist for some special retrial queueing systems with assumptions on some characteristics such as the retrial times distribution, the number of servers, the customers homogeneity ..., while for many others, the performance evaluation is limited to numerical algorithms [28,31], approximation methods [17,34] and simulation.

In this paper, we show a method of modeling and analyzing retrial queueing systems, using the Generalized Stochastic Petri nets (GSPNs). The method provides several benefits both for the qualitative and the quantitative analysis

of these queueing systems. In particular, it offers the possibility of using results and software tools developed within the Stochastic Petri net framework, to obtain performance indices either with analytic means or by other methods.

On the other hand, the GSPNs have been extensively applied to represent and analyze production, communication and computer systems and have been proved to be a powerful tool for modeling and analyzing the performance of parallel systems. Another advantage of using the generalized stochastic Petri net formalism to model retrial queueing systems, is that we can analyze systems that may be somewhat difficult to analyze by more conventional methods. For example, a retrial queueing system with multiple servers that may take vacations, and in which the externals customers arrive in batches and the retrial times are generally distributed.

The article is organized as follows : First, the basics of stochastic and generalized stochastic Petri nets are briefly reviewed. In section 3, we give the GSPN model of retrial queueing systems. An example of this model is studied in section 4. Next, we present the different analysis methods of large GSPNs. Finally, conclusion is presented in section 5.

2 Stochastic and generalized stochastic Petri nets

Petri nets [8,27,30] are an important graphical and mathematical tool used to study the behavior of many systems. This model is a promising tool for describing and studying systems that are characterized as being concurrent, asynchronous, distributed, parallel, non-deterministic and/or stochastic.

A Petri net is a directed bipartite graph that consists of two types of nodes called places (represented by circles) and transitions (represented by bars). The directed arcs connect places to transitions and transitions to places. If an arc exists from a place to a transition, then the place is called an "input place" of the transition. Similarly, if an arc exists from a transition to a place, then the place is called an "output place" of that transition. Places may contain tokens (represented by dots).

The state of a Petri net is defined by the number of tokens in each place, and is represented by a vector M = (M1,M2,..., Mn) called the marking, where Mi is the number of tokens in place ,PiJ and 'n' is the number of places. A formal definition of a Petri net is thus the following : [30]

(1) PN = (P, T, A, W, M0)

P = {P1; P2,..., Pn} is the set of places, T = {t1 ,t2,...,tm} is the set of transitions,

A C {P x T}U{T x P} is the set of arcs (standard and inhibitor), W : A ^ N is the weighting function which associates to each arc a weight, and N is the set of natural numbers. If the weight equals 1, it can be omitted from the Petri net representation. M0 : is the initial marking.

The marking in general, can be regarded as a mapping from the set of places to N. That is M : P ^ N, where M(Pi) = Mi , i = 1,...., n.

The set of input (resp. output) transitions of a place p E P is denoted by p (resp. p'). Similarly, the set of input (resp. output) places of a transition t E T is denoted by't (resp. t').

A transition t E T is said to be enabled at a marking M, if each of its input places contains at least as many tokens as arc weight connecting the place to the transition.

Formally, t E T is enabled at M if and only if Wp E't, M(p) > W(p, t). A transition may fire if it is enabled. The firing of a transition't' at marking M removes W(p,t) tokens from each of its input places 'p' and put W(t,q) tokens to each of its output places 'q', thus leads to a new marking.

In some applications, a transition may be inhibited from firing under certain conditions. This feature is captured by defining an input "inhibitor" arc which has a rounded rather than arrow-pointed head.

A transition with inhibitor input places is enabled if and only if the number of tokens in each of the inhibitor input places is less than the weight of the inhibitor input arc.

Each firing of a transition changes the distribution of tokens on places and, thus creates a new state.

A marking Mj is said to be reachable from a marking Mi, if starting from Mi, there is a sequence of transitions whose firing generates Mj. The set of all markings reachable from initial marking M0, denoted by R(M0) is called the "reachability set".

The "reachability graph" associated with the reachability set can be constructed as follows : Represent each state by a vertex and place a directed edge from vertex Vi to vertex Vj if the state Vj can result from the firing of some transition enabled in Vi.

