A Polynomial Factorization Approach for the Discrete Time GIX/>G/1/K Queue

This paper proposes a polynomial factorization approach for queue length distribution of discrete time GI ^X /G/1 and GI _X /G/1/K queues. They are analyzed by using a two-component state model at the arrival and departure instants of customers. The equilibrium state-transition equations of state probabilities are solved by a polynomial factorization method. Finally, the queue length distributions are then obtained as linear combinations of geometric series, whose parameters are evaluated from roots of a characteristic polynomial.     

On the Stationary Distribution of the GI X /M Y /1 Queueing System

For the GI ^X /M/1 queue, it has been recently proved that there exist geometric distributions that are stochastic lower and upper bounds for the stationary distribution of the embedded Markov chain at arrival epochs. In this note we observe that this is also true for the GI ^X /M ^Y /1 queue. Moreover, we prove that the stationary distribution of its embedded Markov chain is asymptotically geometric. It is noteworthy that the asymptotic geometric parameter is the same as the geometric parameter of the upper bound. This fact justifies previous numerical findings about the quality of the bounds.     

Analyzing the Steady-State Queue <Emphasis Type="Italic">GI</Emphasis><Superscript><Emphasis Type="Italic">X</Emphasis></Superscript><Emphasis Type="Italic">/G/</Emphasis>1

It is proved in this note that the delay in the queue GI ^X /G/1 can be expressed as the sum of two independent components, such that known results of the queue GI/G/1 (e.g. approximations) can be readily applied. Based on this result, closed-form expressions are also derived for other performance measures of interest.     

Mean Waiting Time Approximations in the G/G/1 Queue

It is known that correlations in an arrival stream offered to a single-server queue profoundly affect mean waiting times as compared to a corresponding renewal stream offered to the same server. Nonetheless, this paper uses appropriately constructed GI/G/1 models to create viable approximations for queues with correlated arrivals. The constructed renewal arrival process, called PMRS (Peakedness Matched Renewal Stream), preserves the peakedness of the original stream and its arrival rate; furthermore, the squared coefficient of variation of the constructed PMRS equals the index of dispersion of the original stream. Accordingly, the GI/G/1 approximation is termed PMRQ (Peakedness Matched Renewal Queue). To test the efficacy of the PMRQ approximation, we employed a simple variant of the TES⁺ process as the autocorrelated arrival stream, and simulated the corresponding TES ⁺/G/1 queue for several service distributions and traffic intensities. Extensive experimentation showed that the proposed PMRQ approximations produced mean waiting times that compared favorably with simulation results of the original systems. Markov-modulated Poisson process (MMPP) is also discussed as a special case.     

A New and Pragmatic Approach to the GIX/Geo/c/N Queues Using Roots

A simple and complete solution to determine the distributions of queue lengths at different observation epochs for the model GI^X/Geo/c/N is presented. In the past, various discrete-time queueing models, particularly the multi-server bulk-arrival queues with finite-buffer have been solved using complicated methods that lead to results in a non-explicit form. The purpose of this paper is to present a simple derivation for the model GI^X/Geo/c/N that leads to a complete solution in an explicit form. The same method can also be used to solve the GI^X/Geo/c/N queues with heavy-tailed inter-batch-arrival time distributions. The roots of the underlying characteristic equation form the basis for all distributions of queue lengths at different time epochs. All queue-length distributions are in the form of sums of geometric terms.

     

Queue-length and waiting-time distributions of discrete-time GI X/Geom/ 1 queueing systems with early and late arrivals

In this paper, we give a unified approach to solving discrete-time GI ^X/Geom/ 1 queues with batch arrivals. The analysis has been carried out for early- and late-arrival systems using the supplementary variable technique. The distributions of numbers in systems at prearrival epochs have been expressed in terms of roots of associated characteristic equations. Furthermore, distributions at arbitrary as well as outside observer's observation epochs have been obtained using the relation derived in this paper. We also present delay analyses for both the systems. Numerical results are presented for various interarrival-time and batch-size distributions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Some results for the queues Mx/M/c and GIx/ G/c

We develop for the queue M^x/M/c an upper bound for the mean queue length and lower bounds for the delay probabilities (that of an arrival group and that of an arbitrary customer in the arrival group). An approximate formula is also developed for the general bulk-arrival queue GI^x/G/c. Preliminary numerical studies have indicated excellent performance of the results.     

