A Multiserver Queue with Thresholds for the Acceptance of Customers into Service

We consider the stationary waiting time distribution under the FIFO discipline for multi-server queues with (possibly heterogeneous) phase type servers and renewal arrivals, and for queues which can be represented as a quasi-birth-and-death process. It is shown that the complementary distribution function of the stationary waiting time in these queues is of the form where θ is positive and the dj are probabilities. A complete characterization of θ and dj , which lends itself to interesting probabilistic interpretations, is given for all these cases. Except in the case of the multiserver GI/PH/c queue which is to be discussed in detail elsewhere, we provide an efficient algorithm to compute the waiting time distribution.

Analysis of voice and low-priority data traffic by means of brisk periods and slack periods

This paper models a k‐unit service system (e.g., a repair, maintenance, or rental facility) with Poisson arrivals, exponential service times, and no queue. If we denote the number of units that are busy as the state of the system, the state‐dependent pricing model formalizes the intuitive notion that when most units are idle, the price (i.e., the service charge per unit time) should be low, and when most units are busy, the price should be higher than the average. A computationally efficient algorithm based on a nonlinear programming formulation of the problem is provided for determination of the optimal state‐dependent prices. The procedure ultimately reduces to the search on a single variable in an interval to determine the unique intersection point of a concave increasing function and a linear decreasing function. The algorithm takes, on the average, only about 1/2 second per problem on the IBM 360/65 (FORTRAN G Compiler). A discrete optimal‐control approach to the problem is shown to result in essentially the same procedure as the nonlinear‐programming formulation. Several properties of the optimal state‐dependent prices are given. Comparisons of the optimal values of the objective function for the state‐dependent and state‐independent pricing policies show that the former is on the average, only about 0.7% better than the latter, which may explain partly why state‐dependent pricing is not prevalent in many service systems. Potential generalizations of the model are discussed.

Markov chains with periodic graphs arise frequently in a wide range of modelling experiments. Application areas range from flexible manufacturing systems in which pallets are treated in a cyclic manner to computer communication networks. The primary goal of this paper is to show how advantage may be taken of this property to reduce the amount of computer memory and computation time needed to compute stationary probability vectors of periodic Markov chains. After reviewing some basic properties of Markov chains whose associated graph is periodic, we introduce a ‘reduced scheme’ in which only a subset of the probability vector need be computed. We consider the effect of the application of both direct and iterative methods to the original Markov chain (permuted to normal periodic form) as well as to the reduced system. We show how the periodicity of the Markov chain may be efficiently computed as the chain itself is being generated. Finally, some numerical experiments to illustrate the theory, are presented.

Convergence theory of some classes of iterative aggregation/disaggregation methods for computing stationary probability vectors of stochastic matrices

The goal of this research is to implement network analysis using Markov chains in Python.Networks exist almost everywhere in life.There are networks of computers, people, articles, posts, etc. Network analysis is used to understand the structure, function, and performance of the network.Markov chains method is used to predict the future state based on the present state and not on the previous states.The basic steps of network analysis using Markov chains are explained: defining network (states, transition matrix, and distribution vector), performing matrix multiplication (computing stationary distribution vector and computing stationary transition vector), performing random walk (computing stationary distribution vector), comparing results, and plotting charts.The developed program was tested on an experimental data.The program has successfully performed the basic steps of network analysis using Markov chains and provided the required results.

