Busy Period Density Research Articles

Busy period density of M//1 queueing system through lattice path approach: a special case

We consider M/ / lqueueing system which is a queueing system with arrival rate follows a Poisson distribution, service rate follows an exponential distribution which is approximated by Coxian 2-phase distribution, service is performed in b-size, and the number of service facility is one. This paper aims to derive the properties of M/ / 1 queueing system when the system is in busy pea riod using latticea path combinatorics approach. We consider a special case which is called pure incomplete busy period (PIBP) Through this approach, the queue system is represented in the form of lattice path. Furthermore, we have determined the number of lattice paths and finally, the probability function in the queue system M/ / 1 in PIBP is derived.

The busy period of an M/M/1 queue with balking and reneging

It has been shown by (R.O. Al-Seedy, A.A. El-Sherbiny, S.A. El-Shehawy, S.I. Ammar, Transient solution of the M/M/c queue with balking and reneging, Comput. Math. Appl. 57 (2009) 1280–1285) that a generating function technique can be successfully applied to derive the transient solution for an M/M/c queueing system. In this paper, we further illustrate how this technique can be used to obtain the busy period density function of an M/M/1 queue with balking and reneging. Finally, numerical calculations are presented.

Busy period analysis of [formula omitted] queue: Lattice path approach

In this paper, we find the busy period density of C 2 / C 2 b / 1 queues in explicit computational form, through lattice path (LP) approach. Both the arrival and service time distributions are approximated by 2-phase Cox distribution C 2 , which has a Markovian property enabling us to use LP combinatorics. Since any distribution with rational Laplace–Stieltjes transform (LST) and square coefficient of variation ( C V 2 ) lying in [ 1 / 2 , ∞ ) can be approximated by a C 2 ([M. Agarwal, K. Sen, B. Borkakaty, Busy period density of queueing system C 3 / M / 1 , Journal of Combinatorics, Information and Systems Sciences 31 (1–4) (2006) 127–161]), the results obtained would be applicable to a very wide class of distributions occurring in real life.

Lattice path approach to busy period density of queueing system C<SUB align=right>2/C<SUB align=right>2/1

The paper aims at busy period analysis of non-Markovian queuing system GI/G/1 starting initially with i0 customers, through lattice path approach. Both interarrival and service time distributions are approximated by two-phase Cox distributions, C2, that have Markovian property, amenable to the application of lattice paths combinatorial analysis. Distributions having rational Laplace?Stieltjes transform and square coefficient of variation lying in [1/2∞) form a very wide class of distributions. As any distribution of this class can be approximated by C2, therefore, the use of C2 has led us to achieve results applicable to almost any real life queuing system GI/G/1 occurring in computer systems, communication systems, manufacturing systems, etc. Numerical computations have been performed for different sets of values of the parameters involved using software package Mathematica and represented graphically.

Lattice path approach for busy period density of M/G/1 queues using C3 Coxian distribution

The paper aims at busy period analysis of non-Markovian queuing system M/G/1 starting initially with i0 customers, through lattice path approach. The general distribution involved is approximated by a 3-phase Cox distribution, C3. Distributions having rational Laplace–Stieltjes transforms and square coefficient of variation lying in [1/3, ∞) form a very wide class of distributions. As any distribution of this class can be approximated by a C3, that has Markovian property, amenable to the application of lattice paths combinatorial analysis, the use of C3 therefore has led us to achieve results applicable to almost any real life queuing system M/G/1.

A new formula for the busy period of a non-empty multiserver queueing system

In a recent paper Sharma and Tarabia [Sankhyâ 62 (2) (2000) 273–281] obtain a density function of a busy period for an empty multichannel Markovian queue in series form. In this paper a simple approach is applied to obtain the busy period density function by taking an arbitrary number (say i) of customers present in the system initially. Also a general form for the kth order moment of the busy period distribution is given.

Lattice path approach to transient analysis of M/G/1/N non-Markovian queues using Cox distributions

In this paper, on approximating the general service time distribution by Coxian two-phase distribution, C 2 , explicit busy period density function for non-Markovian finite queues M/G/1/ N has been obtained by using lattice paths combinatorics (LPC). The results obtained are specifically important as it seems that no explicit transient solutions are available for finite queues. Two-phase Cox distribution, C 2 , has Markovian property and is thus amenable to lattice path (LP) analysis. The transient results obtained are applicable to almost any real life finite queuing system M/G/1/ N .

