A single server feedback retrial queue with collisions

A Markovian single server feedback retrial queue with linear retrial rate and collisions of customers is studied. Using generating function technique, the joint distribution of the server state and the orbit length under steady-state is investigated. Some interesting and important performance measures of the system are obtained. Finally, numerical illustrations are provided.     

A single server queue with a varying gate mechanism

A Markovian single server queue in a varying environment with gate mechanism is studied. Exact expressions for the duration of the nth batch and the number served are obtained by exploiting the connection between queues, branching processes and continued fractions. Numerical illustrations are presented.     

On a single server queue with two-stage heterogeneous service and deterministic server vacations

We analyse a single server queue with Poisson arrivals, two stages of heterogeneous service with different general (arbitrary) service time distributions and binomial schedule server vacations with deterministic (constant) vacation periods. After first-stage service the server must provide the second stage service. However, after the second stage service, he may take a vacation or may decide to stay on in the system. For convenience, we designate our model as M/G ₁, G ₂/D/1 queue. We obtain steady state probability generating function of the queue length for various states of the server. Results for some particular cases of interest such as M/E_{k 1} , E_{k 2} /D/1, M/M ₁, M ₂/D/1, M/E _k /D/1 and M/G ₁, G ₂/1 have been obtained from the main results and some known results including M/E_k /1 and M/G/1 have been derived as particular cases of our particular cases.     

A priority queue with interruptions of service permitted after a time quantum

Summary The queueing theory is applied to analyse a model of a multiprogramming operating system in which preemptive priorities are used for scheduling the service of concurrent streams of requests. Preemptions are permitted at the end of each service quantum. Mean waiting times and higher statistical moments in the M/G/1 system analysed are computed.     

A discrete-time retrial queue with negative customers and unreliable server

This paper treats a discrete-time single-server retrial queue with geometrical arrivals of both positive and negative customers in which the server is subject to breakdowns and repairs. Positive customers who find sever busy or down are obliged to leave the service area and join the retrial orbit. They request service again after some random time. If the server is found idle or busy, the arrival of a negative customer will break the server down and simultaneously kill the positive customer under service if any. But the negative customer has no effect on the system if the server is down. The failed server is sent to repair immediately and after repair it is assumed as good as new. We analyze the Markov chain underlying the queueing system and obtain its ergodicity condition. The generating functions of the number of customers in the orbit and in the system are also obtained along with the marginal distributions of the orbit size when the server is idle, busy or down. Finally, some numerical examples show the influence of the parameters on some crucial performance characteristics of the system.     

Location of a boundary for a diffusion equation of a single server queue

A diffusion-equation model of the M/E_s/1 queue is derived from forward Kolmogorov equations for stages. The derivation results in a new boundary condition. Namely, it is established that an abscissa of a reflecting barrier equals minus squared variance coefficient of service time. That result is then obtained for E_r/M/1 and E_r/E_s/1 queues. Its validity is also illustrated for other systems.     

A numerical procedure for the selection of the constant interarrival time to a single server queue

We study a single server queue with deterministric arrivals to find the optimal interarrival time. No analytically tractable solution is available, except in the particular D/M/1 case, but under the assumption of a phase type service time distribution, the model has a tractable algorithmic solution.

The purpose of this paper is to present the algorithm for the D/PH/1 queue and to demonstrate its implementation through interactive computation. The algorithm provides enough information about the system to be useful to a variety of problems in engineering design.

Potential applications are in assembly line industrial processes, particularly those in computer-controlled, fully automated factories, and also in the selection of a good appointment system.     

Analysis of an infinite buffer system with random server interruptions

An infinite buffer with general arrival process, synchronous transmission, one single output channel and random server interruptions is considered.As opposed to previous analyses the interruption process of the output line is kept rather general, i.e. the server is assumed to be in one of two states, “available” or “blocked”, where the sojourn time of the blocked state is arbitrarily distributed and the sojourn time of the available state has a density function which is a mixture of a finite number of geometric densities. For this general case the probability generating function of the buffer occupancy at various time instants is derived.The results of the study are applied to the case where the server interruptions are due to the presence of speech at the input of the transmission channel of an integrated voice-data system. Some considerable deviations from earlier results are found.     

