Rational Laplace-Stieltjes Research Articles

Efficient computational analysis of non-exhaustive service vacation queues: BMAP/R/1/N(∞) under gated-limited discipline

• A finite-buffer vacation queue with batch Markovian arrival process has been analyzed. • The server is subjected to serve under gated-limited service discipline. • Proposed analysis is based on the successive substitution and the supplementary variable method. • The results have been matched with the corresponding infinite-buffer queue. • Numerical results are presented for different service- and vacation-time distributions. This paper analyzes the finite-buffer single server queue with vacation(s). It is assumed that the arrivals follow a batch Markovian arrival process ( BMAP ) and the server serves customers according to a non-exhaustive type gated-limited service discipline. It has been also considered that the service and vacation distributions possess rational Laplace-Stieltjes transformation (LST) as these types of distributions may approximate many other distributions appeared in queueing literature. Among several batch acceptance/rejection strategies, the partial batch acceptance strategy is discussed in this paper. The service limit L (1 ≤ L ≤ N ) is considered to be fixed, where N is the buffer-capacity excluding the one in service. It is assumed that in each busy period the server continues to serve until either L customers out of those that were waiting at the start of the busy period are served or the queue empties, whichever occurs first. The queue-length distribution at vacation termination/service completion epochs is determined by solving a set of linear simultaneous equations . The successive substitution method is used in the steady-state equations embedded at vacation termination/service completion epochs. The distribution of the queue-length at an arbitrary epoch has been obtained using the supplementary variable technique. The queue-length distributions at pre-arrival and post-departure epoch are also obtained. The results of the corresponding infinite-buffer queueing model have been analyzed briefly and matched with the previous model. Net profit function per unit of time is derived and an optimal service limit and buffer-capacity are obtained from a maximal expected profit. Some numerical results are presented in tabular and graphical forms.

Stationary distributions of the R[X]/R/1 cross-correlated queue

ABSTRACTWe consider an infinite buffer single server queue wherein batch interarrival and service times are correlated having a bivariate mixture of rational (R) distributions, where R denotes the class of distributions with rational Laplace–Stieltjes transform (LST), i.e., ratio of a polynomial of degree at most n to a polynomial of degree n. The LST of actual waiting time distribution has been obtained using Wiener–Hopf factorization of the characteristic equation. The virtual waiting time, idle period (actual and virtual) distributions, as well as inter-departure time distribution between two successive customers have been presented. We derive an approximate stationary queue-length distribution at different time epochs using the Markovian assumption of the service time distribution. We also derive the exact steady-state queue-length distribution at an arbitrary epoch using distributional form of Little’s law. Finally, some numerical results have been presented in the form of tables and figures.

Detailed computational analysis of queueing-time distributions of the BMAP/G/1 queue using roots

Abstract In this paper we present closed-form expressions for the distribution of the virtual (actual) queueing time for the BMAP/R/1 and BMAP/D/1 queues, where `R' represents a class of distributions having rational Laplace‒Stieltjes transforms. The closed-form analysis is based on the roots of the underlying characteristic equation. Numerical aspects have been tested for a variety of arrival and service-time distributions and results are matched with those obtained using the matrix-analytic method (MAM). Further, a comparative study of computation time of the proposed method with the MAM has been carried out. Finally, we also present closed-form expressions for the distribution of the virtual (actual) system time. The proposed method is analytically quite simple and easy to implement.

An alternative method for computing system-length distributions of BMAP/R/1 and BMAP/D/1 queues using roots

In this paper, we present closed-form expressions for system-length distributions in terms of roots outside the unit disk of the characteristic equation of the BMAP/R/1 queue, where arrival process is batch Markovian arrival process (BMAP) and R represents a class of distributions having rational Laplace–Stieltjes transform. The unknown boundary vector has been evaluated using the roots (inside and on the unit disk) of the characteristic equation. Several numerical results are presented for a variety of arrival and service-time distributions including phase-type (PH) and matrix-exponential (ME) which cover a wide variety of distributions that arise in applications. Using an approximated representation of ME distribution, results for BMAP/D/1 queue are also presented. We compared our result with the results obtained using classical matrix analytic method as well as cyclic reduction algorithm. It is shown that the computation-time of the proposed method does not depend upon the parameters like traffic intensity and correlation co-efficient. The method is analytically quite simple and easy to implement.

