scholarly journals Analytically Explicit Results for the Distribution of the Number of Customers Served during a Busy Period for Special Cases of the M/G/1 Queue

2019 ◽  
Vol 2019 ◽  
pp. 1-15
Author(s):  
M. L. Chaudhry ◽  
Veena Goswami
Keyword(s):  
Poisson Process ◽  
Busy Period ◽  
Inverse Gaussian ◽  
Chi Square ◽  
Markov Modulated Poisson Process ◽  
Special Cases ◽  
Phase Type ◽  
The Laplace Transform ◽  
Markov Modulated ◽  
Number Of Customers

This paper presents analytically explicit results for the distribution of the number of customers served during a busy period for special cases of the M/G/1 queues when initiated with m customers. The functional equation for the Laplace transform of the number of customers served during a busy period is widely known, but several researchers state that, in general, it is not easy to invert it except for some simple cases such as M/M/1 and M/D/1 queues. Using the Lagrange inversion theorem, we give an elegant solution to this equation. We obtain the distribution of the number of customers served during a busy period for various service-time distributions such as exponential, deterministic, Erlang-k, gamma, chi-square, inverse Gaussian, generalized Erlang, matrix exponential, hyperexponential, uniform, Coxian, phase-type, Markov-modulated Poisson process, and interrupted Poisson process. Further, we also provide computational results using our method. The derivations are very fast and robust due to the lucidity of the expressions.

scholarly journals Explicit results for the distribution of the number of customers served during a busy period for $M^X/PH/1$ queue

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Veena Goswami ◽  
M. L. Chaudhry
Keyword(s):  
Functional Equation ◽  
Busy Period ◽  
Stochastic Modelling ◽  
Lagrange Inversion ◽  
Special Cases ◽  
Service Distribution ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Laplace Transform ◽  
Number Of Customers

<p style='text-indent:20px;'>We give analytically explicit solutions for the distribution of the number of customers served during a busy period for the <inline-formula><tex-math id="M1">\begin{document}$ M^X/PH/1 $\end{document}</tex-math></inline-formula> queues when initiated with <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula> customers. When customers arrive in batches, we present the functional equation for the Laplace transform of the number of customers served during a busy period. Applying the Lagrange inversion theorem, we provide a refined result to this functional equation. From a phase-type service distribution, we obtain the distribution of the number of customers served during a busy period for various special cases such as exponential, Erlang-k, generalized Erlang, hyperexponential, Coxian, and interrupted Poisson process. The results are exact, rapid and vigorous, owing to the clarity of the expressions. Moreover, we also consider computational results for several service-time distributions using our method. Phase-type distributions can approximate any non-negative valued distribution arbitrarily close, making them a useful practical stochastic modelling tool. These distributions have eloquent properties which make them beneficial in the computation of performance models.</p>

Algorithms for a continuous-review (s, S) inventory system

1981 ◽  
Vol 18 (02) ◽  
pp. 461-472
Author(s):  
V. Ramaswami
Keyword(s):  
Steady State ◽  
Inventory System ◽  
Steady State Distribution ◽  
State Distribution ◽  
Continuous Review ◽  
Markov Modulated Poisson Process ◽  
Special Cases ◽  
Phase Type ◽  
Markov Modulated ◽  
Stimulation Of

The steady-state distribution of the inventory position for a continuous-review (s, S) inventory system is derived in a computationally tractable form. Demands for items in inventory are assumed to form an N-process which is the ‘versatile Markovian point process' introduced by Neuts (1979). The N-process includes the phase-type renewal process, Markov-modulated Poisson process etc., as special cases and is especially useful in modelling a wide variety of qualitative phenomena such as peaked arrivals, interruptions, inhibition or stimulation of arrivals by certain events etc.

