Stationary analysis of the shortest queue first service policy (original) (raw)

Analysis of the Shortest Queue First service discipline with two classes

Proceedings of the 7th International Conference on Performance Evaluation Methodologies and Tools, 2014

We analyze the so-called Shortest Queue First (SQF) queueing discipline whereby a unique server addresses queues in parallel by serving at any time that queue with the smallest workload. Considering a stationary system composed of two parallel queues and assuming Poisson arrivals and general service time distributions, we first establish the functional equations satisfied by the Laplace transforms of the workloads in each queue. We further specialize these equations to the so-called "symmetric case", with same arrival rates and identical exponential service time distributions at each queue; we then obtain a functional equation

An Analytic Approach to a General Class of G/G/s Queueing Systems

Operations Research, 1990

We solve the queueing system (QS) Ck/Cm/s, where Ck is the class of Coxian probability density functions (pdfs) of order k, which is a subset of the pdfs that have rational Laplace transform (R). We formulate the model as a continuous-time, infinite-space Markov chain by generalizing the method of stages. By using a generating function technique, we solve an infinite system of partial difference equations and find closed form expressions for the systemsize, general-time, pre-arrival, post-departure probability distributions and the usual performance measures. In particular, we prove that the probability of n customers being in the system, when it is "saturated" (n s) is a linear s+m-1 combination of exactly (s) geometric terms. The closed form expressions involve a solution of a system of nonlinear equations that involves only the Laplace transforms of the interarrival and service time distributions. We conjecture that this result holds for the more general model GIRls. Following these theoretical results we propose an exact algorithm for finding the systemsize distribution and system's performance measures, which has an algorithmic complexity of O(k3(+' s)3). We examine special cases and apply this method for solving numerically the QS C 2 /C 2 /s and Ek/C 2 /s.

State-dependent M/G/1 type queueing analysis for congestion control in data networks

Computer Networks, 2002

We study in this paper a TCP-like linear-increase multiplicative-decrease flow control mechanism. We consider congestion signals that arrive in batches according to a Poisson process. We focus on the case when the transmission rate cannot exceed a certain maximum value. We write the Kolmogorov equations and we use Laplace Transforms to calculate the distribution of the transmission rate in the steady state as well as its moments. Our model is particularly useful to study the behavior of TCP, the congestion control mechanism in the Internet. By a simple transformation, the problem can be reformulated in terms of an equivalent M/G/1 queue, where the transmission rate in the original model corresponds to the workload in the 'dual' queue. The service times in the queueing model are not i.i.d., and they depend on the workload in the system.

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.

A queueing model with the bulk arrival rate depending upon the nature of service

Applied Mathematical Modelling, 1996

In this paper we study the time-dependent analysis of a limited capacity queueing model with the bulk arrival rate depending upon the nature of service available in the system. The customers arrive in the system in batches of size x, which is a random variable, and the service consists of two stages, one is essential (first stage) while the other may be inessential. The decision to offer the inessential service depends upon the size of the system. However, if this inessential service is temporarily suspended, the arrival rate of the customers decreases. Laplace transforms (in time) of the different probability generating functions describing the system size under various conditions of service and the expected system size are derived. Steady-state results consequently follow.

Finite-Buffer Queues with Workload-Dependent Service and Arrival Rates

Queueing Systems, 2005

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.

A M/M/ queue in a semi-Markovian environment

ACM SIGMETRICS Performance Evaluation Review, 2001

We consider an M/M/1 queue in a semi-Markovian environment. The environment is modeled by a two-state semi-Markov process with arbitrary sojourn time distributions F 0 ( x ) and F 1 ( x ). When in state i = 0, 1, customers are generated according to a Poisson process with intensity λ i and customers are served according to an exponential distribution with rate μ i . Using the theory of Riemann-Hilbert boundary value problems we compute the z -transform of the queue-length distribution when either F 0 ( x ) or F 1 ( x ) has a rational Laplace-Stieltjes transform and the other may be a general --- possibly heavy-tailed --- distribution. The arrival process can be used to model bursty traffic and/or traffic exhibiting long-range dependence, a situation which is commonly encountered in networking. The closed-form results lend themselves for numerical evaluation of performance measures, in particular the mean queue-length.

Queueing models for the analysis of communication systems

Top, 2014

Queueing models can be used to model and analyze the performance of various subsystems in telecommunication networks; for instance, to estimate the packet loss and packet delay in network routers. Since time is usually synchronized, discretetime models come natural. We start this paper with a review of suitable discrete-time queueing models for communication systems. We pay special attention to two important characteristics of communication systems. First, traffic usually arrives in bursts, making the classic modeling of the arrival streams by Poisson processes inadequate and requiring the use of more advanced correlated arrival models. Second, different applications have different quality-of-service requirements (packet loss, packet delay, jitter, etc.). Consequently, the common first-come-first-served (FCFS) scheduling is not satisfactory and more elaborate scheduling disciplines are required. Both properties make common memoryless queueing models (M/M/1-type models) inadequate. After the review, we therefore concentrate on a discrete-time queueing analysis with two traffic classes, heterogeneous train arrivals and a priority scheduling discipline, as an example analysis where both time correlation and heterogeneity in the arrival This invited paper is discussed in the comments available

Rejoinder on: Queueing models for the analysis of communication systems

TOP, 2014

Queueing models can be used to model and analyze the performance of various subsystems in telecommunication networks, for instance to estimate the packet loss and packet delay in network routers. Since time is usually synchronized, discretetime models come natural. We start this paper with a review of suitable discretetime queueing models for communication systems. We pay special attention to two important characteristics of communication systems. First, traffic usually arrives in bursts, making the classic modeling of the arrival streams by Poisson processes inadequate and requiring the use of more advanced correlated arrival models. Second, different applications have different quality-of-service requirements (packet loss, packet delay, jitter,. . .). Consequently, the common first-come-first-served (FCFS) scheduling is not satisfactory and more elaborate scheduling disciplines are required. Both properties make common memoryless queueing models (M/M/1-type models) inadequate. After the review, we therefore concentrate on a discrete-time queueing analysis with two traffic classes, heterogeneous train arrivals and a priority scheduling discipline, as an example analysis where both time correlation and heterogeneity in the arrival process as well as non-FCFS scheduling are taken into account. Focus is on delay performance measures, such as the mean delay experienced by both types of packets and probability tails of these delays.

Taylor series solution of the M/M/1 queueing system

Journal of Computational and Applied Mathematics, 1992

Krinik, A., Taylor series solution of the M/M/l queueing system, Journal of Computational and Applied Mathematics 44 (1992) 371-380. A Taylor series method for determining the transient probabilities of the classical single server queueing system is presented. The method is direct, practical and avoids Bessel function theory, Laplace transform theory and complex analysis. The resulting Taylor series are proved to converge for all time under arbitrary initial conditions. A method to obtain explicit Taylor series representations is demonstrated.