Parametric nonlinear programming approach to fuzzy queues with bulk service

This paper proposes a procedure to construct the membership functions of the performance measures in bulk service queuing systems with the arrival rate and service rate are fuzzy numbers. The basic idea is to transform a fuzzy queue with bulk service to a family of conventional crisp queues with bulk service by applying the α-cut approach. On the basis of α-cut representation and the extension principle, a pair of parametric nonlinear programs is formulated to describe that family of crisp bulk service queues, via which the membership functions of the performance measures are derived. To demonstrate the validity of the proposed procedure, two fuzzy queues often encountered in transportation management are exemplified. Since the performance measures are expressed by membership functions rather than by crisp values, they completely conserve the fuzziness of input information when some data of bulk-service queuing systems are ambiguous. Thus the proposed approach for vague systems can represent the system more accurately, and more information is provided for designing queuing systems in real life. By extending to fuzzy environment, the bulk service queuing models would have wider applications.     

Combined Elapsed Time and Matrix-Analytic Method for the Discrete Time GI/G/1 and GI X /G/1 Systems

In this paper, we show that the discrete GI/G/1 system can be easily analysed as a QBD process with infinite blocks by using the elapsed time approach in conjunction with the Matrix-geometric approach. The positive recurrence of the resulting Markov chain is more easily established when compared with the remaining time approach. The G-measure associated with this Markov chain has a special structure which is usefully exploited. Most importantly, we show that this approach can be extended to the analysis of the GI ^X /G/1 system. We also obtain the distributions of the queue length, busy period and waiting times under the FIFO rule. Exact results, based on computational approach, are obtained for the cases of input parameters with finite support – these situations are more commonly encountered in practical problems.     

Loss probability in an overloaded discrete-time GI/G/1/K system with very large K

We present a simple semi-explicit formula for estimating the loss probability in a discrete-time GI/G/1/K system (with large K) which is operating under an overload condition. The method relaxes the lower boundary and then studies the upper boundary only. The idea is extended to the GI^X/G/1/K system.     

On some relations between stationary time and customer state probabilities for qneueing systems G/GI/s/r

By means of a general formula for stochastic processes with imbedded marked point processes (PMP) some necessary and sufficient condition is given for the validity of a relationship, which is well-known in the case of exponentially distributed service times, between stationary time and customer state probabilities for loss systems G/GI/s/O (Theorem 3). A result of Miyazawa for the GI/GI/l/∞ queue is generalized to the case of non-recurrent interarrival times (Theorem 4)-. Furthermore, bounds are derived for the mean increment of the waiting time process at arrival epochs and for the mean actual waiting time in multi-server queues.     

On the distribution of the number stranded in bulk-arrival, bulk-service queues of the M/G/1 form

Bulk-arrival queues with single servers that provide bulk service are widespread in the real world, e.g., elevators in buildings, people-movers in amusement parks, air-cargo delivery planes, and automated guided vehicles. Much of the literature on this topic focusses on the development of the theory for waiting time and number in such queues. We develop the theory for the number stranded, i.e., the number of customers left behind after each service, in queues of the M/G/1 form, where there is single server, the arrival process is Poisson, the service is of a bulk nature, and the service time is a random variable. For the homogenous Poisson case, in our model the service time can have any given distribution. For the non-homogenous Poisson arrivals, due to a technicality, we assume that the service time is a discrete random variable. Our analysis is not only useful for performance analysis of bulk queues but also in designing server capacity when the aim is to reduce the frequency of stranding. Past attempts in the literature to study this problem have been hindered by the use of Laplace transforms, which pose severe numerical difficulties. Our approach is based on using a discrete-time Markov chain, which bypasses the need for Laplace transforms and is numerically tractable. We perform an extensive numerical analysis of our models to demonstrate their usefulness. To the best of our knowledge, this is the first attempt in the literature to study this problem in a comprehensive manner providing numerical solutions.     

