Sojourn time in an M/M/1 processor sharing queue with permanent customers

We explicitly compute the sojourn time distribution of an arbitrary customer in an M/M/1 processor sharing (PS) queue with permanent customers. We notably exhibit the orthogonal structure associated with this queuing system and we show how sieved Pollaczek polynomials and their associated orthogonality measure can be used to obtain an explicit representation for the complementary cumulative distribution function of the sojourn time of a customer. This explicit formula subsequently allows us to compute the two first moments of this random variable and to study the asymptotic behavior of its distribution. The most salient result is that the decay rate depends on the load of the system and the number K of permanent customers. When the load is above a certain threshold depending on K, the decay rate is identical to that of a regular M/M/1 PS queue.     

Stationary distributions of the R[X]/R/1 cross-correlated queue

We consider an infinite buffer single server queue wherein batch interarrival and service times are correlated having a bivariate mixture of rational (R) distributions, where R denotes the class of distributions with rational Laplace–Stieltjes transform (LST), i.e., ratio of a polynomial of degree at most n to a polynomial of degree n. The LST of actual waiting time distribution has been obtained using Wiener–Hopf factorization of the characteristic equation. The virtual waiting time, idle period (actual and virtual) distributions, as well as inter-departure time distribution between two successive customers have been presented. We derive an approximate stationary queue-length distribution at different time epochs using the Markovian assumption of the service time distribution. We also derive the exact steady-state queue-length distribution at an arbitrary epoch using distributional form of Little’s law. Finally, some numerical results have been presented in the form of tables and figures.     

SOJOURN TIMES IN NON-HOMOGENEOUS QBD PROCESSES WITH PROCESSOR SHARING

We study sojourn times of customers in a processor sharing queue with a service rate that varies over time, depending on the number of customers and on the state of a random environment. An explicit expression is derived for the Laplace–Stieltjes transform of the sojourn time conditional on the state upon arrival and the amount of work brought into the system. Particular attention is paid to the conditional mean sojourn time of a customer as a function of his required amount of work, and we establish the existence of an asymptote as the amount of work tends to infinity. The method of random time change is then extended to include the possibility of a varying service rate. By means of this method, we explain the well-established proportionality between the conditional mean sojourn time and required amount of work in processor sharing queues without random environment. Based on numerical experiments, we propose an approximation for the conditional mean sojourn time. Although first presented for exponentially distributed service requirements, the analysis is shown to extend to phase-type services. The service discipline of discriminatory processor sharing is also shown to fall within the framework.     

Analysis of a two incongruent arrivals and services M/G/1 queue with random feedback

A queuing system with two incongruent arrivals and services is considered. Two kinds of customers enter the system by Poisson process and the service times are assumed to have general distribution. After first kind service completion, it may feedback to repeat the first service, leave the system or go to give second service. The same policy is applied for the other kind of customer. All stochastic processes involved in this system are independent. We derive the probability generating function for each kind and for the system that yield the performance measures. Some numerical approaches examined the validity of the results.     

Stationary analysis of a BMAP/R/1 queue with R-type multiple working vacations

We consider an infinite-buffer single server queue with batch Markovian arrival process (BMAP) and exhaustive service discipline under multiple working vacation policy. The service time during a working vacation is generally distributed random variable which is independent of the service times during a normal busy period as well as the arrival process. Duration of service times during a normal busy period and duration of working vacation times follow the class of distributions whose Laplace-Stieltjes transforms are rational functions (R-type distributions). The service time during a normal busy period, working vacation time, and the service time during a working vacation are independent of each other as well as of the arrival process. If a working vacation terminates while service is going on for a customer at head of the queue in vacation mode then, the server switches to normal mode and the customer at head of the queue is entitled to receive a full service time in the normal busy period irrespective of the amount of service received by the customer at head of the queue during the previous working vacation period. We obtain system-length distributions at various epoch, such as post-departure, pre-arrival, arbitrary, and pre-service. The proposed analysis is based on the use of matrix-analytic procedure to obtain system-length distribution at post-departure epoch. Later, we use supplementary variable technique and simple algebraic manipulations to obtain system-length distribution at arbitrary epoch using the system-length distribution at post-departure epoch. Some important performance measures, such as mean system lengths and mean waiting time have been obtained. Finally, some numerical results have been presented in the form of tables and graphs to show the applicability of the results obtained in this article. The model has potential application in areas of computer and communication networks, such as ethernet passive optical network (EPON).     

Computational analysis of the queue with working breakdowns and delaying repair under a Bernoulli-schedule-controlled policy

This paper considers a single server queueing system with working breakdowns and delaying repair under a Bernoulli-schedule-controlled policy. At a breakdown instant, the system either goes to repair period immediately with probability p, or continues to provide auxiliary service for the current customers with probability q = 1 ? p. While the system resides in the auxiliary service period, it may go to repair period if there is no customer at the epoch of service completion or the occurrence of breakdown. By using the matrix analytic method and the spectral expansion method, we respectively obtain the steady state distribution to make the straightforward computation of performance measures and the Laplace-Stieltjes transform of the stationary sojourn time of an arbitrary customer. In addition, some numerical examples are presented to show the impact of parameters on the performance measures.     