On Quadrature Approximation for the Busy Period Density Function of an M/M/1 Queueing System

ABSTRACTIn this paper we obtain a quadrature approximation for the busy period density function of an M/M/1 queue. By using these values, the computation of the transient probability values of M/M/1 and M/M/l/N queues are discussed.

Calculating the M/G/1 busy-period density and LIFO waiting-time distribution by direct numerical transform inversion

It is well known that the M/G/1 busy-period density can be characterized by the Kendall functional equation for its Laplace transform. The Kendall functional equation can be solved iteratively to obtain transform values to use in numerical inversion algorithms. However, we show that the busy-period density can also be numerically inverted directly, without iterating a functional equation, exploiting a contour integral representation due to Cox and Smith (1961). The contour integral representation was originally proposed as a basis for asymptotic approximations. We derive heavy-traffic expansions for the asymptotic parameters appearing there. We also use the integral representation to derive explicit series representations of the busy-period density for serval service-time distributions. In addition, we discuss related contour integral representations for the probability of emptiness, which is directly related to the waiting-time distribution with the LIFO discipline. The asymptotics and the numerical inversion reveal the striking difference between the waiting-time distributions for the FIFO and LIFO disciplines.

A note on the alternative form of the busy period density for an m/m/1/∞ queue

Using probability arguments a series form for the density function of M/M/I/∞ queue is obtained.

The transient solution to a class of Markovian queues

In this paper, we study the transient behavior of Markovian queues such as M/ M/ s queues and bulk-arrival M/ M/1 queues. It will be shown that the transition probabilities of a birth-death process on the non-negative integers, governed by parameters { λ n , μ n } n=0 ∞ such that λ n−1 = λ and μ n = μ for all n ≥ N and for some N ≥ 1 with μ 0 = 0, can be represented in terms of the busy period density of an M/ M/1 queue having the arrival rate λ and the service rate μ and some exponential functions. The transition probabilities of a bulk-arrival M/ M/1 queue can also be expressed in terms of its busy period density.

Decompositions of theM/M/1 transition function

Two decompositions are established for the probability transition function of the queue length process in the M/M/1 queue by a simple probabilistic argument. The transition function is expressed in terms of a zero-avoiding probability and a transition probability to zero in two different ways. As a consequence, the M/M/1 transition function can be represented as a positive linear combination of convolutions of the busy-period density. These relations provide insight into the transient behavior and facilitate establishing related results, such as inequalities and asymptotic behavior.

Simple spectral representations for the M/M/1 queue

This paper shows that certain basic descriptions of the time-dependent behavior of the M/M/1 queue have very simple representations as mixtures of exponentials. In particular, this is true for the busy-period density, the probability that the server is busy starting at zero, the expected queue length starting at zero and the autocorrelation function of the stationary queue-length process. In each case the mixing density is a minor modification of a beta density. The last two representations also apply to regulated or reflected Brownian motion (RBM) by virtue of the heavy-traffic limit. Connections are also established to the classical spectral representations of Ledermann and Reuter (1954) and Karlin and McGregor (1958) and the associated trigonometric integral representations of Ledermann and Reuter, Vaulot (1954), Morse (1955), Riordan (1961) and Takacs (1962). Overall, this paper aims to provide a more unified view of the M/M/1 transient behavior and show how several different approaches are related.

The structure of transient distribution functions for time-shared computer systems

The transient behavior of a single-server queuing system, e.g., a computer, having two classes of customers, of high and low priority, is difficult. When the customer service times are Poisson and exponentially distributed, one is coping with a random walk in two dimensions of known intractability. The server busy period distribution for the M/G/1 queue is of related interest. Even though this distribution is known explicitly in real time for arbitrary service-time distribution, the form of the answer as an infinite series involving convolutions provides limited insight. For the two Poisson streams with exponential service times, the effective service-time distribution of an equivalent single stream is completely monotone. It will be shown that the busy-period density of the server is itself completely monotone, and that this and related results provided improved understanding of the system behavior. Some interest is focused on the spectral distribution function of the completely monotone density as a working tool.

Optimal Operating Policies for the Finite-Source Queuing Process

This paper evaluates the queue-length distribution, the busy-period density and other characteristics of the finite-source queuing process in which the server remains idle until a queue of n units is built up. It also determines values of n that optimize the net profits for discounted, as well as undiscounted, infinite-horizon models.