M/M/c Retrial Queue with Multiclass of Customers

We consider the M/M/c retrial queues with multiclass of customers. We show that the stationary joint distribution for the number of customers in service facility and orbit converges to those of the ordinary M/M/c with discriminatory random order service (DROS) policy as retrial rate tends to infinity. Approximation formulae for the distributions of the number of customers in service facility, the mean number of customers in orbit and the sojourn time distribution of a customer are presented. The approximations are compared with exact and simulation results.     

Comparisons between observable and unobservable M/M/1 queues with respect to optimal customer behavior

We consider an M/M/1 queueing system in which the queue length may or may not be observable by a customer upon entering the system. The “observable” and “unobservable” models are compared with respect to system properties and performance measures under two different types of optimal customer behavior, which we refer to as “selfishly optimal” and “socially optimal”. We consider average customer throughput rates and show that, under both types of optimal customer behavior, the equality of effective queue-joining rates between the observable and unobservable systems results in differences with respect to other performance measures such as mean busy periods and waiting times. We also show that the equality of selfishly optimal queue-joining rates between the two types of system precludes the equality of socially optimal joining rates, and vice versa.     

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.     

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.     

Approximation of M/M/c retrial queue with PH-retrial times

We consider the M/M/c retrial queues with PH-retrial times. Approximation formulae for the distribution of the number of customers in service facility and the mean number of customers in orbit are presented. Some numerical results are presented.     

The MAP/(PH/PH)/1 queue with self-generation of priorities and non-preemptive service

Customers arriving according to a Markovian arrival process are served at a single server facility. Waiting customers generate priority at a constant rate γ$\gamma$; such a customer waits in a waiting space of capacity 1 if this waiting space is not already occupied by a priority generated customer; else it leaves the system. A customer in service will be completely served before the priority generated customer is taken for service (non-preemptive service discipline). Only one priority generated customer can wait at a time and a customer generating into priority at that time will have to leave the system in search of emergency service elsewhere. The service times of ordinary and priority generated customers follow PH-distributions. The matrix analytic method is used to compute the steady state distribution. Performance measures such as the probability of n consecutive services of priority generated customers, the probability of the same for ordinary customers, and the mean waiting time of a tagged customer are found by approximating them by their corresponding values in a truncated system. All these results are supported numerically.     

Transient analysis of an M/G/1 retrial queue subject to disasters and server failures

An M/G/1 retrial queueing system with disasters and unreliable server is investigated in this paper. Primary customers arrive in the system according to a Poisson process, and they receive service immediately if the server is available upon their arrivals. Otherwise, they will enter a retrial orbit and try their luck after a random time interval. We assume the catastrophes occur following a Poisson stream, and if a catastrophe occurs, all customers in the system are deleted immediately and it also causes the server’s breakdown. Besides, the server has an exponential lifetime in addition to the catastrophe process. Whenever the server breaks down, it is sent for repair immediately. It is assumed that the service time and two kinds of repair time of the server are all arbitrarily distributed. By applying the supplementary variables method, we obtain the Laplace transforms of the transient solutions and also the steady-state solutions for both queueing measures and reliability quantities of interest. Finally, numerical inversion of Laplace transforms is carried out for the blocking probability of the system, and the effects of several system parameters on the blocking probability are illustrated by numerical inversion results.     

An M/G/1 queue with two phases of service subject to the server breakdown and delayed repair

This paper deals with the steady-state behaviour of an M/G/1 queue with an additional second phase of optional service subject to breakdowns occurring randomly at any instant while serving the customers and delayed repair. This model generalizes both the classical M/G/1 queue subject to random breakdown and delayed repair as well as M/G/1 queue with second optional service and server breakdowns. For this model, we first derive the joint distributions of state of the server and queue size, which is one of chief objectives of the paper. Secondly, we derive the probability generating function of the stationary queue size distribution at a departure epoch as a classical generalization of Pollaczek–Khinchin formula. Next, we derive Laplace Stieltjes transform of busy period distribution and waiting time distribution. Finally, we obtain some important performance measures and reliability indices of this model.     

Optimal incentive-compatible pricing for M/G/1 queues

This paper extends the applicability of the pricing results of Mendelson and Whang (Oper. Res. 38 (5) (1990) 870) and Balachandran and Radhakrishnan (Management Sci. 40 (10) (1994) 353) for congested service facilities by considering general, class-dependent, service time distributions. Two theorems for nonpreemptive M/G/1 queues and preemptive-resume M/G/1 queues are presented.     

On the Two-Class M/M/1 System under Preemptive Resume and Impatience of the Prioritized Customers

The paper deals with the two-class priority M/M/1 system, where the prioritized class-1 customers are served under FCFS preemptive resume discipline and may become impatient during their waiting for service with generally distributed maximal waiting times. The class-2 customers have no impatience. The required mean service times may depend on the class of the customer. As the dynamics of class-1 customers are related to the well analyzed M/M/1+GI system, our aim is to derive characteristics for class-2 customers and for the whole system. The solution of the balance equations for the partial probability generating functions of the detailed system state process is given in terms of the weak solution of a family of boundary value problems for ordinary differential equations, where the latter can be solved explicitly only for particular distributions of the maximal waiting times. By means of this solution formulae for the joint occupancy distribution and for the sojourn and waiting times of class-2 customers are derived generalizing corresponding results recently obtained by Choi et al. in case of deterministic maximal waiting times. The latter case is dealt as an example in our paper.     