A transition 't' is said to be live if for any marking M E R(M0), there exists a sequence of transitions firable from M which contains the transition 't'. A Petri net is said to be live if all the transitions are live.

A place 'Pi in a PN is said to be bounded if there exists a positive integer 'k' such that M(Pi) < k for all M E R(Mo). A PN defined to be bounded if each place p E P is bounded. Hence, the reachability graph of the Petri net will be finite.

Early studies on Petri nets were limited to analyzing the reachability set for such qualitative properties as liveness, safeness and boundedness. No timing considerations were included and as such their usefulness in performance evaluation was limited.

The inclusion of time in a PN model can be done by associating with each transition a fixed time duration which is the time that elapses from the enabling to the firing of the transition.

The stochastic Petri nets (SPNs) [26] are obtained by associating with each transition in a PN an exponentially distributed firing time.

A formal definition of a SPN is thus the following : SPN = (P, T, A, W, M0, A) where P,T,A,W and M0 are as in (1), and A = {A1;A2,...,Xm} is the set of firing rates (possibly marking dependent) associated with the PN transitions. An enabled transition 'tj' can fire after an exponentially distributed time delay equals 1/Aj elapses.

The firing rate of a transition 'tj' can be marking dependent. Thus, if there are 'k' tokens in the input place p', then the firing rate of 'tj' is 'kAj'. The condition of marking dependent firing is represented by the symbol ' # ' placed next to 'tj'.

Live and bounded SPNs are isomorphic to continuous-time Markov chains (CTMC) due to the memoryless property of exponential distributions [26]. The states of the CTMC are the markings in the reachability graph, and the state transition rates are the exponential firing rates of the transitions in the SPN.

The transition rate from marking Mi to marking Mj is Y1 ken- Ak ,where Hij is the set of transitions enabled by Mi and whose firing generates Mj. Hence, the infinitesimal generator Q (transition rates matrix) of the Markov chain is easily derived, and a vector of steady-state probabilities n can be computed using : n.Q = 0 where the summation of all the elements of n equals 1. Then, performance indices can be deduced using these state probabilities.

A generalized stochastic Petri net (GSPN) [2,3] is basically a SPN with transitions that are either timed (to describe the execution of time consuming activities) or immediate (to describe some logical behavior of the model). Pictorially, stochastic timed transitions are represented by rectangular boxes and immediate transitions are represented by thick bars. Timed transitions behave as in SPNs, whereas the immediate transitions have an infinite firing rate and fire in zero time. Markings in which at least one immediate transition is enabled are called vanishing markings. On the other hand, markings in which only exponential transitions are enabled are called tangible markings.

The firing of transitions in a GSPN depends on whether we are examining a tangible or a vanishing marking. In the case of tangible marking, any enabled transition can fire next. For a vanishing marking, only the enabled immediate transitions are allowed to fire because the lowest priority level is reserved for timed transitions.

A GSPN that contains tangible and vanishing markings is still equivalent to a Markov chain [15]. In this case, the equivalent chain is called an embedded Markov chain. If we remove the vanishing markings, then we have a reduced embedded Markov chain. This chain, which contains only the tangible markings, is the chain we use for computing the steady-state probabilities and performance indices.

The Generalized stochastic Petri nets in which the firing times of all timed transitions are exponentially distributed are said : "markovian GSPNs", otherwise if the firing delays of timed transitions are deterministic or generally

distributed, the GSPN is said: "non-markovian".

3 Generalized stochastic Petri net models of retrial qu-eueing systems

Because of the complexity of retrial queueing systems, analytic results are generally difficult to obtain and frequently the last resort remain numerical algorithms [28,31], approximation methods [17,34] and simulation. In fact, analytical results exist for special retrial queueing systems with assumptions on some characteristics such as :

• The population and the capacity of the system are infinite.

• The number of servers : Multiserver retrial queues have attracted considerable attention. However, the analytical studies of these models are limited, and in many cases restricted to two-server systems [17,4]. Due to the complexity of the analysis of the general case with more than two servers, only simulation, approximation [13] and numerical [28] methods are applied [21].

• The retrial times distributions : Most of the papers considered retrial queues with retrial times exponentially distributed. However, there has been very little research on retrial queues with general retrial time distribution, which models realistic problems of daily life. This is partly because of the complexity of the problem. To obtain results about these systems with generally distributed retrial times, simulation and approximation methods seem to be the only way and no analytical solutions are available [21].