In the last chapter, we used Markov process models for queueing systems with Poisson arrivals and exponential service times. To model a system as a Markov process, we should be able to give complete distribution characteristics of the process beyond time t, using what we know about the process at t and changes that may occur after t, without referring back to the events before t. When arrivals are Poisson and service times are exponential, because of the memoryless property of the exponential distribution we are able to use the Markov process as a model. If the arrival rate is λ and service rate is μ, at any time point t, time to next arrival has the exponential distribution with rate λ, and if a service is in progress, the remaining service time has the exponential distribution with rate μ. If one or both of the arrival and service distributions are non-exponential, the memoryless property does not hold and a Markov model of the type discussed in the last chapter does not work. In this chapter, we discuss a method by which a Markov model can be constructed, not for all t, but for specific time points on the time axis.KeywordsMarkov ChainService TimeTransition Probability MatrixProbability Generate FunctionService Time DistributionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper deals with multi-server queues with a finite buffer of size N in which units waiting for service generate into priority at a constant rate, independently of other units in the buffer. At the epoch of a unit's priority generation, the unit is immediately taken for service if there is one unit in service that did not generate into priority while waiting; otherwise such a unit leaves the system in search of immediate service elsewhere. The arrival stream of units is a Markovian arrival process (MAP) and service requirements are of phase (PH) type. Our interest is in the continuous-time Markov chain describing the state of the queue at arbitrary times, which constitutes a finite quasi-birth-and-death (QBD) process. We give formulas well suited for numerical computation for a variety of performance measures, including the blocking probability, the departure process, and the stationary distributions of the system state at pre-arrival epochs, at post-departure epochs and at epochs at which arriving units are lost. Illustrative numerical examples show the effect of several parameters on certain probabilistic descriptors of the queue for various levels of congestion.

In 1969 H. and D. Stoyan showed that the stationary waiting-time distribution in a GI/G/1 queue increases in the ordering determined by the expected value of all non-decreasing convex functions when the interarrival-time and service-time distributions become more variable, as expressed in the ordering determined by the expected value of all convex functions. Ross (1978) and Wolff (1977) showed by counterexample that this conclusion does not extend to all GI/G/s queues. Here it is shown that this conclusion does hold for all GI/G/s queues for several other measures of congestion which coincide with the waiting time in single-server systems. One such alternate measure of congestion is the clearing time, the time required after the arrival epoch of the nth customer for the system to serve all customers in the system at that time, excluding the nth customer. The stochastic comparisons also imply an ordering for the expected waiting times in M/G/s queues.

AQM (Active Queue Management) is an effective scheme to enhance congestion control. Most of the mobile networks today use queue management to manage and improve network performance. In this paper, we deploy AQM mechanism in Mobile Ad Hoc networks to manage the packets queue in the network nodes. By modeling a node with multi-receiver in Ad Hoc networks, we develop an analytical model for a multi-server and finite capacity queuing system with AQM mechanisms under multi-classes of traffic. We consider multiple classes of traffic modeled by Poisson arrival and exponentially service processes. It deduced the performance metrics including the average queue length, system throughput, probability of drop packet and the average waiting time. Then, it illustrated the relationship between above metrics and the number of servers. Finally, by demonstrating the effective number of servers, it illuminate that we should deploy appropriate number of servers for a certain condition.

In this paper, we use the Markov chain censoring technique to study infinite state Markov chains whose transition matrices possess block-repeating entries. We demonstrate that a number of important probabilistic measures are invariant under censoring. Informally speaking, these measures involve first passage times or expected numbers of visits to certain levels where other levels are taboo;they are closely related to the so-called fundamental matrix of the Markov chain which is also studied here. Factorization theorems for the characteristic equation of the blocks of the transition matrix are obtained. Necessary and sufficient conditions are derived for such a Markov chain to be positive recurrent, null recurrent, or transient based either on spectral analysis, or on a property of the fundamental matrix. Explicit expressions are obtained for key probabilistic measures, including the stationary probability vector and the fundamental matrix, which could be potentially used to develop various recursivealgorithms for computing these measures.

