A discrete-time queue with customers with geometric deadlines

Applied Mathematical Modelling, 1996

In this paper we study the time-dependent analysis of a limited capacity queueing model with the bulk arrival rate depending upon the nature of service available in the system. The customers arrive in the system in batches of size x, which is a random variable, and the service consists of two stages, one is essential (first stage) while the other may be inessential. The decision to offer the inessential service depends upon the size of the system. However, if this inessential service is temporarily suspended, the arrival rate of the customers decreases. Laplace transforms (in time) of the different probability generating functions describing the system size under various conditions of service and the expected system size are derived. Steady-state results consequently follow.

In this paper, a discrete-time single server queueing system with infinite buffer size and geometrically distributed arrivals is considered. We derive the functional equations and analyze the distribution of the number of customers served during a busy period for geometrically distributed service time as well as for deterministic service time. We also show that in the limiting case the results obtained in this paper are consistent with the corresponding continuous-time counterparts by Medhi [1].

This paper deals with the steady-state solution of the queueing system: M X /H k /1/N with reneging in which (i) units arrive in batches of random size with the interarrival times of batches following negative exponential distribution, (ii) the batches are served in order of their arrival; and (iii) the service time distribution is hyperexponential with k branches. Recurrence relations connecting the various probabilities introduced are found. Some measures of effectiveness are deduced and some special cases are also obtained. Keywords: queueing system: M X /H k /1/N, hyperexponential DESCRIPTION OF THE SYSTEM Morse [4] discussed the steady-state queueing system in which the service channel consists of two branches, the units arrive singly and the capacity of the waiting space is infinite. Gupta and Goyal [1] studied a similar system by using the generating functions with k branches in the service channel, the units arrive singly and the capacity of the waiting space is finite. Habib [3] and Gupta and Goyal [2] treated the system M X /Hk/1. White et al. [6] solved the system: M/H2 /2/2 numerically. All the previous studies are without balking and reneging. In the present system, it is assumed that the units arrive at the system in batches of random size X, i. e., at each moment of arrival, there is a probability Cj = Pr (X= j) that j units arrive simultaneously, and the interarrival times of batches follow a negative exponential distribution with time independent parameter . Let  Cj t, (j = 1, 2,..., N), be the first order probability that a batches of j units comes in time t. The service channel is busy if a unit is present in any one of the k branches and in this case the arrival units form a queue and the capacity of the system is N. The unit at the head of the queue requires service in the r th branch with probability *. The service time distribution in the r th branch is * The variation of the subscripts i,j,r,s is from 1 to k, unless otherwise explicitly mentioned.

Operations Research, 1990

We solve the queueing system (QS) Ck/Cm/s, where Ck is the class of Coxian probability density functions (pdfs) of order k, which is a subset of the pdfs that have rational Laplace transform (R). We formulate the model as a continuous-time, infinite-space Markov chain by generalizing the method of stages. By using a generating function technique, we solve an infinite system of partial difference equations and find closed form expressions for the systemsize, general-time, pre-arrival, post-departure probability distributions and the usual performance measures. In particular, we prove that the probability of n customers being in the system, when it is "saturated" (n s) is a linear s+m-1 combination of exactly (s) geometric terms. The closed form expressions involve a solution of a system of nonlinear equations that involves only the Laplace transforms of the interarrival and service time distributions. We conjecture that this result holds for the more general model GIRls. Following these theoretical results we propose an exact algorithm for finding the systemsize distribution and system's performance measures, which has an algorithmic complexity of O(k3(+' s)3). We examine special cases and apply this method for solving numerically the QS C 2 /C 2 /s and Ek/C 2 /s.

Operations Research, 1990

In this paper we consider the MX/G/I queueing system with batch arrivals. We give simple approximations for the waiting-time probabilities of individual customers. These approximations are checked numerically and they are found to perform very well for a wide variety of batch-size and service-time distributions. Batch-arrival queueing models can be used in 1) many practical situations, such as the analysis of message packetization in data communication systems. In general it is difficult, if not impossible, to find tractable expressions for the waiting-time probabilities of individual customers. It is, therefore, useful to have easily computable approximations for these probabilities. This paper gives such approximations for the single server MX/G/1 model. Exact methods for the computation of the waitingtime distribution in the MX/G/1 queue are discussed in Eikeboom and Tijms (1987), cf. also Chaudhry and Templeton (1983), Neuts (1981) and Tijms (1986). However, these methods apply only for special servicetime distributions and are, in general, not suited for routine calculations in practice. A simple approximation for the tail probabilities of the waiting time was given in Eikeboom and Tijms by using interpolation of the asymptotic expansions for the particular cases of deterministic and exponential services. This approximate approach uses only the first two moments of the service time. This paper presents an alternative approach that uses the actual service-time distribution rather than just its first two moments. This alternative approach starts with the asymptotic expansion of the waitingtime distribution. In Van Ommeren (1988) it is shown that the complementary waiting-time probability of an arbitrary customer in the MX/G/1 queue has an exponentially fast decreasing tail under some mild assumptions. By calculating the decay parameter and the amplitude factor, we get the asymptotic expansion of the waiting-time distribution. Such asymptotic expansions already provide a very powerful tool in practical queueing analysis, cf. Cromie, Chaudhry and Grassman (1979) and Tijms. It turns out that for Subject classifications: Queues, approximations: approximations based on asymptotic analysis. Queues, batch/bulk: approximations.