• Batch arrivals : Generally, it is assumed that at every arrival epoch one customer arrive to the system. However, retrial queues could be used to model systems in which customers arrive in batches. Retrial queues with batch arrivals are quite common in telecommunication networks, but relatively little attention has been paid to this kind of queueing model. In these systems, if an incoming batch finds the server idle, one of the members of the batch begins service and the remaining members join the orbit. If the incoming batch finds the server busy, the entire batch joins the orbit. The members of the orbit, individually and independently, try their luck after a random amount of time.

• Multiclass systems : In most retrial queues, it is assumed that the customers characteristics such as inter-arrival times, service times and retrial times are homogeneous. However, in practice, these characteristics may differ widely for different customers groups. This leads us to multiclass retrial queues or queues with multiple types of customers. In Particular, retrial queues with two types of calls have been widely used to model and solve various practical problems occurring in telecommunication networks, telephone switching systems and digital cellular mobile networks. On the other hand, in multiclass retrial queueing systems, it is natural to consider that

each customer class has its own priority level [12]. From the mathematical point of view, such models are essentially more difficult than retrial queues with single type of calls (customers).

• Vacation systems : In the retrial queueing systems with vacation [6,22], the server takes a vacation, ie. becomes unavailable to the customers for a random period of time. These vacation periods are usually introduced in order to exploit the idle time of the server for other secondary jobs : maintenance and/or inspection tasks, servicing customers of another system, improvement of the quality of service and so on. For example, processors in computer and communication systems do considerable testing and maintenance besides doing their primary functions (processing telephone calls, processing interactive and batch jobs, receiving and transmitting data, etc). The testing and maintenance periods which are mainly to preserve the sanity of the system and to provide high reliability, may be regarded as server vacations. Once again, the machine breakdowns which may occur randomly, can also be regarded as server vacations.

In this paper, we show a method of modeling and analyzing retrial queue-ing systems, using the Generalized Stochastic Petri nets (GSPNs). The method provides several benefits both for the qualitative and the quantitative analysis of these queueing systems. In particular, it offers the possibility of using results and tools developed within the GSPN framework.

We consider a single server retrial queueing system at which customers arrive according to a Poisson process with rate Xa. An arriving customer receives immediately service if the server is idle. The service times are exponentially distributed with rate Xs. Otherwise, if the server is busy, the customer leaves the service station immediately and joins the orbit to retry for service after a random delay. The duration of the customer's retrial is assumed to be exponentially distributed with rate Xr. If this retrial to get service finds the server free, then one customer of the orbit (the customer at the head or another one chosen randomly) occupies the server.

We also assume that the system has a finite population (N) and finite capacity (L). We finally assume that the input flow of external arrivals, intervals between repeated attempts (retrials) and service times are mutually independent. The above queueing system is denoted by : M/M/1/L/N. Figure 2 shows the GSPN model of the above system.

• The place 'Pa' contains N tokens, which represents the condition that none of the N customers has arrived for service.

• The place 'PS' represents the condition that a customer is ready for service.

• The place 'P0' represents the orbit.

• Currently, the marking (initial marking) of the net is

Mo = {M(Pa),M(Ps),M(Po)} = {N, 0, 0}

Fig. 2. GSPN for the M/M/1/L/N retrial queueing system

which implies that none of the customers has arrived for service and that the orbit is empty.

• The firing of the transition 'ta' indicates the arrival of a customer. This firing is marking dependent. Thus, the firing rate of 'ta' is 'kAa'.

• The immediate transition 'S' is enabled when the place 'PS' contains at least 2 tokens, which represents the fact that the server is in service thus 'S' fires and the arriving customer must join the orbit. This is represented by the fact that one token is deposited in 'Po\ Once in orbit, the customers behave independent and each of them tries again to be served after a mean delay

• When the timed transition 'tR' fires, which represents the end of the waiting time of a customer in orbit, one token is taken from 'Po' and deposited in 'PS'. Thus, the firing rate of transition 'tR' should depend on the marking of the place 'Po', because a constant rate means that only one customer at a time tries again to be served. So, if there are 'k' tokens in 'Po', then the firing rate is 'kAR'

• The timed transition 'ts' is enabled when 'PS' contains at least one token. When it fires, one token is deposited in 'Pa\ which represents the condition that the customer has returned to be the idle state.

