Level-Phase Independence for Fluid Queues

ABSTRACT

A Markov-modulated fluid queue is a two-dimensional Markov process; the first dimension is continuous and is usually called the level, and the second is the state of a Markov process that determines the evolution of the level, it is usually called the phase. We show that it is always possible to modify the transition rules at the boundary level of the fluid queue in order to obtain independence between the level and the phase under the stationary distribution. We obtain this result by exploiting the similarity between fluid queues and Quasi-Birth-and-Death (QBD) processes.     

A MAP/G/1 Queue with an Underlying Birth–Death Process

Abstract

In this paper, we define a birth–death‐modulated Markovian arrival process (BDMMAP) as a Markovian arrival process (MAP) with an underlying birth–death process. It is proved that the zeros of det(zI ? A(z)) in the unit disk are real and simple. In order to analyze a BDMMAP/G/1 queue, two spectral methods are proposed. The first one is a bisection method for calculation of the zeros from which the boundary vector is derived. The second one is the Fourier inversion transform of the probability generating function for the calculation of the stationary probability distribution of the queue length. Eigenvalues required in this calculation are obtained by the Duran–Kerner–Aberth (DKA) method. For numerical examples, the stationary probability distribution of the queue length is calculated by using the spectral methods. Comparisons of the spectral methods with the currently best methods available are discussed.     

BMAP/G/1 Queue Under D-Policy: Queue Length Analysis

ABSTRACT

We study the queue length distribution of a queueing system with BMAP arrivals under D-policy. The idle server begins to serve the customers only when the sum of the service times of all waiting customers exceeds some fixed threshold D. We derive the vector generating functions of the queue lengths both at a departure and at an arbitrary point of time. Mean queue lengths are derived and a numerical example is presented.     

An RG-Factorization Approach for a BMAP/M/1 Generalized Processor-Sharing Queue

ABSTRACT

In this paper, we study a BMAP/M/1 generalized processor-sharing queue. We propose an RG-factorization approach, which can be applied to a wider class of Markovian block-structured processor-sharing queues. We obtain the expressions for both the distribution of the stationary queue length and the Laplace transform of the sojourn time distribution. From these two expressions, we develop an algorithm to compute the mean and variance of the sojourn time approximately.     

A Tandem Queue with Server Slow-Down and Blocking

Abstract

We consider two variants of a two-station tandem network with blocking. In both variants the first server ceases to work when the queue length at the second station hits a ‘blocking threshold.’ In addition, in variant 2 the first server decreases its service rate when the second queue exceeds a ‘slow-down threshold, ’ which is smaller than the blocking level. In both variants the arrival process is Poisson and the service times at both stations are exponentially distributed. Note, however, that in case of slow-downs, server 1 works at a high rate, a slow rate, or not at all, depending on whether the second queue is below or above the slow-down threshold or at the blocking threshold, respectively. For variant 1, i.e., only blocking, we concentrate on the geometric decay rate of the number of jobs in the first buffer and prove that for increasing blocking thresholds the sequence of decay rates decreases monotonically and at least geometrically fast to max{ρ₁, ρ₂}, where ρ _i is the load at server i. The methods used in the proof also allow us to clarify the asymptotic queue length distribution at the second station. Then we generalize the analysis to variant 2, i.e., slow-down and blocking, and establish analogous results.     

Queue Length Analysis of MAP/G/1 Queue Under D-Policy

Abstract

We study the queue length distribution of a queueing system with MAP arrivals under D-policy. The idle server begins to serve the customers only when the sum of the service times of all waiting customers exceeds some fixed threshold D. We derive the vector generating functions of the queue lengths both at a departure and at an arbitrary point of time. Mean queue lengths will be derived from these transform results. A numerical example is provided.     

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.     

A Two‐Queue Polling Model with Regularly Varying Service and/or Switchover Times

Abstract

We consider the cyclic polling system with two queues. One queue is severed according to the exhaustive discipline, and the other queue is served according to the 1‐limited discipline. At least one of the service and/or switchover times has a regularly varying tail. We obtain the tail behavior of the waiting time distributions. When one of the service and/or switchover times has an infinite second moment, we derive the heavy‐traffic behavior of the waiting time distribution at the 1‐limited queue.     

Approximating M/G/1 Waiting Time Tail Probabilities

Abstract

We propose a new approximation formula for the waiting time tail probability of the M/G/1 queue with FIFO discipline and unlimited waiting space. The aim is to address the difficulty of obtaining good estimates when the tail probability has non-exponential asymptotics. We show that the waiting time tail probability can be expressed in terms of the waiting time tail probability of a notional M/G/1 queue with truncated service time distribution plus the tail probability of an extreme order statistic. The Cramér–Lundberg approximation is applied to approximate the tail probability of the notional queue. In essence, our technique extends the applicability of the Cramér–Lundberg approximation to cases where the standard Lundberg condition does not hold. We propose a simple moment-based technique for estimating the parameters of the approximation; numerical results demonstrate that our approximation can yield very good estimates over the whole range of the argument.     

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.     