Annals of Operations Research, 2013

In this paper, we present a basic discrete-time queueing model whereby the service process is decomposed in two (variable) components: the demand of each customer, expressed in a number of work units needed to provide full service of the customer, and the capacity of the server, i.e., the number of work units that the service facility is able to perform per time unit. The model is closely related to multi-server queueing models with server interruptions, in the sense that the service facility is able to deliver more than one unit of work per time unit, and that the number of work units that can be executed per time unit is not constant over time. Although multi-server queueing models with server interruptions-to some extentallow us to study the concept of variable capacity, these models have a major disadvantage. The models are notoriously hard to analyze and even when explicit expressions are obtained, these contain various unknown probabilities that have to be calculated numerically, which makes the expressions difficult to interpret. For the model in this paper, on the other hand, we are able to obtain explicit closed-form expressions for the main performance measures of interest. Possible applications of this type of queueing model are numerous: the variable service capacity could model variable available bandwidths in communication networks, a varying production capacity in factories, a variable number of workers in an HR-environment, varying capacity in road traffic, etc.

2007

We consider an extension of the standard G/G/1 queue, described by the equation W D = max{0, B − A + Y W }, where P[Y = 1] = p and P[Y = −1] = 1 − p. For p = 1 this model reduces to the classical Lindley equation for the waiting time in the G/G/1 queue, whereas for p = 0 it describes the waiting time of the server in an alternating service model.

Performance Evaluation, 2017

We consider a discrete-time queueing system having two distinct servers: one server, the "regular" server, is permanently available, while the second server, referred to as the "extra" server, is only allocated to the system intermittently. Apart from their availability, the two servers are identical, in the sense that the customers have deterministic service times equal to 1 fixed-length time slot each, regardless of the server that processes them. In this paper, we assume that the extra server is available during random "up-periods", whereas it is unavailable during random "down-periods". Up-periods and down-periods occur alternately on the time axis. The up-periods have geometrically distributed lengths (expressed in time slots), whereas the distribution of the lengths of the down-periods is general, at least in the first instance. Customers enter the system according to a general independent arrival process, i.e., the numbers of arrivals during consecutive time slots are i.i.d. random variables with arbitrary distribution. For this queueing model, we are able to derive closed-form expressions for the steadystate probability generating functions (pgfs) and the expected values of the numbers of customers in the system at various observation epochs, such as the start of an up-period, the start of a down-period and the beginning of an arbitrary time slot. At first sight, these formulas, however, appear to contain an infinite number of unknown constants. One major issue of the mathematical analysis turns out to be the determination of these constants. In the paper, we show that restricting the pgf of the down-periods to be a rational function of its argument, brings about the crucial simplification that the original infinite number of unknown constants appearing in the formulas can be expressed in terms of a finite number of independent unknowns. The latter can then be adequately determined based on the bounded nature of pgfs inside the complex unit disk, and an extensive use of properties of polynomials. Various special cases, both from the perspective of the arrival distribution and the down-period distribution, are discussed. The results are also illustrated by means of relevant numerical examples. Possible applications of this type of queueing model are numerous: the extra server could be the regular server of another similar queue, helping whenever an idle period occurs in its own queue; a geometric distribution for these idle times is then a very natural modelling assumption. A typical example would be the situation at the check-in counter at a gate in an airport: the regular server serves customers with a low-fare ticket, while the extra server gives priority to the business-class and first-class customers, but helps checking regular customers, whenever the priority line is empty.

2011

Many-Server Queues with Time-Varying Arrivals, Customer A bandonment and Non-Exponential Distributions Yunan Liu This thesis develops deterministic heavy-traffic fluid appr oximations for many-server stochastic queueing models. The queueing models, with many homogenous servers working independently in parallel, are intended to model largescale service systems such as call centers and health care systems. Such models also have been e mploy d to study communication, computing and manufacturing systems. The heavytraffic approximations yield relatively simple formulas for quantities describing syst em performance, such as the expected number of customers waiting in the queue. The new performance approximations are valuable because, i n the generality considered, these complex systems are not amenable to exact mathem atical analysis. Since the approximate performance measures can be computed quite rap idly, they usefully complement more cumbersome computer simulation. Thus these heavy -t...

