The Discrete-Time GI/Geo/1 Queue with Multiple Vacations

We study a discrete-time GI/Geo/1 queue with server vacations. In this queueing system, the server takes vacations when the system does not have any waiting customers at a service completion instant or a vacation completion instant. This type of discrete-time queueing model has potential applications in computer or telecommunication network systems. Using matrix-geometric method, we obtain the explicit expressions for the stationary distributions of queue length and waiting time and demonstrate the conditional stochastic decomposition property of the queue length and waiting time in this system.     

Analysis of the GI/Geo/1 queue with N-policy

We consider a discrete-time single server N  -policy GI/Geo/1$\mathit{GI}/\mathit{Geo}/1$ queueing system. The server stops servicing whenever the system becomes empty, and resumes its service as soon as the number of waiting customers in the queue reaches N. Using an embedded Markov chain and a trial solution approach, the stationary queue length distribution at arrival epochs is obtained. Furthermore, we obtain the stationary queue length distribution at arbitrary epochs by using the preceding result and a semi-Markov process. The sojourn time distribution is also presented.     

A discrete-time Geo[X]/G/1 retrial queue with control of admission

This paper analyses a discrete-time Geo/G/1 retrial queue with batch arrivals in which individual arriving customers have a control of admission. We study the underlying Markov chain at the epochs immediately after the slot boundaries making emphasis on the computation of its steady-state distribution. To this end we employ numerical inversion and maximum entropy techniques. We also establish a stochastic decomposition property and prove that the continuous-time M/G/1 retrial queue with batch arrivals and control of admission can be approximated by our discrete-time system. The outcomes agree with known results for special cases.     

Algorithmic approximations for the busy period distribution of the M/M/c retrial queue

In this paper we deal with the main multiserver retrial queue of M/M/c type with exponential repeated attempts. This model is known to be analytically intractable due to the spatial heterogeneity of the underlying Markov chain, caused by the retrial feature. For this reason several models have been proposed for approximating its stationary distribution, that lead to satisfactory numerical implementations. This paper extends these studies by developing efficient algorithmic procedures for calculating the busy period distribution of the main approximation models of Wilkinson [Wilkinson, R.I., 1956. Theories for toll traffic engineering in the USA, The Bell System Technical Journal 35, 421–514], Falin [Falin, G.I., 1983. Calculations of probability characteristics of a multiline system with repeated calls, Moscow University Computational Mathematics and Cybernetics 1, 43–49] and Neuts and Rao [Neuts, M.F., Rao, B.M., 1990. Numerical investigation of a multiserver retrial model, Queueing Systems 7, 169–190]. Moreover, we develop stable recursive schemes for the computation of the busy period moments. The corresponding distributions for the total number of customers served during a busy period are also studied. Several numerical results illustrate the efficiency of the methods and reveal interesting facts concerning the behavior of the M/M/c retrial queue.     

The N threshold policy for the GI/M/1 queue

We study a GI/M/1 queue with an N threshold policy. In this system, the server stops attending the queue when the system becomes empty and resumes serving the queue when the number of customers reaches a threshold value N. Using the embeded Markov chain method, we obtain the stationary distributions of queue length and waiting time and prove the stochastic decomposition properties.     

A Discrete-Time Geo /G/1 Retrial Queue with General Retrial Times

We consider a discrete-time Geo/G/1 retrial queue in which the retrial time has a general distribution and the server, after each service completion, begins a process of search in order to find the following customer to be served. We study the Markov chain underlying the considered queueing system and its ergodicity condition. We find the generating function of the number of customers in the orbit and in the system. We derive the stochastic decomposition law and as an application we give bounds for the proximity between the steady-state distributions for our queueing system and its corresponding standard system. Also, we develop recursive formulae for calculating the steady-state distribution of the orbit and system sizes. Besides, we prove that the M/G/1 retrial queue with general retrial times can be approximated by our corresponding discrete-time system. Finally, we give numerical examples to illustrate the effect of the parameters on several performance characteristics.     

Tail asymptotics for the queue size distribution in a discrete-time Geo/G/1 retrial queue

We consider a discrete-time Geo/G/1 retrial queue where the service time distribution has a finite exponential moment. We show that the tail of the queue size distribution is asymptotically geometric. Remarkably, the result is inconsistent with the corresponding result in the continuous-time counterpart, the M/G/1 retrial queue, where the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function.     

A Discrete-Time Geo/G/1 retrial queue with the server subject to starting failures

This paper studies a discrete-time Geo/G/1 retrial queue where the server is subject to starting failures. We analyse the Markov chain underlying the regarded queueing system and present some performance measures of the system in steady-state. Then, we give two stochastic decomposition laws and find a measure of the proximity between the system size distributions of our model and the corresponding model without retrials. We also develop a procedure for calculating the distributions of the orbit and system size as well as the marginal distributions of the orbit size when the server is idle, busy or down. Besides, we prove that the M/G/1 retrial queue with starting failures can be approximated by its discrete-time counterpart. Finally, some numerical examples show the influence of the parameters on several performance characteristics. This work is supported by the DGINV through the project BFM2002-02189.     

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.