Large deviations of multiclass M/G/1 queues

Consider a multiclass M/G/1 queue where queued customers are served in their order of arrival at a rate which depends on the customer class. We model this system using a chain with states represented by a tree. Since the service time distribution depends on the customer class, the stationary distribution is not of product form so there is no simple expression for the stationary distribution. Nevertheless, we can find a harmonic function on this chain which provides information about the asymptotics of this stationary distribution. The associated h‐transformation produces a change of measure that increases the arrival rate of customers and decreases the departure rate thus making large deviations common. The Canadian Journal of Statistics 37: 327–346; 2009 © 2009 Statistical Society of Canada     

Response Time Distribution in a D-MAP/PH/1 Queue with General Customer Impatience

ABSTRACT

This paper presents two methods to calculate the response time distribution of impatient customers in a discrete-time queue with Markovian arrivals and phase-type services, in which the customers’ patience is generally distributed (i.e., the D-MAP/PH/1 queue). The first approach uses a GI/M/1 type Markov chain and may be regarded as a generalization of the procedure presented in Van Houdt ^[14] Van Houdt , B. ; Lenin , R. B. ; Blondia , C. Delay distribution of (im)patient customers in a discrete time D-MAP/PH/1 queue with age dependent service times Queueing Systems and Applications 2003 , 45 1 , 59 – 73 . [CROSSREF]  [Google Scholar] for the D-MAP/PH/1 queue, where every customer has the same amount of patience. The key construction in order to obtain the response time distribution is to set up a Markov chain based on the age of the customer being served, together with the state of the D-MAP process immediately after the arrival of this customer. As a by-product, we can also easily obtain the queue length distribution from the steady state of this Markov chain.

We consider three different situations: (i) customers leave the system due to impatience regardless of whether they are being served or not, possibly wasting some service capacity, (ii) a customer is only allowed to enter the server if he is able to complete his service before reaching his critical age and (iii) customers become patient as soon as they are allowed to enter the server. In the second part of the paper, we reduce the GI/M/1 type Markov chain to a Quasi-Birth-Death (QBD) process. As a result, the time needed, in general, to calculate the response time distribution is reduced significantly, while only a relatively small amount of additional memory is needed in comparison with the GI/M/1 approach. We also include some numerical examples in which we apply the procedures being discussed.     

An Exact Analysis of the Multi‐class M/M/k Priority Queue with Partial Blocking

Abstract

We consider a multi‐server queuing model with two priority classes that consist of multiple customer types. The customers belonging to one priority class customers are lost if they cannot be served immediately upon arrival. Each customer type has its own Poisson arrival and exponential service rate. We derive an exact method to calculate the steady state probabilities for both preemptive and nonpreemptive priority disciplines. Based on these probabilities, we can derive exact expressions for a wide range of relevant performance characteristics for each customer type, such as the moments of the number of customers in the queue and in the system, the expected postponement time and the blocking probability. We illustrate our method with some numerical examples.     

Performance analysis of queue with two-stage vacation policy

In this paper, we investigate an M/M/1 queue with a two-stage vacation policy which comprises of single working vacation and single vacation. Using the matrix-analytic method, we obtain the distribution of stationary system size, and then the decomposition structures of the stationary system size and the sojourn time are demonstrated. Furthermore, we study the waiting time by first-passage time analysis. Meanwhile, the busy-cycle analysis is provided by the limiting theorem of alternative renewal process. Finally, several numerical examples are presented in the paper.     

Equilibrium strategies in a constant retrial queue with setup time and the N-policy

Abstract

In this article, customers’ strategic behavior and social optimation in a constant retrial queue with setup time and the N-policy are investigated. Customers who find the server isn’t idle either leave forever or enter an orbit. After a service, the server will seek a customer from the orbit at a constant rate. The server is closed whenever the system becomes empty, and is activated when the number of waitlisted customers reaches a threshold. We obtain the equilibrium arrival rates in different states. There exist both Follow-the-Crowd (FTC) and Avoid-the-Crowd (ATC) behaviors. Through the Particle Swarm Optimization (PSO) algorithm, we numerically obtain the optimal solution of the social welfare maximization problem. Finally, numerical examples are presented to illustrate the sensitivity of system performance measures.     

ANALYSIS OF A GI/G Y /1 SYSTEM IN DISCRETE-TIME

ABSTRACT

We consider a class of single server queueing systems in which customers arrive singly and service is provided in batches, depending on the number of customers waiting when the server becomes free. Service is independent of the batch size. This system could also be considered as a batch service queue in which a server visits the queue at arbitrary times and collects a batch of waiting customers for service, or waits for a customer to arrive if there are no waiting customers. A waiting server immediately collects and processes the first arriving customer. The system is considered in discrete time. The interarrival times of customers and the inter-visit times of the server, which we call the service time, have general distributions and are represented as remaining time Markov chains. We analyze this system using the matrix-geometric method and show that the resulting R matrix can be determined explicitly in some special cases and the stationary distributions are known semi-explicitly in some other special cases.     