IEEE Journal on Selected Areas in Communications, 1991

This paper is concerned with an M / G / I FCFS queue with twd types of customers, viz. (1) ordinary customers who arrive according to a Poisson process, and (2) permanent customers, who immediately return to the end of the queue after having received a service. The influence of the permanent customers on queue length and sojourn times of the Poisson customers is studied, using results from queueing theory and from the theory of branching processes. In particular it is shown, for the case that the service time distributions of the Poisson customers and all K permanent customers are negative exponential with identical means, that the queue length and sojourn time distributions of the Poisson customers are the ( K + 1 )-fold convolution of those for the case without permanent customers.

Mathematics

In this paper, we discuss the waiting-time distribution for a finite-space, single-server queueing system, in which customers arrive singly following a Poisson process and the server operates under (a,b)-bulk service rule. The queueing system has a finite-buffer capacity ‘N’ excluding the batch in service. The service-time distribution of batches follows a general distribution, which is independent of the arrival process. We first develop an alternative approach of obtaining the probability distribution for the queue length at a post-departure epoch of a batch and, subsequently, the probability distribution for the queue length at a random epoch using an embedded Markov chain, Markov renewal theory and the semi-Markov process. The waiting-time distribution (in the queue) of a random customer is derived using the functional relation between the probability generating function (pgf) for the queue-length distribution and the Laplace–Stieltjes transform (LST) of the queueing-time distri...

Operations Research, 1996

We continue to focus on simple exponential approximations for steady-state tail probabilities in queues based on asymptotics. For the G/GI/1 model with i.i.d. service times that are independent of an arbitrary stationary arrival process, we relate the asymptotics for the steady-state waiting time, sojourn time, and workload. We show that the three asymptotic decay rates coincide and that the three asymptotic constants are simply related. We evaluate the exponential approximations based on the exact asymptotic parameters and their approximations by making comparisons with exact numerical results for BMAP/G/1 queues, which have batch Markovian arrival processes. Numerical examples show that the exponential approximations for the tail probabilities are remarkably accurate at the 90th percentile and beyond. Thus, these exponential approximations appear very promising for applications.

Performance Evaluation, 1997

A new approximate method is developed for finding the waiting and sojourn time distributions in a class of multi-queue systems served in cyclic order at discrete intervals. An immediate application for such a model is in communication networks where a number of different traffic sources compete to access a group of transmission channels operating under a time-slotted sharing policy. This system maps naturally onto a model in which the inter-visit time has a probability mass function of phase-type. We derive a set of matrix equations with easily tractable iterative procedures for their solution and controllable accuracy in their numerical evaluation. We then validate the analytical model against simulation and discuss the validity of the assumptions. This methodology can be extended to several other polling strategies. © 1997 Elsevier Science B.V. a stochastic decomposition of the system's unfinished work. E:cact expressions for weighted sums of mean waiting times were derived using this method (see [i]). Several other methods have been developed for computing the mean delay, the mean queue length for each queue, the amount of work of the server and the cycle time. A survey of these methods is given by Levy and Sidi (see [ 10]). This paper tackles the problem of finding an approximate method for evaluating the waiting time pi ,'Jability density function in multi-queue systems with discrete service time. More abstractly, it also considers the sojourn ~.ime distribution of systems with exceptional first service times. A direct application for such systems can be found in time-slotted medium access protocols. In these systems several traffic sources compete to use a group of transmission channels. These transmitters are available only at the beginning of fixed length time-slots. At each discrete interval a single server starts visiting queues in cyclic order and assigning packets to the transmitters.

In this paper, a discrete-time single server queueing system with infinite buffer size and geometrically distributed arrivals is considered. We derive the functional equations and analyze the distribution of the number of customers served during a busy period for geometrically distributed service time as well as for deterministic service time. We also show that in the limiting case the results obtained in this paper are consistent with the corresponding continuous-time counterparts by Medhi [1].

Operations Research, 2004

For a broad class of discrete-and continuous-time queueing systems, we show that the stationary number of customers in system (queue plus servers) is the sum of two independent random variables, one of which is the stationary number of customers in queue and the other is the number of customers that arrive during the time a customer spends in service. We call this relation an invariance relation in the sense that it does not change for a variety of single-sever queues (with batch arrivals and batch services) and some of multi-server queues (with batch arrivals and deterministic service times) that satisfy a certain set of assumptions. Making use of this relation, we also present a simple method of deriving the stationary distributions of the numbers in queue and in system as well as some of their properties. This is illustrated by several examples, which show that new simple derivations of old results as well as new results can be obtained in a unified manner. Furthermore, we show that the invariance relation and the method we are presenting are easily generalized to analyze queues with BMAP (Batch Markovian Arrival Process) arrivals. Most of the results are presented under the discrete-time setting. The corresponding continuous-time results, however, are covered as well because deriving the results for continuous-time queues runs exactly parallel to that for their discrete-time counterparts.