Characterizing the BMAP/MAP/1 Departure Process via the ETAQA Truncation

Abstract

We propose a family of finite approximations for the departure process of a BMAP/MAP/1 queue. The departure process approximations are derived via an exact aggregate solution technique (called ETAQA) applied to M/G/1-type Markov processes. The proposed approximations are indexed by a parameter n(n > 1), which determines the size of the output model as n + 1 block levels of the M/G/1-type process. This output approximation preserves exactly the marginal distribution of the true departure process and the lag correlations of the interdeparture times up to lag n ? 2. Experimental results support the applicability of the proposed approximation in traffic-based decomposition of queueing networks.     

A computational algorithm for the loss probability in the M/G/1+PH queue

This article develops a computational algorithm for the loss probability in the stationary M/G/1 queue with impatient customers whose impatience times follow a phase-type distribution (M/G/1+PH). The algorithm outputs the loss probability, along with an upper-bound of its numerical error due to truncation, and it is readily applicable to the M/D/1+PH, M/PH/1+PH, and M/Pareto/1+PH queues.     

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.     

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.     

REMARKS ON OPTIMAL INFERENCE FOR MARKOV BRANCHING PROCESSES: A SEQUENTIAL APPROACH

Use of a suitable stopping rule yields exact uniformly most powerful tests and minimum variance unbiased estimators of various parameters of a Markov branching model with or without immigration. The population model discussed includes the pure birth, simple epidemic, immigration-death, M/M/ 1 queue, linear birth-death and a branching diffusion process, among others, as special cases.     

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.     

A transient analysis of M/G/1 queues with N-policy

We consider an M/G/1 queueing model with N-policy operating. This means, that the server will start up only if a queue of a prescribed length has built up. For this model the time dependent distribution of the queue length is given by simple renewal arguments without resorting to integral transform techniques.     

Lattice paths combinatorics applied to transient queue length distribution of C2/M/1 queues and busy period analysis of bulk queues C2/M/1

This paper aims at deriving explicit transient queue length distribution for GI/M/1 system and busy period analysis of bulk queue GI^b/M/1 through lattice paths (LPs) combinatorics. The general interarrival time distribution is approximated by two-phase Cox distribution, C₂, that has Markovian property, enabling us to represent the processes by two-dimensional LPs. As distributions C₂ cover a wide class of distributions that have rational Laplace–Stieltjes transforms (LSTs) with square coefficient of variation lying in , the results obtained are applicable to a large class of real life situations. Some numerical results for the C₂^b/M/1 model are also given.     

Estimation of Traffic Intensity Based on Queue Length in a Single M/M/1 Queue

In this article, maximum likelihood estimator (MLE) as well as Bayes estimator of traffic intensity (ρ) in an M/M/1/∞ queueing model in equilibrium based on number of customers present in the queue at successive departure epochs have been worked out. Estimates of some functions of ρ which provide measures of effectiveness of the queue have also been derived. A comprehensive simulation study starting with the transition probability matrix has been carried out in the last section.     

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.     

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.     

Estimation in M/Er /1 queueing systems

In this paper, the maximum likelihood estimates of the parameters for the M/E_r /1 queueing model are derived when the queue size at each departure point is observed. A numerical example is generated by simulating a finite Markov chain to illustrate the methodology for estimating the parameters with variable Erlang service time distribution. The problem of hypothesis testing and simultaneous Confidence regions of the parameter is also investigated.0     

Optimal sequential estimation for semi-markov and markov renewal processes

We consider the problem of obtaining efficient estimators and sampling plans for semi-Markov and Markov-renewal processes. A lower bound for the variance of an unbiased estimator of a function of the parameters is obtained under a sequential scheme and we characterize the parametric functions and sampling plans which admit minimum variance unbiased estimators.     

Commuting Matrices in the Queue Length and Sojourn Time Analysis of MAP/MAP/1 Queues

Queues with Markovian arrival and service processes, i.e., MAP/MAP/1 queues, have been useful in the analysis of computer and communication systems and different representations for their stationary sojourn time and queue length distribution have been derived. More specifically, the class of MAP/MAP/1 queues lies at the intersection of the class of QBD queues and the class of semi-Markovian queues. While QBD queues have a matrix exponential representation for their queue length and sojourn time distribution of order N and N², respectively, where N is the size of the background continuous time Markov chain, the reverse is true for a semi-Markovian queue. As the class of MAP/MAP/1 queues lies at the intersection, both the queue length and sojourn time distribution of a MAP/MAP/1 queue has an order N matrix exponential representation. The aim of this article is to understand why the order N² distributions of the sojourn time of a QBD queue and the queue length of a semi-Markovian queue can be reduced to an order N distribution in the specific case of a MAP/MAP/1 queue. We show that the key observation exists in establishing the commutativity of some fundamental matrices involved in the analysis of the MAP/MAP/1 queue.     

Corrected Phase-Type Approximations of Heavy-Tailed Queueing Models in a Markovian Environment

We develop accurate approximations for the delay distribution of the MArP/G/1 queue that capture the exact tail behavior and provide bounded relative errors. Motivated by statistical analysis, we consider the service times as a mixture of a phase-type and a heavy-tailed distribution. With the aid of perturbation analysis, we derive corrected phase-type approximations as a sum of the delay in a MArP/PH/1 queue and a heavy-tailed component depending on the perturbation parameter. We exhibit their performance with numerical examples.