Stationary system-length distribution of Markovian bulk service queue with modified bulk service rule and dynamic service rates

2021 ◽  
pp. 1-0
Author(s):  
Gagandeep Singh ◽  
Anjana Kumari ◽  
U. C. Gupta
Keyword(s):  
Length Distribution ◽  
Stationary System ◽  
Bulk Service ◽  
Service Rates ◽  
System Length

ANALYTICALLY EXPLICIT RESULTS FOR THE GI/C-MSP/1/∞ QUEUEING SYSTEM USING ROOTS

2012 ◽  
Vol 26 (2) ◽  
pp. 221-244 ◽  
Author(s):  
M. L. Chaudhry ◽  
S. K. Samanta ◽  
A. Pacheco
Keyword(s):  
Length Distribution ◽  
Renewal Theory ◽  
Computationally Efficient ◽  
Form Analysis ◽  
Matrix Geometric Method ◽  
Alternative Approach ◽  
The Matrix ◽  
Sojourn Time Distribution ◽  
System Length ◽  
Markov Renewal Theory

In this paper, we present (in terms of roots) a simple closed-form analysis for evaluating system-length distribution at prearrival epochs of the GI/C-MSP/1 queue. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of system-length distribution. We also provide the steady-state system-length distribution at an arbitrary epoch by using the classical argument based on Markov renewal theory. The sojourn-time distribution has also been investigated. The prearrival epoch probabilities have been obtained using the method of roots which is an alternative approach to the matrix-geometric method and the spectral method. Numerical aspects have been tested for a variety of arrival- and service-time distributions and a sample of numerical outputs is presented. The proposed method not only gives an alternative solution to the existing methods, but it is also analytically simple, easy to implement, and computationally efficient. It is hoped that the results obtained will prove beneficial to both theoreticians and practitioners.

scholarly journals Steady-State Analysis of a Flexible Markovian Queue with Server Breakdowns

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 259 ◽  
Author(s):  
Messaoud Bounkhel ◽  
Lotfi Tadj ◽  
Ramdane Hedjar
Keyword(s):  
Steady State ◽  
Probability Generating Function ◽  
Threshold Level ◽  
System Size ◽  
Single Server ◽  
Markovian Queue ◽  
Breakdown Rates ◽  
Bulk Service ◽  
Service Rates ◽  
Number Of Customers

A flexible single-server queueing system is considered in this paper. The server adapts to the system size by using a strategy where the service provided can be either single or bulk depending on some threshold level c. If the number of customers in the system is less than c, then the server provides service to one customer at a time. If the number of customers in the system is greater than or equal to c, then the server provides service to a group of c customers. The service times are exponential and the service rates of single and bulk service are different. While providing service to either a single or a group of customers, the server may break down and goes through a repair phase. The breakdowns follow a Poisson distribution and the breakdown rates during single and bulk service are different. Also, repair times are exponential and repair rates during single and bulk service are different. The probability generating function and linear operator approaches are used to derive the system size steady-state probabilities.

A Short Note on the System-Length Distribution in a Finite-Buffer $$GI^X/C$$-MSP/1/N Queue Using Roots

2021 ◽  
pp. 396-410
Author(s):  
Abhijit Datta Banik ◽  
Mohan L. Chaudhry ◽  
Sabine Wittevrongel ◽  
Herwig Bruneel
Keyword(s):  
Short Note ◽  
Length Distribution ◽  
Finite Buffer ◽  
System Length

On the system queue length distributions of LCFS-P queues with arbitrary acceptance and restarting policies

1992 ◽  
Vol 29 (02) ◽  
pp. 430-440 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Stationary Distribution ◽  
Queue Length ◽  
Length Distribution ◽  
Stationary System ◽  
Service Discipline ◽  
Acceptance Probability ◽  
Queue Length Distribution ◽  
Poisson Arrival ◽  
History Of

Shanthikumar and Sumita (1986) proved that the stationary system queue length distribution just after a departure instant is geometric forGI/GI/1 with LCFS-P/H service discipline and with a constant acceptance probability of an arriving customer, where P denotes preemptive and H is a restarting policy which may depend on the history of preemption. They also got interesting relationships among characteristics. We generalize those results forG/G/1 with an arbitrary restarting LCFS-P and with an arbitrary acceptance policy. Several corollaries are obtained. Fakinos' (1987) and Yamazaki's (1990) expressions for the system queue length distribution are extended. For a Poisson arrival case, we extend the well-known insensitivity for LCFS-P/resume, and discuss the stationary distribution for LCFS-P/repeat.

Time-dependent solution and optimal control of a bulk service queue

1997 ◽  
Vol 34 (01) ◽  
pp. 258-266
Author(s):  
Shokri Z. Selim
Keyword(s):  
Queueing System ◽  
Maximum Size ◽  
Time Dependent ◽  
Optimal Service ◽  
Bulk Service ◽  
Service Rates ◽  
Finite Time Horizon ◽  
Number Of Customers ◽  
Time Dependent Solution ◽  
Dependent Probability

We consider the queueing system denoted by M/MN /1/N where customers are served in batches of maximum size N. The model is motivated by a traffic application. The time-dependent probability distribution for the number of customers in the system is obtained in closed form. The solution is used to predict the optimal service rates during a finite time horizon.

ON THE PROBABILITY DISTRIBUTION OF JOIN QUEUE LENGTH IN A FORK-JOIN MODEL

2010 ◽  
Vol 24 (4) ◽  
pp. 473-483
Author(s):  
Jun Li ◽  
Yiqiang Q. Zhao
Keyword(s):  
Probability Distribution ◽  
Queue Length ◽  
Length Distribution ◽  
Arrival Process ◽  
Queue Length Distribution ◽  
Poisson Arrival ◽  
Service Rates ◽  
Queue Lengths ◽  
Heterogeneous Service ◽  
Geometric Decay

In this article, we consider the two-node fork-join model with a Poisson arrival process and exponential service times of heterogeneous service rates. Using a mapping from the queue lengths in the parallel nodes to the join queue length, we first derive the probability distribution function of the join queue length in terms of joint probabilities in the parallel nodes and then study the exact tail asymptotics of the join queue length distribution. Although the asymptotics of the joint distribution of the queue lengths in the parallel nodes have three types of characterizations, our results show that the asymptotics of the join queue length distribution are characterized by two scenarios: (1) an exact geometric decay and (2) a geometric decay with the prefactor n−1/2.

Time-dependent solution and optimal control of a bulk service queue

1997 ◽  
Vol 34 (1) ◽  
pp. 258-266 ◽  
Author(s):  
Shokri Z. Selim
Keyword(s):  
Queueing System ◽  
Maximum Size ◽  
Time Dependent ◽  
Optimal Service ◽  
Bulk Service ◽  
Service Rates ◽  
Finite Time Horizon ◽  
Number Of Customers ◽  
Time Dependent Solution ◽  
Dependent Probability

We consider the queueing system denoted by M/MN/1/N where customers are served in batches of maximum size N. The model is motivated by a traffic application. The time-dependent probability distribution for the number of customers in the system is obtained in closed form. The solution is used to predict the optimal service rates during a finite time horizon.

Sign in / Sign up

Export Citation Format

Share Document