Simultaneous arrival of customers to two different queues and modeling dependence via copula approach

ABSTRACT

In classical queueing systems, a customer is allowed to wait only in one queue to receive the service. In practice, when there exist a number of queues rendering the same service, some customers may tend to simultaneously take turn in more than one queue with the aim to receive the service sooner and thus reduce their waiting time. In this article, we introduce such a model and put forward a methodology to deal with the situation. In this regard, we consider two queues and assume that if a customer, who has turn in both queues, receives the service from one of the queues, the other turn is automatically withdrawn. This circumstance for the model brings about some abandonment in each queue as some customers receive the service from the other one. We study the customer’s waiting time in the mentioned model, which is defined as the minimum of waiting times in both queues and obtain probability density function of this random variable. Our approach to obtain probability density function of each of the waiting time random variables is to rely on the existing results for the abandonment case. We examine the situation for the cases of independence and dependence of the waiting time random variables. The latter is treated via a copula approach.     

Analysis of an unreliable MX/G1G2/1/repeated service queue with delayed repair under randomized vacation policy

Abstract

In this article we consider an unreliable M^X/G/1 queue with two types of general heterogeneous service and optional repeated service subject to server’s break down and delayed repair under randomized vacation policy. We assume that customer arrive to the system according to a compound Poisson process. The server provides two types of general heterogeneous service and a customer can choose either type of service before its service start. After the completion of either type of service, the customer has the further option to repeat the same type of service once again. While the server is working with any types of service or repeated service, it may breakdown at any instant. Further the concept of randomized vacation is also introduced. For this model, we first derive the joint distribution of state of the server and queue size by considering both elapsed and remaining time, which is one of the objective of this article. Next, we derive Laplace Stieltjes transform of busy period distribution. Finally, we obtain some important performance measure and reliability indices of this model.     

Multi-class M/PH/1 queues with deterministic impatience times

This article considers computational procedures for the waiting time and queue length distributions in stationary multi-class first-come, first-served single-server queues with deterministic impatience times. There are several classes of customers, which are distinguished by deterministic impatience times (i.e., maximum allowable waiting times). We assume that customers in each class arrive according to an independent Poisson process and a single server serves customers on a first-come, first-served basis. Service times of customers in each class are independent and identically distributed according to a phase-type distribution that may differ for different classes. We first consider the stationary distribution of the virtual waiting time and then derive numerically feasible formulas for the actual waiting time distribution and loss probability. We also analyze the joint queue length distribution and provide an algorithmic procedure for computing the probability mass function of the stationary joint queue length.     

The analysis of MX/M/1 queue with two-stage vacations policy

Abstract

In this article, we consider a batch arrival M^X/M/1 queue with two-stage vacations policy that comprises of single working vacation and multiple vacations, denoted by M^X/M/1/SWV?+?MV. Using the matrix analytic method, we derive the probability generating function (PGF) of the stationary system size and investigate the stochastic decomposition structure of stationary system size. Further, we obtain the Laplace–Stieltjes transform (LST) of stationary sojourn time of a customer by the first passage time analysis. At last, we illustrate the effects of various parameters on the performance measures numerically and graphically by some numerical examples.     

The virtual waiting time of the M/G/1 queue with customers of n types of impatience

We consider the $M/G/1$ queue in which the customers are classified into $n+1$ classes by their impatience times. First, we analyze the model with two types of customers; one is the customer with constant impatience time $k$ and the other is the patient customer whose impatience time is $\infty$. The expected busy period of the server and the limiting distribution of the virtual waiting time process are obtained. Then, the model is generalized to the one in which the impatience time of each customer is anyone in $\{k_1,k_2,\dots ,k_n,\infty \}$.     

ML and Bayes estimation in a two-phase tandem queue with a second optional service and random feedback

ABSTRACT

In this article, we consider a two-phase tandem queueing model with a second optional service and random feedback. The first phase of service is essential for all customers and after the completion of the first phase of service, any customer receives the second phase of service with probability α, feedback to the tail of the first queue with probability β if the service is not successful and leaves the system with probability 1 ? α ? β. In this model, our main purpose is to estimate the parameters of the model, traffic intensity, and mean system size, in the steady state, via maximum likelihood and Bayesian methods. Furthermore, we find asymptotic confidence intervals for mean system size. Finally, by a simulation study, we compute the confidence levels and mean length for asymptotic confidence intervals of mean system size with a nominal level 0.95.     

Analysis of a batch arrival queue with vacation policy and exceptional service

ABSTRACT

We consider the distributions of operating characteristics of an M^[x]/G/1 queue under vacation policies, where the first customer of each busy period receives an exceptional service. When all the customers are served in the system exhaustively, the server deactivates and operates one of two vacation policies: (1) multiple vacation policy and (2) single vacation policy. We develop the performance measures for both systems. Finally, some numerical illustrations are also given. These two vacation models have potential applications in day-to-day life, such as post offices, banks, hospitals, etc.