Structural interpretation and derivation of necessary and sufficient conditions for delay moments in FIFO multiserver queues

Scheller-Wolf [12] established necessary and sufficient conditions for finite stationary delay moments in stable FIFO GI/GI/s queues that incorporate the interaction between service time distribution, traffic intensity (ρ) and the number of servers in the queue. These conditions can be used to show that when the service time has finite first but infinite αth moment, s slow servers can give lower delays than one fast server. In this paper, we derive an alternative derivation of these moment results: Both upper bounds, that serve as sufficient conditions, and lower bounds, that serve as necessary conditions are presented. In addition, we extend the class of service time distributions for which the necessary conditions are valid. Our new derivations provide a structural interpretation of the moment bounds, giving intuition into their origin: We show that FIFO GI/GI/s delay can be represented as the minimum of (s − k) i.i.d. GI/GI/1 delays, when ρ satisfies k < ρ < k+1. AMS Subject Classification 60K25     

Some remarks about the duality relation in queues

The dual of a queue is derived by inter-changing the arrival and service processes. In this paper some general relations relating the dual queues to the parent queues have been brought out. The queue considered is of type GI/G/1. Presented by A. Rényi     

Increasing convex ordering of queue length in bulk queues

This paper considers single-server bulk queues M^(X)/G^(Y)/1 and G^(X)/M^(Y)/1. In the former queue, service times and service capacity are dependent, while in the latter queue, inter-arriving times and arriving group size are dependent. We show that stronger dependence between those leads to shorter queue lengths in the increasing convex ordering sense.     

The software catalog: Minicomputers,Fall 1983: Produced from the International Software Database,Elsevier, Amsterdam, 1983, ix + 780 pages,Dfl. 310.00

This note develops some expected value formulas for the queue GI^X/M/1 and analyzes them all computationally.     

A note on conservation laws for a multiclass service queueing system with setup times

This paper derives a conservation law for mean waiting times in a single-server multi-class service queueing system (M ^X/G/1 type queue) with setup times which may be dependent on multiple customer classes and its arrival batch size by using the work decomposition property in the queueing system with vacations.     

Diffusion Approximations for Queues with Markovian Bases

Consider a base family of state-dependent queues whose queue-length process can be formulated by a continuous-time Markov process. In this paper, we develop a piecewise-constant diffusion model for an enlarged family of queues, each of whose members has arrival and service distributions generalized from those of the associated queue in the base. The enlarged family covers many standard queueing systems with finite waiting spaces, finite sources and so on. We provide a unifying explicit expression for the steady-state distribution, which is consistent with the exact result when the arrival and service distributions are those of the base. The model is an extension as well as a refinement of the M/M/s-consistent diffusion model for the GI/G/s queue developed by Kimura [13] where the base was a birth-and-death process. As a typical base, we still focus on birth-and-death processes, but we also consider a class of continuous-time Markov processes with lower-triangular infinitesimal generators.     

On a multivariate renewal-reward process involving time delays and discounting: applications to IBNR processes and infinite server queues

This paper considers a particular renewal-reward process with multivariate discounted rewards (inputs) where the arrival epochs are adjusted by adding some random delays. Then, this accumulated reward can be regarded as multivariate discounted Incurred But Not Reported claims in actuarial science and some important quantities studied in queueing theory such as the number of customers in \(G/G/\infty \) queues with correlated batch arrivals. We study the long-term behaviour of this process as well as its moments. Asymptotic expressions and bounds for quantities of interest, and also convergence for the distribution of this process after renormalization, are studied, when interarrival times and time delays are light tailed. Next, assuming exponentially distributed delays, we derive some explicit and numerically feasible expressions for the limiting joint moments. In such a case, for an infinite server queue with a renewal arrival process, we obtain limiting results on the expectation of the workload, and the covariance of queue size and workload. Finally, some queueing theoretic applications are provided.     

Managing marketing information: Nigel Piercy and Martin Evans Croom Helm,London, 1983, 223 pages

This note reports in a unified way some analytic results on the queuing system GI^X/M/1. It also corrects some erroneous results reported by Jensen and Paulson, and Krakowski.