On queues with Markov modulated service rates (original) (raw)
Related papers
On Markov-Krein Characterization of Mean Sojourn Time in Queueing Systems
2011
We present a new analytical tool for three queueing systems which have defied exact analysis so far: (i) the classical M/G/k multi-server system, (ii) queueing systems with fluctuating arrival and service rates, and (iii) the M/G/1 round-robin queue. We argue that rather than looking for exact expressions for the mean response time as a function of the job size distribution, a more fruitful approach is to find distributions which minimize or maximize the mean response time given the first n moments of the job size distribution. We prove that for the M/G/k system in light-traffic asymptote and given first n (= 2, 3) moments of the job size distribution, analogous to the classical Markov-Krein Theorem, these 'extremal' distributions are given by the principal representations of the moment sequence. Furthermore, if we restrict the distributions to lie in the class of Completely Monotone (CM) distributions, then for all the three queueing systems, for any n, the extremal distributions under the appropriate "light traffic" asymptotics are hyper-exponential distributions with finite number of phases. We conjecture that the property of extremality should be invariant to the system load, and thus our light traffic results should hold for general load as well, and propose potential strategies for a unified approach to finding moments-based bounds for queueing systems. By identifying the extremal distributions, our results allow numerically obtaining tight bounds on the performance of these queueing systems.
Fundamental Characteristics of Queues With Fluctuating Load
Proceedings of the joint …, 2006
Systems whose arrival or service rates fluctuate over time are very common, but are still not well understood analytically. Stationary formulas are poor predictors of systems with fluctuating load. When the arrival and service processes fluctuate in a Markovian manner, computational methods, such as Matrix-analytic and spectral analysis, have been instrumental in the numerical evaluation of quantities like mean response time. However, such computational tools provide only limited insight into the functional behavior of the system with respect to its primitive input parameters: the arrival rates, service rates, and rate of fluctuation. For example, the shape of the function that maps rate of fluctuation to mean response time is not well understood, even for an M/M/1 system. Is this function increasing, decreasing, monotonic? How is its shape affected by the primitive input parameters? Is there a simple closed-form approximation for the shape of this curve? Turning to user experience: How is the performance experienced by a user arriving into a "high load" period different from that of a user arriving into a "low load" period, or simply a random user. Are there stochastic relations between these? In this work, we provide the first answers to these fundamental questions.
Queues with Workload-Dependent Arrival and Service Rates
Queueing Systems, 2004
We consider two types of queues with workload-dependent arrival rate and service speed. Our study is motivated by queueing scenarios where the arrival rate and/or speed of the server depends on the amount of work present, like production systems and the Internet.
Computing the Performance Measures in Queueing Models via the Method of Order Statistics
Journal of Applied Mathematics, 2011
This paper focuses on new measures of performance in single-server Markovian queueing system. These measures depend on the moments of order statistics. The expected value and the variance of the maximum minimum number of customers in the system as well as the expected value and the variance of the minimum maximum waiting time are presented. Application to an M/M/1 model is given to illustrate the idea and the applicability of the proposed measures.
An approximate analysis of a queueing system with markov-modulated arrivals
Electronics and Communications in Japan (Part I: Communications), 1990
ABSTRACT This paper presents an excellent two-moment approximation via an N-environment diffusion model of a queueing system with Markov modulated Poisson process (MMPP) arrivals and general service time distribution. By choosing residual workload as a performance criterion, an approximation error of less than 3 percent can be achieved.We give a closed form solution for all moments of the waiting time distribution with two-state MMPP arrivals. We also propose an algorithm for the approximate solution of single server queueing systems with n-state MMPP arrivals. The proposed sufficiency condition for the existence of a stationary solution has not been proved for the n-state case with n ≥ 3.
Transient analysis of non-markovian queues via markov regenerative processes
1996
In this chapter, we develop computational techniques for the time-dependent solution of the queue length distribution for a class of non-Markovian queueing systems. The class of systems we consider are those for which the queue length process is Markov regenerative. We consider standard single server nite capacity queues such as the M/G/1/K and the GI/M/1/K queues. We also show how these algorithms can be extended to more general arrival processes such as the Batch Markovian Arrival Process (BMAP) and to multiclass queueing systems.
An Invariance Relation and a Unified Method to Derive Stationary Queue-Length Distributions
Operations Research, 2004
For a broad class of discrete-and continuous-time queueing systems, we show that the stationary number of customers in system (queue plus servers) is the sum of two independent random variables, one of which is the stationary number of customers in queue and the other is the number of customers that arrive during the time a customer spends in service. We call this relation an invariance relation in the sense that it does not change for a variety of single-sever queues (with batch arrivals and batch services) and some of multi-server queues (with batch arrivals and deterministic service times) that satisfy a certain set of assumptions. Making use of this relation, we also present a simple method of deriving the stationary distributions of the numbers in queue and in system as well as some of their properties. This is illustrated by several examples, which show that new simple derivations of old results as well as new results can be obtained in a unified manner. Furthermore, we show that the invariance relation and the method we are presenting are easily generalized to analyze queues with BMAP (Batch Markovian Arrival Process) arrivals. Most of the results are presented under the discrete-time setting. The corresponding continuous-time results, however, are covered as well because deriving the results for continuous-time queues runs exactly parallel to that for their discrete-time counterparts.
Computers & Industrial Engineering, 2006
The independence of processes in queueing systems is generally assumed when developing queueing models. However, real systems often involve several process dependencies, and failure to take these into consideration can lead to serious underestimation of the performance measures. We consider herein a single server queueing system with a Markov renewal process (MRP) for its arrival process and a general service time distribution, and derive the distribution function and correlation coefficient of the departure process. Since the departure process also often corresponds to an arrival process in downstream queues, the results obtained here can be used to derive a better approximation of the performance measures of a non-product form general queueing network.