• The inhibitor arc from 'Po' to 'ta' is of multiplicity L. Thus, new customers may arrive (ie. 'ta' can fire) only when the marking of ''Po is less than L (ie. the system capacity including the one receiving service is less than L). We assume that L < N; that is, the capacity of the system cannot exceed the population size.

4 Example

As an example, we consider an M/M/1/2/3 retrial queueing system, which

has a finite capacity of size 2 and a finite population of size 3.

1 2 0 M2 M4

M7 1 0 2

M3 1 1 1

Fig. 3. Reachability tree for the GSPN of Figure 2 (N=3, L=2)

Given that a customer has not arrived yet, the time until he arrives is exponentially distributed with mean 1/Aa. The duration of the customer's retrial and customer's service are assumed to be exponentially distributed with mean 1/\r and 1/Xs respectively.

The GSPN model of this system is the same that the one which was given in Fig. 2 with N = 3 and L = 2.

Figure 3 shows the reachability tree for this GSPN. It allows us to compute all possible future markings starting from the initial one. The tree represents the firing sequence of the transitions for a given initial marking. The tree is pruned when a previous marking is obtained. The label on each directed edge represents the transition whose firing produced the successor marking.

Fig. 4. CTMC for the GSPN of figure 2(N = 3, L = 2)

The reachability tree contains two vanishing markings : M2 and M4 and six tangibles markings : Mo, Mi, M3, M5, M6 and M7. A vanishing marking is one in which at least one immediate transition is enabled, and a tangible marking is one in which only timed transitions are enabled.

A continuous-time Markov chain can now be constructed from the reachability tree. The states of the MC are the tangible markings of the tree. The vanishing markings are merged with their successor tangible markings since it takes zero time to go through a vanishing marking. Thus, M2 is merged with M3 and M4 with M6. The transition rates of the CTMC are the firing rates of the transitions of the Petri net.

Figure 4 shows the CTMC for the GSPN in figure 2 (with N = 3 and L = 2).

The infinitesimal generator (transition rates matrix) Q of the CTMC is given by :

■3A„ 3A„ 0 0 0 0

A, -(A, + 2Aa) 2Aa 0 0 0

0 0 -(A, + Aa) A, Aa 0

0 AR 2Aa -(Ar + 2Aa) 0 0

0 0 0 0 -A, A,

0 0 2Ar 0 0 -2A

The vector of steady-state probabilities n = (n0,n1,n3,n5,n6,n7) is the solution of the system : n.Q = 0 and Y1 i ni = 1, where nj denotes the steady-state probability that in Fig. 4 the process is in state Mj. Having the steady-state probabilities n, we can easily compute several performance measures, particularly :

• The effective customer arrival rate:(n)

V =^2 *a(Mj ).nj

(2) = A„.[3no + 2(ni + + na]

where :

• SMj is the set of markings where the transition HJ is enabled.

• \a(Mj) is the firing rate associated to the transition 'ta' in Mj.

• The average number of customers in orbit:(n0)

This correspond to the average number of tokens in the place 'P0' which models the orbit.

(3) no = Y^ Mj (Po).nj = n + n + 2(n + rn)

where : Mj (Po) is the number of tokens in the place 'Po' in the marking 'Mj' and the sum concerns all accessible markings.

• The average number of customers in the system:(ns)

This represents the average number of customers in service and in orbit.

(4) ns = ^2 n.[Mj(Po) + Mj(Ps)] = ni + n + 2(na + n7) + 3n6

• The main response time:(r)

From Little's law [23], we obtain the main response time (t).

(5) t = —

In this example, we have considered the case of systems with finite population and finite capacity. However, retrial queueing systems with infinite capacity can be modeled by the same GSPN of Fig. 2 in which the inhibitor arc from the place 'Po' to the transition 'ta' is omitted.

On the other hand, we can emulate an infinite population retrial queueing system by marking the value of 'N' to be very large. This is equivalent to say that there are enough tokens in 'Pa to cause 'ta' to be constantly enabled. In this case, the firing rate of 'ta' will no longer be marking-dependent but constant, as long as there is a token in 'PA'.