Characterizing matrix-exponential distributions of order 4 (abstract only)

The Laplace-Stieltjes transform of a matrix-exponential distribution is a rational function. If there are no common factors between the numerator and denominator polynomials, then the order of the matrix-exponential distribution is the degree of the denominator polynomial. Given a rational Laplace-Stieltjes transform, it is unknown, in general, when it corresponds to a matrix-exponential distribution. Matrix-exponential distributions of order 3 have been completely characterized in this manner, but in this talk we look at the problem of characterizing matrix-exponential distributions of order 4.

An alternative characterization for matrix exponential distributions

A necessary condition for a rational Laplace–Stieltjes transform to correspond to a matrix exponential distribution is that the pole of maximal real part is real and negative. Given a rational Laplace–Stieltjes transform with such a pole, we present a method to determine whether or not the numerator polynomial admits a transform that corresponds to a matrix exponential distribution. The method relies on the minimization of a continuous function of one variable over the nonnegative real numbers. Using this approach, we give an alternative characterization for all matrix exponential distributions of order three.

An alternative characterization for matrix exponential distributions

A necessary condition for a rational Laplace–Stieltjes transform to correspond to a matrix exponential distribution is that the pole of maximal real part is real and negative. Given a rational Laplace–Stieltjes transform with such a pole, we present a method to determine whether or not the numerator polynomial admits a transform that corresponds to a matrix exponential distribution. The method relies on the minimization of a continuous function of one variable over the nonnegative real numbers. Using this approach, we give an alternative characterization for all matrix exponential distributions of order three.

Lattice path approach for busy period density of [formula omitted] queues using [formula omitted] Coxian distributions

In this paper busy period analysis of non-Markovian queueing system GI a / G b / 1 , starting initially with i 0 batches of customers, is carried out via lattice path approach. Both interarrival and service time distributions are approximated by 2-phase Cox distributions, C 2 , that have Markovian property, amenable to the application of lattice paths combinatorial analysis. Arrivals occur in batches of size a and services occur in batches of size b , a and b are co-prime. 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 a C 2 , the use of C 2 , therefore, has led us to achieve results applicable to almost any real life queueing system GI a / G b / 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 Mathematica and presented graphically.

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.

Characterization of Matrix-Exponential Distributions

The class of matrix-exponential distributions can be equivalently defined as the class of all distributions with rational Laplace–Stieltjes transform. An immediate question that arises is: when does a rational Laplace–Stieltjes transform correspond to a matrix-exponential distribution? For a rational Laplace–Stieltjes transform that has a pole of maximal real part that is real and negative, we give a geometric description of all admissible numerator polynomials that give rise to matrix-exponential distributions. Using this approach we give a complete characterization for all matrix-exponential distributions of order three.

Lattice Path Approach for Busy Period Density ofGIb/G/1 Queues UsingC2Coxian Distributions

In this paper busy period analysis of non-Markovian queuing system GIb/G/1, starting initially with i0 batches of customers, is carried out via lattice path approach. Both interarrival and service time distributions are approximated by 2-phase Cox distributions, C2, that have Markovian property, amenable to the application of lattice paths combinatorial analysis. Arrivals occur in batches of size b. 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 a C2, the use of C2, therefore, has led us to achieve results applicable to almost any real life queuing system GIb/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 Mathematica and presented 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 Queue with Semi-Markovian Batch Plus Poisson Arrivals with Application to the MPEG Frame Sequence

We consider a queueing system with a single server having a mixture of a semi-Markov process (SMP) and a Poisson process as the arrival process, where each SMP arrival contains a batch of customers. The service times are exponentially distributed. We derive the distributions of the queue length of both SMP and Poisson customers when the sojourn time distributions of the SMP have rational Laplace–Stieltjes transforms. We prove that the number of unknown constants contained in the generating function for the queue length distribution equals the number of zeros of the denominator of this generating function in the case where the sojourn times of the SMP follow exponential distributions. The linear independence of the equations generated by those zeros is discussed for the same case with additional assumption. The necessary and sufficient condition for the stability of the system is also analyzed. The distributions of the waiting times of both SMP and Poisson customers are derived. The results are applied to the case in which the SMP arrivals correspond to the exact sequence of Motion Picture Experts Group (MPEG) frames. Poisson arrivals are regarded as interfering traffic. In the numerical examples, the mean and variance of the waiting time of the ATM cells generated from the MPEG frames of real video data are evaluated.