Algorithms for a continuous-review (s, S) inventory system

1981 ◽  
Vol 18 (2) ◽  
pp. 461-472 ◽  
Author(s):  
V. Ramaswami
Keyword(s):  
Steady State ◽  
Inventory System ◽  
Steady State Distribution ◽  
State Distribution ◽  
Continuous Review ◽  
Markov Modulated Poisson Process ◽  
Special Cases ◽  
Phase Type ◽  
Markov Modulated ◽  
Stimulation Of

The steady-state distribution of the inventory position for a continuous-review (s, S) inventory system is derived in a computationally tractable form. Demands for items in inventory are assumed to form an N-process which is the ‘versatile Markovian point process' introduced by Neuts (1979). The N-process includes the phase-type renewal process, Markov-modulated Poisson process etc., as special cases and is especially useful in modelling a wide variety of qualitative phenomena such as peaked arrivals, interruptions, inhibition or stimulation of arrivals by certain events etc.

scholarly journals A bivariate two-state Markov modulated Poisson process for failure modeling

2021 ◽  
Vol 208 ◽  
pp. 107318
Author(s):  
Yoel G. Yera ◽  
Rosa E. Lillo ◽  
Bo F. Nielsen ◽  
Pepa Ramírez-Cobo ◽  
Fabrizio Ruggeri
Keyword(s):  
Poisson Process ◽  
Failure Modeling ◽  
Markov Modulated Poisson Process ◽  
Markov Modulated

scholarly journals Estimating the parameters of a seasonal Markov-modulated Poisson process

2015 ◽  
Vol 26 ◽  
pp. 103-123 ◽  
Author(s):  
Armelle Guillou ◽  
Stéphane Loisel ◽  
Gilles Stupfler
Keyword(s):  
Poisson Process ◽  
Markov Modulated Poisson Process ◽  
Markov Modulated

Distribution of the busy period in a controllable M/M/2 queue operating under the triadic (0, K, N, M) policy

1990 ◽  
Vol 27 (02) ◽  
pp. 425-432
Author(s):  
Hahn-Kyou Rhee ◽  
B. D. Sivazlian
Keyword(s):  
Steady State ◽  
Busy Period ◽  
Queueing System ◽  
Service Completion ◽  
Special Cases ◽  
Service Stations ◽  
Gambler’S Ruin ◽  
Gambler's Ruin Problem ◽  
Number Of Customers ◽  
System Operating

We consider an M/M/2 queueing system with removable service stations operating under steady-state conditions. We assume that the number of operating service stations can be adjusted at customers' arrival or service completion epochs depending on the number of customers in the system. The objective of this paper is to obtain the distribution of the busy period using the theory of the gambler's ruin problem. As special cases, the distributions of the busy periods for the ordinary M/M/2 queueing system, the M/M/1 queueing system operating under the N policy and the ordinary M/M/1 queueing system are obtained.

Busy period in GIX/G/∞

1996 ◽  
Vol 33 (3) ◽  
pp. 815-829 ◽  
Author(s):  
Liming Liu ◽  
Ding-Hua Shi
Keyword(s):  
Service Time ◽  
Busy Period ◽  
Random Variables ◽  
Batch Service ◽  
Idle Period ◽  
Special Cases ◽  
Infinite Server Queues ◽  
General Relations ◽  
Number Of Customers ◽  
Busy Cycle

Busy period problems in infinite server queues are studied systematically, starting from the batch service time. General relations are given for the lengths of the busy cycle, busy period and idle period, and for the number of customers served in a busy period. These relations show that the idle period is the most difficult while the busy cycle is the simplest of the four random variables. Renewal arguments are used to derive explicit results for both general and special cases.

A monotonicity result for a single-server queue subject to a Markov-modulated Poisson process

1995 ◽  
Vol 32 (4) ◽  
pp. 1103-1111 ◽  
Author(s):  
Qing Du
Keyword(s):  
Poisson Process ◽  
Arrival Rate ◽  
Loss Probability ◽  
Arrival Process ◽  
Single Server ◽  
Single Server Queue ◽  
Markov Modulated Poisson Process ◽  
Server Queue ◽  
Monotonicity Result ◽  
Markov Modulated

Consider a single-server queue with zero buffer. The arrival process is a three-level Markov modulated Poisson process with an arbitrary transition matrix. The time the system remains at level i (i = 1, 2, 3) is exponentially distributed with rate cα i. The arrival rate at level i is λ i and the service time is exponentially distributed with rate μ i. In this paper we first derive an explicit formula for the loss probability and then prove that it is decreasing in the parameter c. This proves a conjecture of Ross and Rolski's for a single-server queue with zero buffer.

Sign in / Sign up

Export Citation Format

Share Document