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a customer, who has turn in both queues, receives the service from one of the queues, the other turn is automatically withdrawn.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.     

A Retrial Queue with a Constant Retrial Rate,Server Downs and Impatient Customers

ABSTRACT

In this paper, we consider a retrial queueing system consisting of a waiting line of infinite capacity in front of a single server subject to breakdowns. A customer upon arrival may join the queue (waiting line) or go to the retrial orbit (another queue) to retry for service after a random time. Only the customer at the head of the retrial orbit is allowed to retry for service. Upon retrial, the customer enters the service if the server is idle; otherwise, it may go back to the retrial orbit or leave the system (become impatient). All the interarrival times, service times, server up times, server down times and retrial times are exponential, and all the necessary independence conditions in these variables are assumed. For this system, we provide sufficient conditions under which, for any given number of customers in the orbit, the stationary probability of the number of customers in the waiting line decays geometrically. We also provide explicitly an expression for the decay parameter.     

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.     

Limit theorems for single server queues with mixed renewal and poisson inputs

The following queuing system is considered:Two independent recurrent input streams (streams 1 and 2) arrive at a server. It is assumed that stream 1 is of Poisson type. Three priority disciplines are studied in case that these customers have priority:head-of-the-line, preemptive-resume, and preemptive-repeat discipline. Formulas derived for the limiting distribution functions of the actual and the virtual waiting time of low priority customers and of the number of these customers in the system, by using of independences of certain random processes when the time tends to infinity.     

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.     

The snowball effect of customer slowdown in critical many-server systems

Customer slowdown describes the phenomenon that a customer’s service requirement increases with experienced delay. In healthcare settings, there is substantial empirical evidence for slowdown, particularly when a patient’s delay exceeds a certain threshold. For such threshold slowdown situations, we design and analyze a many-server system that leads to a two-dimensional Markov process. Analysis of this system leads to insights into the potentially detrimental effects of slowdown, especially in heavy-traffic conditions. We quantify the consequences of underprovisioning due to neglecting slowdown, demonstrate the presence of a subtle bistable system behavior, and discuss in detail the snowball effect: A delayed customer has an increased service requirement, causing longer delays for other customers, who in turn due to slowdown might require longer service times.     

Approximate queue length distribution of a discriminatory processor sharing queue with impatient customers

We consider a two-class processor sharing queueing system with impatient customers. The system operates under the discriminatory processor sharing (DPS) scheduling. The arrival process of each class customers is the Poisson process and the service requirement of a customer is exponentially distributed. The reneging rate of a customer is a constant. To analyze the performance of the system, we develop a time scale decomposition approach to approximate the joint queue-length distribution of each class customers. Via a numerical experiment, we show that the time scale decomposition approach gives a fairly good approximation of the queue-length distribution and the expected queue length.     

The existence of waiting time moments for single server queues with mixed renewal and poisson inputs

The following queuing system is considered: Two independent recurrent input streams (streams 1 and 2) arrive at a server. It is assumed that stream 1 is of Poisson type. Three priority disciplines are studied in case that customers of type 1 have priority: head-of-the-line, preemptive-resume, and preemptive-repeat discipline. For all three cases, the limiting distribution function of actual waiting times of low-priority customers is considered, and conditions are given for the existence of moments related to these limiting distributions.     

Functionals of Markovian Branching D-BMAPS

Abstract

We consider a stochastic system in which Markovian customer attribute processes are initiated at customer arrivals in a discrete batch Markovian arrival process (D-BMAP). We call the aggregate a Markovian branching D-BMAP. Each customer attribute process is an absorbing discrete time Markov chain whose parameters depend both on the phase transition, of the driving D-BMAP, and the number of arrivals taking place at the customer's arrival instant. We investigate functionals of Markovian branching D-BMAPs that may be interpreted as cumulative rewards collected over time for the various customers that arrive to the system, in the transient and asymptotic cases. This is achieved through the derivation of recurrence relations for expected values and Laplace transforms in the former case, and Little's law in the latter case.     