Single server retrial queue with group admission of customers

We consider a retrial queueing system with a single server and novel customer׳s admission discipline. The input flow is described by a Markov Arrival Process. If an arriving customer meets the server providing the service, it goes to the orbit and repeats attempts to get service in random time intervals whose duration has exponential distribution with parameter dependent on the customers number in orbit. Server operates as follows. After a service completion epoch, customers admission interval starts. Duration of this interval has phase type distribution. During this interval, primary customers and customers from the orbit are accepted to the pool of customers which will get service after the admission interval. Capacity of this pool is limited and after the moment when the pool becomes full before completion of admission interval all arriving customers move to the orbit. After completion of an admission interval, all customers in the pool are served simultaneously by the server during the time having phase type distribution depending on the customers number in the pool. Using results known for Asymptotically Quasi-Toeplitz Markov Chains, we derive stability condition of the system, compute the stationary distribution of the system states, derive formulas for the main performance measures and numerically show advantages of the considered customer׳s admission discipline (higher throughput, smaller average number of customers in the system, higher probability to get a service without visiting the orbit) in case of proper choice of the capacity of the pool and the admission period duration.     

Stability analysis of multiserver discrete-time queueing systems with renewal-type server interruptions

For many queueing systems, the server is not continuously available. Service interruptions may result from repair times after server failures, planned maintenance periods or periods during which customers from other queues are being served. These service interruptions cause an overall performance degradation which is most striking when interruptions can start while a customer is being served and his service has to start all over after the interruption. This is the so-called preemptive repeat service discipline. This paper investigates stability conditions for discrete-time queueing systems with preemptive server interruptions. Under renewal assumptions for arrival, service and interruption processes, sufficient conditions for the positive recurrence of the single-server and multiserver queueing processes are established for the preemptive repeat different and the preemptive resume service disciplines.     

Switched poisson process/g/1 queue with service interruptions

In this paper, we develop an expression for the expected waiting time in a single server queueing system subject to interruptions with alternately varying Poisson arrival and renewal service rates. This queueing system is useful to model situations in production, computer and telecommunication systems in which customer arrivals and service requirements differ depending on whether the server is working or not. We develop an expression for the expected waiting time by approximating the virtual delay process by a Brownian motion. Our approximation for the expected waiting time involves only the means and variances and does not depend on any assumptions regarding the interarrival, service or switching time distributions. We present simulation results to illustrate the quality of our approximations.     

A batch arrival queue under randomised multi-vacation policy with unreliable server and repair

This article examines an M^[x]/G/1 queueing system with an unreliable server and a repair, in which the server operates a randomised vacation policy with multiple available vacations. Upon the system being found to be empty, the server immediately takes a vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with probability p or leaves for another vacation with probability 1???p. When one or more customers arrive when the server is idle, the server immediately starts providing service for the arrivals. It is possible that an unpredictable breakdown may occur in the server, in which case a repair time is requested. For such a system, we derive the distributions of several important system characteristics, such as the system size distribution at a random epoch and at a departure epoch, the system size distribution at the busy period initiation epoch, and the distribution of the idle and busy periods. We perform a numerical analysis for changes in the system characteristics, along with changes in specific values of the system parameters. A cost effectiveness maximisation model is constructed to show the benefits of such a queueing system.     