The new results are (C), (D) and (F) below. Let $X_0, X_1, \cdots$ be a Markov chain with countable state space $I$ and stationary transitions. Suppose $I$ is a positive recurrent class, with stationary probability vector $p$. Let $f$ be a real-valued function on $I$. Fix a reference state $s \varepsilon I$, and let $0 \leqq t_1 < t_2 < \cdots$ be the times $n$ at which $X_n = s$. Let $Y_j = \sum \{f(X_n):t_j \leqq n < t_{j + 1}\}$ and $U_j = \sum \{|f(X_n)|:t_j \leqq n < t_{j + 1}\}.$ Let $V_m = \sum^m_{j = 1} Y_j$ and $S_n = \sum^n_{j = 0}f(X_j)$. For (C) and (D) below, assume (A) $\sum_{i \varepsilon I} p_if(i) = 0$; and (B) $U^2_j$ has finite expectation. Then: (C) Theorem. $n^{-\frac{1}{2}} \max \{|S_j - V_{jp_s}|: 1 \leqq j \leqq n\} \rightarrow 0$ in probability; and (D) Theorem. $(n \log \log n)^{-\frac{1}{2}} \max \{|S_j - V_{jp_s}|: 1 \leqq j \leqq n\} \rightarrow 0$ almost everywhere. For (F), do not assume (A) and (B), but assume (E) $Y_j$ differs from 0 with positive probability. Let $v_m$ (respectively, $s_n$) be 1 or 0 according as $V_m$ (respectively, $S_n$) is positive or non-positive. Then (F) Theorem. $n^{-1} \sum \{s_j : 1 \leqq j \leqq n\} - p^{-1}_sn^{-1} \sum \{v_j : 1 \leqq j \leqq np_s\} \rightarrow 0$ almost everywhere. I do not believe the convergence in (C) is a.e., but have no counter-example.

It is shown that, in a multiserver queue with interarrival and service-time distributions of phase type (PH/PH/c), the waiting-time distribution W(x) has an asymptotically exponential tail, i.e., 1 – W(x) ∽ Ke–ckx. The parameter k is the unique positive number satisfying T*(ck) S*(–k) = 1, where T*(s) and S*(s) are the Laplace–Stieltjes transforms of the interarrival and the service-time distributions. It is also shown that the queue-length distribution has an asymptotically geometric tail with the rate of decay η = T*(ck). The proofs of these results are based on the matrix-geometric form of the state probabilities of the system in the steady state.The equation for k shows interesting relations between single- and multiserver queues in the rates of decay of the tails of the waiting-time and the queue-length distributions.The parameters k and η can be easily computed by solving an algebraic equation. The multiplicative constant K is not so easy to compute. In order to obtain its numerical value we have to solve the balance equations or estimate it from simulation.

The multi-server queue with non-homogeneous Poisson arrivals and customer abandonment is a fundamental dynamic rate queueing model for large scale service systems such as call centers and hospitals. Scaling the arrival rates and number of servers arises naturally when a manager updates a staffing schedule in response to a forecast of increased customer demand. Mathematically, this type of scaling ultimately gives us the fluid and diffusion limits as found in Mandelbaum et al., Queueing Syst 30:149---201 (1998) for Markovian service networks. The asymptotics used here reduce to the Halfin and Whitt, Oper Res 29:567---588 (1981) scaling for multi-server queues. The diffusion limit suggests a Gaussian approximation to the stochastic behavior of this queueing process. The mean and variance are easily computed from a two-dimensional dynamical system for the fluid and diffusion limiting processes. Recent work by Ko and Gautam, INFORMS J Comput, to appear (2012) found that a modified version of these differential equations yield better Gaussian estimates of the original queueing system distribution. In this paper, we introduce a new three-dimensional dynamical system that is based on estimating the mean, variance, and third cumulant moment. This improves on the previous approaches by fitting the distribution from a quadratic function of a Gaussian random variable.

A discrete time single-line queueing system with independent geometric streams of regular and negative claims, infinite buffer, and geometric service is considered. A negative claim pushes a regular claim out of the buffer queue and moves it to a bunker of infinite capacity. If the buffer is empty, a negative claim leaves the system without any effect. After servicing a claim, the system receives the next claim from the buffer, if it is not empty, or from the bunker. We obtain relations for stationary waiting and sojourn time distributions of the regular claim.