On a general mixed priority queue with server discretion

We consider a single-server queueing system which attends to N priority classes that are classified into two distinct types: (i) urgent: classes which have preemptive resume priority over at least one lower priority class, and (ii) non-urgent: classes which only have non-preemptive priority among lower priority classes. While urgent customers have preemptive priority, the ultimate decision on whether to interrupt a current service is based on certain discretionary rules. An accumulating prioritization is also incorporated. The marginal waiting time distributions are obtained and numerical examples comparing the new model to other similar priority queueing systems are provided.     

Performance measures of M/G/1 retrial queues with recurrent customers,breakdowns, and general delays

In this paper, an M/G/1 retrial system with two classes of customers: transit and recurrent customers is studied. After service completion, recurrent customers always return to the orbit and transit customers leave the system forever. The server is subject to breakdowns and delayed repairs. The customer whose service is interrupted stays in the service, waiting for delay and repair of the server. After repair this customer completes his service. The study of the system concerns the joint generating function of the server state and the queue length in steady state. Some performance measures of the system are then derived and some numerical results are presented to illustrate the effect of the system parameters on the developed performance measures.     

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 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).     

Estimation of Measures in M/M/1 Queue

Maximum likelihood and uniform minimum variance unbiased estimators of steady-state probability distribution of system size, probability of at least ? customers in the system in steady state, and certain steady-state measures of effectiveness in the M/M/1 queue are obtained/derived based on observations on X, the number of customer arrivals during a service time. The estimators are compared using Asympotic Expected Deficiency (AED) criterion leading to recommendation of uniform minimum variance unbiased estimators over maximum likelihood estimators for some measures.     

Predicting effective customer lifetime: an application of survival analysis for telecommunication industry

Abstract

Innovation helps brands in making customer’s lives better and more meaningful. The purpose of brand management is to create an impact that makes differences to its customers. This study describes certain demographic factors that usually impact the continuing probability of customers and thereby can help management in identifying the effective lifetime of customers towards a particular offering. For this purpose the technique of Survival Analysis has been used and the data has been collected from customers of telecommunication industries. Results depicted that the attitudinal aspect of customers for continuing a particular product is significantly impacted by the factors under consideration.     

Performance analysis of the retrial queues with finite number of sources and service interruptions

This paper aims at presenting an analytic approach for investigating a single-server retrial queue with finite population of customers where the server is subject to interruptions. A free source may generate a primary call to request service. If the server is free upon arrival, the call starts to be served and the service times are independent, generally distributed random variables. During the service time the source cannot generate a new primary call. After service the source moves into the free state and can generate a new primary call. There is no waiting space in front of the server, and a call who finds the server unavailable upon arrival joins an orbit of unsatisfied customers. The server is subject to interruptions during the service processes. When the server is interrupted, the call being served just before server interruption goes to the retrial orbit and will retry its luck after a random amount of time until it finds the server available. The recovery times of the interrupted server are assumed to be generally distributed. Our analysis extends previous work on this topic and includes the analysis of the arriving customer’s distribution, the busy period, and the waiting time process.     

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.     

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.     

Unit-Root Tests and the Statistical Pitfalls of Seasonal Adjustment: The Case of U.S. Postwar Real Gross National Product

An often-used scenario in marketing is that of individuals purchasing in a Poisson manner with their purchasing rates distributed gamma across the population of customers. Ehrenberg (1959) introduced the marketing community to this story and the resulting negative binomial distribution (NBD), and during the past 30 years the NBD model has been shown to work quite well. But the basic gamma/Poisson assumptions lack some face validity. In many product categories, customers purchase more regularly than the exponential. There are some individuals who will never purchase. The purpose of this article is to review briefly the literature that addresses these and other issues. The tractable results presented arise when the basic gamma/Poisson assumptions are relaxed one issue at a time. Some conjectures will be made about the robustness of the NBD when multiple deviations occur together. The NBD may work, but there are still opportunities for working on variations of the NBD theme.