Matrix‐Exponential Distributions: Calculus and Interpretations via Flows

By considering randomly stopped deterministic flow models, we develop an intuitively appealing way to generate probability distributions with rational Laplace–Stieltjes transforms on [0,∞). That approach includes and generalizes the formalism of PH-distributions. That generalization results in the class of matrix-exponential probability distributions. To illustrate the novel way of thinking that is required to use these in stochastic models, we retrace the derivations of some results from matrix-exponential renewal theory and prove a new extension of a result from risk theory. Essentially the flow models allows for keeping track of the dynamics of a mechanism that generates matrix-exponential distributions in a similar way to the probabilistic arguments used for phase-type distributions involving transition rates. We also sketch a generalization of the Markovian arrival process (MAP) to the setting of matrix-exponential distribution. That process is known as the Rational arrival process (RAP).

Distribution of number served during a busy period of GI/M/1/N queues-lattice path approach

Abstract This paper aims at transient analysis of finite queue GI/M/1/N by deriving the distribution of the number of customers served in a busy period. Lattice paths combinatorics is used, in which the process is split up at suitable renewable epochs and thus can be represented by a LP. The interarrival time distribution is approximated by two-phase Cox distribution, C2, that has Markovian property amenable to LP analysis. As distributions having rational Laplace–Stieltjes transforms and square coefficient of variation lying in [ 1 2 ,∞[ , forming a very wide class of distributions, can be approximated by C2, the use of C2 has yielded transient results applicable to almost any real life queueing system GI/M/1/N.

Waiting-Time Asymptotics for the M / G /2 Queue with Heterogeneous Servers

This paper considers a heterogeneous M/G/2 queue. The service times at server 1 are exponentially distributed, and at server 2 they have a general distribution B(⋅). We present an exact analysis of the queue length and waiting time distribution in case B(⋅) has a rational Laplace–Stieltjes transform. When B(⋅) is regularly varying at infinity of index −ν, we determine the tail behaviour of the waiting time distribution. This tail is shown to be semi-exponential if the arrival rate is lower than the service rate of the exponential server, and regularly varying at infinity of index 1−ν if the arrival rate is higher than that service rate.

Lattice paths combinatorics applied to transient queue length distribution of C2/M/1 queues and busy period analysis of bulk queues C2b/M/1

Abstract This paper aims at deriving explicit transient queue length distribution for GI/M/1 system and busy period analysis of bulk queue GIb/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 [ 1 2 ,∞) , the results obtained are applicable to a large class of real life situations. Some numerical results for the C2b/M/1 model are also given.

The M/G/1 queue with two service speeds

We consider an M/G/1 queue with the special feature that the speed of the server alternates between two constant values sL and sH > sL. The high-speed periods are exponentially distributed, and the low-speed periods have a general distribution. Our main results are: (i) for the case that the distribution of the low-speed periods has a rational Laplace–Stieltjes transform, we obtain the joint distribution of the buffer content and the state of the server speed; (ii) for the case that the distribution of the low-speed periods and/or the service request distribution is regularly varying at infinity, we obtain explicit asymptotics for the tail of the buffer content distribution. The two cases in which the offered traffic load is smaller or larger than the low service speed are shown to result in completely different asymptotics.

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

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.

The M / G /1 queue with two service speeds

We consider an M/G/1 queue with the special feature that the speed of the server alternates between two constant values SL and SH > SL. The high-speed periods are exponentially distributed, and the low-speed periods have a general distribution. Our main results are: (i) For the case that the distribution of the low-speed periods has a rational Laplace-Stieltjes transform, we obtain the joint distribution of the buffer content and the state of the server speed. (ii) For the case that the distribution of the low-speed periods and/or the service request distribution is regularly varying at infinity, we obtain explicit asymptotics for the tail of the buffer content distribution. The two cases in which the offered traffic load is smaller respectively larger than the low service speed are shown to result in completely different asymptotics.