Transient Analysis of Averaged Queue Length in Markovian Queues

Abstract

We consider a Markovian queue and its associated exponentially averaged length. The set of partial differential equations satisfied by the joint distribution of the queue and the averaged queue length is given. We obtain a recursive expression for the moments of the averaged queue length, and develop a stable algorithm to compute them. These results are illustrated through numerical examples.     

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.     

Lindley first-order autoregressive model with applications

ABSTRACT

A new stationary first-order autoregressive process with Lindley marginal distribution, denoted as LAR(1) is introduced. We derive the probability function for the innovation process. We consider many properties of this process, involving spectral density, some multi-step ahead conditional measures, run probabilities, stationary solution, uniqueness and ergodicity. We estimate the unknown parameters of the process using three methods of estimation and investigate properties of the estimators with some numerical results to illustrate them. Some applications of the process are discussed to two real data sets and it is shown that the LAR(1) model fits better than other known non Gaussian AR(1) models.     

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.     

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.     

AN INFINITE-PHASE QUASI-BIRTH-AND-DEATH MODEL FOR THE NON-PREEMPTIVE PRIORITY M/PH/1 QUEUE

This paper considers a single server queue that handles arrivals from N classes of customers on a non-preemptive priority basis. Each of the N classes of customers features arrivals from a Poisson process at rate λ _i and class-dependent phase type service. To analyze the queue length and waiting time processes of this queue, we derive a matrix geometric solution for the stationary distribution of the underlying Markov chain. A defining characteristic of the paper is the fact that the number of distinct states represented within the sub-level is countably infinite, rather than finite as is usually assumed. Among the results we obtain in the two-priority case are tractable algorithms for the computation of both the joint distribution for the number of customers present and the marginal distribution of low-priority customers, and an explicit solution for the marginal distribution of the number of high-priority customers. This explicit solution can be expressed completely in terms of the arrival rates and parameters of the two service time distributions. These results are followed by algorithms for the stationary waiting time distributions for high- and low-priority customers. We then address the case of an arbitrary number of priority classes, which we solve by relating it to an equivalent three-priority queue. Numerical examples are also presented.     

Asymptotic Throughput in Discrete‐Time Cyclic Networks with Queue‐Length‐Dependent Service Rates

Abstract

For a discrete‐time closed cyclic network of single server queues whose service rates are non‐decreasing in the queue length, we compute the queue‐length distribution at each node in terms of throughputs of related networks. For the asymptotic analysis, we consider sequences of networks where the number of nodes grows to infinity, service rates are taken only from a fixed finite set of non‐decreasing sequences, the ratio of customers to nodes has a limit, and the proportion of nodes for each possible service‐rate sequence has a limit. Under these assumptions, the asymptotic throughput exists and is calculated explicitly. Furthermore, the asymptotic queue‐length distribution at any node can be obtained in terms of the asymptotic throughput. The asymptotic throughput, regarded as a function of the limiting customer‐to‐node ratio, is strictly increasing for ratios up to a threshold value (possibly infinite) and is constant thereafter. For ratios less than the threshold, the asymptotic queue‐length distribution at each node has finite moments of all orders. However, at or above the threshold, bottlenecks (nodes with asymptotically‐infinite mean queue length) do occur, and we completely characterize such nodes.     

Two-queue polling systems with switching policy based on the queue that is not being served

We study a system of two non-identical and separate M/M/1/? queues with capacities (buffers) C₁ < ∞ and C₂ = ∞, respectively, served by a single server that alternates between the queues. The server’s switching policy is threshold-based, and, in contrast to other threshold models, is determined by the state of the queue that is not being served. That is, when neither queue is empty while the server attends Q_i (i = 1, 2), the server switches to the other queue as soon as the latter reaches its threshold. When a served queue becomes empty we consider two switching scenarios: (i) Work-Conserving, and (ii) Non-Work-Conserving. We analyze the two scenarios using Matrix Geometric methods and obtain explicitly the rate matrix R, where its entries are given in terms of the roots of the determinants of two underlying matrices. Numerical examples are presented and extreme cases are investigated.     

Matrix Product-Form Solutions for LCFs Preemptive Service Single-Server Queues with Batch Markovian Arrival Streams

ABSTRACT

This paper considers three variants of last-come first-served (LCFS) preemptive service single-server queues, where customers are served under the LCFS preemptive resume (LCFS-PR), preemptive repeat-different (LCFS-PD), and preemptive repeat-identical (LCFS-PI) disciplines, respectively. These LCFS queues are fed by multiple batch Markovian arrival streams. Service times of customers from each arrival stream are generally distributed and their distributions may differ among different streams. For each of LCFS-PR, LCFS-PD, and LCFS-PI queues, we show that the stationary distribution of the queue string representing enough information to keep track of queueing dynamics has a matrix product-form solution. Further, this paper discusses the stability of LCFS-PD and LCFS-PI queues based on the busy cycle. Finally, by numerical experiment, we examine the impact of the variation of the service time distribution on the mean queue lengths for the three variants of LCFS queues.