Diffusion-equation model of slightly loaded M/M/1 queue

Another derivation of the diffusion approximation of the M/M/1 queue is presented, which results in a new boundary condition. The model proposed approximates the time-dependent behavior of the M/M/1 system for all values of channel utilization.     

Perishable Inventory System at Service Facilities with N Policy

Abstract

This article presents a perishable stochastic inventory system under continuous review at a service facility in which the waiting hall for customers is of finite size M. The service starts only when the customer level reaches N (< M), once the server has become idle for want of customers. The maximum storage capacity is fixed as S. It is assumed that demand for the commodity is of unit size. The arrivals of customers to the service station form a Poisson process with parameter λ. The individual customer is issued a demanded item after a random service time, which is distributed as negative exponential. The items of inventory have exponential life times. It is also assumed that lead time for the reorders is distributed as exponential and is independent of the service time distribution. The demands that occur during stock out periods are lost.The joint probability distribution of the number of customers in the system and the inventory levels is obtained in steady state case. Some measures of system performance in the steady state are derived. The results are illustrated with numerical examples.     

Waiting times for M/M systems under state-dependent processor sharing

We consider a system where the arrivals form a Poisson process and the required service times of the requests are exponentially distributed. The requests are served according to the state-dependent (Cohen’s generalized) processor sharing discipline, where each request in the system receives a service capacity which depends on the actual number of requests in the system. For this system we derive systems of ordinary differential equations for the LST and for the moments of the conditional waiting time of a request with given required service time as well as a stable and fast recursive algorithm for the LST of the second moment of the conditional waiting time, which in particular yields the second moment of the unconditional waiting time. Moreover, asymptotically tight upper bounds for the moments of the conditional waiting time are given. The presented numerical results for the first two moments of the sojourn times in M/M/m?PS systems show that the proposed algorithms work well.     

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.     

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.     

?/M/1: On the equilibrium distribution of customer arrivals

Each day a facility commences service at time zero. All customers arriving prior to time T are served during that day. The queuing discipline is First-Come First-Served. Each day, each person in the population chooses whether or not to visit the facility that day. If he decides to visit, he arrives at an instant of time such that his expected waiting time in the queue is minimal. We investigate the arrival rate of customers in equilibrium, where each customer is fully aware of the characteristics of the system. We show that the arrival rate is constant before opening time, but that in general it is not constant between opening and closing time. For the case of exponential distribution of service time, we develop a set of equations from which the equilibrium queue size distribution and expected waiting time can be numerically computed as functions of time.     

On M/G/1 system under NT policies with breakdowns,startup and closedown

This paper studies the vacation policies of an M/G/1 queueing system with server breakdowns, startup and closedown times, in which the length of the vacation period is controlled either by the number of arrivals during the vacation period, or by a timer. After all the customers are served in the queue exhaustively, the server is shutdown (deactivates) by a closedown time. At the end of the shutdown time, the server immediately takes a vacation and operates two different policies: (i) The server reactivates as soon as the number of arrivals in the queue reaches to a predetermined threshold N or the waiting time of the leading customer reaches T units; and (ii) The server reactivates as soon as the number of arrivals in the queue reaches to a predetermined threshold N or T time units have elapsed since the end of the closedown time. If the timer expires or the number of arrivals exceeds the threshold N, then the server reactivates and requires a startup time before providing the service until the system is empty. If some customers arrive during this closedown time, the service is immediately started without leaving for a vacation and without a startup time. We analyze the system characteristics for each scheme.     

MAP/(PH/PH)/c Queue with Self-Generation of Priorities and Non-Preemptive Service

Abstract

Customers arriving according to a Markovian arrival process are served at a c server facility. Waiting customers generate into priority while waiting in the system (self-generation of priorities), at a constant rate γ; such a customer is immediately taken for service, if at least one of the servers is free. Else it waits at a waiting space of capacity c exclusively for priority generated customers, provided there is vacancy. A customer in service is not preempted to accommodate a priority generated customer. The service times of ordinary and priority generated customers follow distinct PH-distributions. It is proved that the system is always stable. We provide a numerical procedure to compute the optimal number of servers to be employed to minimize the loss to the system. Several performance measures are evaluated.     

An M/G/1 system with startup server and J additional options for service

We consider an M^[x]/G/1 queueing system with a startup time, where all arriving customers demand first the essential service and some of them may further demand one of other optional services: Type 1, Type 2, … , and Type J service. The service times of the essential service and of the Type i  (i=1,2,…,J)$(i=1\text{,}2\text{,}\dots \text{,}J)$ service are assumed to be random variables with arbitrary distributions. The server is turned off each time when the system is empty. As soon as a customer or a batch of customers arrives, the server immediately performs a startup which is needed before starting each busy period. We derive the steady-state results, including system size distribution at a random epoch and at a departure epoch, the distributions of idle and busy periods, and waiting time distribution in the queue. Some special cases are also presented.