Discrete-time queues with generally distributed service times and renewal-type server interruptions

In this contribution, we investigate a discrete-time single-server queue subjected to server interruptions. Server interruptions are modeled as an on/off process with geometrically distributed on-periods and generally distributed off-periods. As message lengths can exceed one time-slot, different operation modes are considered, depending on whether service of an interrupted message continues, partially restarts or completely restarts after an interruption. For all alternatives, we establish expressions for the steady-state probability generating functions (pgf) of the buffer contents at message departure times and random slot boundaries, of the unfinished work at random slot boundaries, the message delay, and the lengths of the idle and busy periods. From these results, closed-form expressions for various performance measures, such as mean and variance of the buffer occupancy and message delay, can be established. As an application, we show that this model is able to assess performance of a multi-class priority scheduling system. We then illustrate our approach with some numerical examples.     

Cost-minimization analysis of a working vacation queue with N-policy and server breakdowns

This paper investigates the N-policy M/M/1 queueing system with working vacation and server breakdowns. As soon as the system becomes empty, the server begins a working vacation. The server works at a lower service rate rather than completely stopping service during a vacation period. The server may break down with different breakdown rates during the idle, working vacation, and normal busy periods. It is assumed that service times, vacation times, and repair times are all exponentially distributed. We analyze this queueing model as a quasi-birth–death process. Furthermore, the equilibrium condition of the system is derived for the steady state. Using the matrix-geometric method, we find the matrix-form expressions for the stationary probability distribution of the number of customers in the system and system performance measures. The expected cost function per unit time is constructed to determine the optimal values of the system decision variables, including the threshold N and mean service rates. We employ the particle swarm optimization algorithm to solve the optimization problem. Finally, numerical results are provided, and an application example is given to demonstrate the applicability of the queueing model.     

Studying the queue length in a single-server queuing system with unreliable server

The unreliable queuing system was studied in both the nonstationary and stationary modes. It was assumed that a Poisson flow of customers arrives to the system; the times of customer servicing and restoration of the server (servicing system) are random variables with arbitrary distributions; the flow of stable failures makes up the restoration process; the interfailure intervals are distributed hyperexponentially; and the waiting and sojourn times as well as the queue length are not limited.Translated from Avtomatika i Telemekhanika, No. 1, 2005, pp. 72–81.Original Russian Text Copyright © 2005 by Mikadze, Khocholava.This paper was recommended for publication by V.V. Rykov, a member of the Editorial Board     

Optimal allocation of customers in a two server queue withresequencing

The problem of optimal allocation of customers in a two server queue with heterogeneous service rates and resequencing is addressed. The resequencing constraint ensures that the customers leave the system in the order in which they entered it. It is shown that the optimal policy that minimizes the average end-to-end delay of customers in the system is independent of the number of customers in the resequencing buffer. It is also shown that the faster server should be kept busy whenever possible     

The optimal service time allocation of a versatile server to queue jobs and stochastically available non-queue jobs of different types

In this paper, we consider a service system in which the server can process N+1$N+1$ different types of jobs. Jobs of type 0 are generated randomly according to a Poisson stream. Jobs of types 1 to N are non-queue types which may or may not be available for completion by the server. To optimally allocate the server's time to these jobs, we formulate a finite state semi-Markov decision process model to this environment. With this model, the optimal stationary policies can be numerically determined via a policy-iteration algorithm. We also discuss the practical applications of this model to tele-service and tele-marketing operations.     

Cost analysis of a finite capacity queue with server breakdowns and threshold-based recovery policy

This paper analyzes a repairable M/M/1/N queueing system under a threshold-based recovery policy. The threshold-based recovery policy means that the server breaks down only if there is at least one customer in the system, and the recovery can be performed when q (1 ≤ q ≤ N) or more customers are present. For this queueing system, a recursive method is applied to obtain steady-state solutions in neat closed-form expressions. We then develop some important system characteristics, such as the number of customers in the system, the probability that the server is busy, the effective arrival rate and the expected waiting time in the system, etc. A cost model is constructed to determine the optimal threshold value, the optimal system capacity and the optimal service rate at a minimum cost. In order to solve this optimization problem, the direct search method and Newton's method are employed. Sensitivity analysis is also conducted with various system parameters. Finally, we present some managerial insights through an application example.