     

Performance analysis of a discrete-time Geo/G/1 queue with randomized vacations and at most J vacations

This paper presents the analysis of a discrete-time Geo/G/1$\mathit{Geo}/G/1$ queue with randomized vacations. Using the probability decomposition theory and renewal process, two variants on this model, namely the late arrival system with delayed access (LAS-DA) and early arrival system (EAS), have been examined. For both the cases, recursive solution for queue length distributions at arbitrary, just before a potential arrival, pre-arrival, immediately after potential departure, and outside observer’s observation epochs are obtained. Further, various performance measures such as potential blocking probability, turned-on period, turned-off period, vacation period, expected length of the turned-on circle period, average queue length and sojourn time, etc. have been presented. It is hoped that the results obtained in this paper may provide useful information to designers of telecommunication systems, practitioners, and others.     

Stationary Distributions of GI/M/c Queue with PH Type Vacations

We study a GI/M/c type queueing system with vacations in which all servers take vacations together when the system becomes empty. These servers keep taking synchronous vacations until they find waiting customers in the system at a vacation completion instant.The vacation time is a phase-type (PH) distributed random variable. Using embedded Markov chain modeling and the matrix geometric solution methods, we obtain explicit expressions for the stationary probability distributions of the queue length at arrivals and the waiting time. To compare the vacation model with the classical GI/M/c queue without vacations, we prove conditional stochastic decomposition properties for the queue length and the waiting time when all servers are busy. Our model is a generalization of several previous studies.     

Analysis of an M/G/∞ queue with batch arrivals and batch-dedicated servers

We analyze an M/G/∞ queue with batch arrivals, where jobs belonging to a batch have to be processed by the same server. The number of jobs in the system is characterized as a compound Poisson random variable through a scaling of the original arrival and batch size processes.     

On the GI/M/1/N queue with multiple working vacations—analytic analysis and computation

We consider finite buffer single server GI/M/1 queue with exhaustive service discipline and multiple working vacations. Service times during a service period, service times during a vacation period and vacation times are exponentially distributed random variables. System size distributions at pre-arrival and arbitrary epoch with some important performance measures such as, probability of blocking, mean waiting time in the system etc. have been obtained. The model has potential application in the area of communication network, computer systems etc. where a single channel is allotted for more than one source.     

Fluid model driven by an M/G/1 queue with multiple exponential vacations

This paper studies a fluid model driven by an M/G/1 queue with multiple exponential vacations. By introducing various vacation strategies to the fluid model, we can provide greater flexibility for the design and control of input rate and output rate. The Laplace transform of the steady-state distribution of the buffer content is expressed through the minimal positive solution to a crucial equation. Then the performance measure-mean buffer content, which is independent of the vacation parameter, is obtained. Finally, with some numerical examples, the parameter effect on the mean buffer content is presented.     

Discrete-Time Geo X /G/1 Queue with Preemptive Repeat Different Priority

This paper considers a discrete-time Geo ^X /G/1 queue accepting two classes of messages with preemptive repeat different priority. Service times of messages of each priority class are i.i.d. according to a general discrete distribution function that may differ between two classes. The completion time and the stability condition for our system are investigated. By using the supplementary variable method and the generating function technique, we derive the joint system contents distributions at various observation instants and also compute the probability distribution for the unfinished work.     

Strong Stability of the Batch Arrival Queueing Systems

In this work, we use the strong stability method to study the batch arrival queue after a perturbation of the batch size distribution. We show that, under some hypotheses, the characteristics of the batch arrival queueing system M ^X /M/1 may be approximated by the correspondent characteristics of the system M ^Geo /M/1.

After clarifying the conditions of approximation, we obtain stability inequalities with an exact computation of constants.     

Discrete-time queue with unreliable server and multiple adaptive delayed vacations

In this paper we consider a discrete-time Geo^X/G/1 queue with unreliable server and multiple adaptive delayed vacations policy in which the vacation time, service time, repair time and the delayed time all follow arbitrary discrete distribution. By using a concise decomposition method, the transient and steady-state distributions of the queue length are studied, and the stochastic decomposition property of steady-state queue length has been proved. Several common vacation policies are special cases of the vacation policy presented in this study. The relationship between the generating functions of steady-state queue length at departure epoch and arbitrary epoch is obtained. Finally, we give some numerical examples to illustrate the effect of the parameters on several performance characteristics.     

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.     

Analysis of Queueing Systems with Synchronous Single Vacation for Some Servers

We study a multi-server M/M/c type queue with a single vacation policy for some idle servers. In this queueing system, if at a service completion instant, any d (d c) servers become idle, these d servers will take one and only one vacation together. During the vacation of d servers, the other c–d servers do not take vacation even if they are idle. Using a quasi-birth-and-death process and the matrix analytic method, we obtain the stationary distribution of the system. Conditional stochastic decomposition properties have been established for the waiting time and the queue length given that all servers are busy.     

Exact Convergence Rate for the Distributions of GI/M/c/K Queue as K Tends to Infinity

We obtain the exact convergence rate of the stationary distribution ^(K) of the embedded Markov chain in GI/M/c/K queue to the stationary distribution of the embedded Markov chain in GI/M/c queue as K. Similar result for the time-stationary distributions of queue size is also included. These generalize Choi and Kim's results of the case c=1 by nontrivial ways. Our results also strengthen the Simonot's results [5].