Generalized stochastic Petri nets can provide a very compact representation of very large systems. However, the difficulty will only lie with the size of the state space of the CTMC. Therefore, a large effort has been devoted to overcome or to alleviate this problem.

Since markovian GSPN are based on the solution of a CTMC, all the techniques that have been explored to handle very large Markov chains can profitably be utilized in connection with GSPN. However, when dealing with large models, not only that the solution of the system becomes difficult, but the model description and the computer representation also become tedious.

With the aim of overcoming the state space explosion phenomenon, in addition to exact numerical solutions [10], simulation and approximate techniques, several approaches have been recently proposed : [7]

• Distributed algorithms : They have been specifically developed for both the generation of the reachability graph from a GSPN and for the solution of the underlying CTMC. Distributed approaches may achieve a significant speed up in the computational time and a considerable extension of the

cardinality of the solvable models.

• Structured representation : In this approach, the generator matrix is represented in a compact form as a combination of smaller component matrices, and this representation is exploited in the solution algorithm.

• Hierarchical models : If an overall system model can be composed from submodels then each submodel is solved separately and results passed to higher level submodel.

• Product form GSPN : Queueing networks with product-form solution are well established and find application in a variety of fields. Several proposals have been recently documented to import the product- form concept into the GSPN arena.

• PN-driven techniques : These techniques deal with the reduction of both memory requirements and time complexity of the solution algorithms of GSPN by using information about the structure of the untimed PN models. Thus, the reduced state space is generated without building the complete reachability graph.

• Performance bounds : A complementary approach to the development of efficient solution techniques for the computation of performance measures, is the search for bounds. Bounds require less computational effort with respect to the cost of exact solution, since they are estimated based on equations at the GSPN level, and do not require the generation of the reachability graph.

• Software packages : The derivation of the reachability graph and the equivalent CTMC model, and the computations of the steady-state probabilities can be automated and incorporated into software packages such as : Stochastic Petri Net Package : SPNP [14], Petri Net based Performability Evaluation Tool (PEPNET) and GRaphical Editor and Analyser for Timed and Stochastic Petri Nets : GreatSPN [11].

For example, in the SPNP, the user enters the GSPN description as a C-based file, and the reachability graph and the equivalent Markov chain are derived automatically. The output describes the steady-state probabilities of the markings.

It is important to note that there is no simulation, Monte Carlo or otherwise, involved in these computations.

In addition to these sequential packages, there exist other distributed GSPN solution tools [25] that are able to capitalize upon the aggregated resources provided by a set of workstations to solve complex GSPN models.

In the example studied, we have concentrated on retrial queues in which

the customer's arrivals, the service times and the retrials are all exponentially

distributed (markovian).

It is important to precise that the modeling of a retrial queueing system with

non markovian distributions is done with the same method and give us a non-

markovian GSPN in which the distribution firing time of timed transitions are deterministic or general. The main advantage of these distributions is the fact that they generalize the exponential distributions.

For the performance evaluation of these retrial queues, the non-markovian GSPN model obtained will be analyzed as the model of any other system. With the aim of specifying these models that are analytically tractable, three main lines of research can be envisaged : [7]

• An approach based on Markov regenerative theory [24].

• An approach based on the use of supplementary variables [19].

• And an approach based on the expansion of the reachability graph of the basic Petri net.

5 Conclusion

Retrial queueing systems have been widely used to model many problems where arriving customers (calls) who find the server busy or not available, join the orbit to try again for their request in random order and at random intervals.

In this paper, we have proposed an approach that allows to model retrial queueing systems, by the Generalized Stochastic Petri nets (GSPNs), and then the analysis of these systems using methods, results and software tools developed within the Generalized Stochastic Petri net framework.

The use of GSPNs as a valid, flexible and powerful tool for modeling and evaluating several retrial queueing systems has been demonstrated on the M/M/1/2/3 queue. We have shown how it can take into account all parameters of the considered system and then may be used to obtain performance measures of interest.

On the other hand, we have observed that this approach can be applied for analyzing non markovian systems in which customer's arrivals, service times and/or retrials are generally distributed.

In conclusion, many problems in retrial queueing systems framework can be simplified by using this approach, and a similar method can be used to evaluate performances of more complex retrial queueing systems.

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