Optimal Thresholds of an Infinite Buffer Discrete-Time Two-Server System with Triadic Policy

2011 ◽  
Vol 2 (4) ◽  
pp. 75-88
Author(s):  
Veena Goswami ◽  
G. B. Mund
Keyword(s):  
Discrete Time ◽  
Queueing System ◽  
Minimum Cost ◽  
Service Rate ◽  
Closed Form Solutions ◽  
Optimal Thresholds ◽  
Optimal Service ◽  
Number Of Customers ◽  
System Operating ◽  
Server System

This paper analyzes a discrete-time infinite-buffer Geo/Geo/2 queue, in which the number of servers can be adjusted depending on the number of customers in the system one at a time at arrival or at service completion epoch. Analytical closed-form solutions of the infinite-buffer Geo/Geo/2 queueing system operating under the triadic (0, Q N, M) policy are derived. The total expected cost function is developed to obtain the optimal operating (0, Q N, M) policy and the optimal service rate at minimum cost using direct search method. Some performance measures and sensitivity analysis have been presented.

Optimal Thresholds of an Infinite Buffer Discrete-Time Two-Server System with Triadic Policy

2012 ◽  
pp. 279-293
Author(s):  
Veena Goswami ◽  
G. B. Mund
Keyword(s):  
Performance Measures ◽  
Discrete Time ◽  
Minimum Cost ◽  
Service Rate ◽  
Closed Form Solutions ◽  
Optimal Thresholds ◽  
Optimal Service ◽  
Number Of Customers ◽  
System Operating ◽  
Server System

This paper analyzes a discrete-time infinite-buffer Geo/Geo/2 queue, in which the number of servers can be adjusted depending on the number of customers in the system one at a time at arrival or at service completion epoch. Analytical closed-form solutions of the infinite-buffer Geo/Geo/2 queueing system operating under the triadic (0, Q N, M) policy are derived. The total expected cost function is developed to obtain the optimal operating (0, Q N, M) policy and the optimal service rate at minimum cost using direct search method. Some performance measures and sensitivity analysis have been presented.

scholarly journals Sojourn Times in A Queueing System with Breakdowns and General Retrial Times

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2882
Author(s):  
Ivan Atencia ◽  
José Luis Galán-García
Keyword(s):  
Steady State ◽  
Discrete Time ◽  
Queueing System ◽  
Retrial Queue ◽  
Stochastic Decomposition ◽  
Discrete Time System ◽  
Main Research ◽  
Extensive Analysis ◽  
Complete Study ◽  
Number Of Customers

This paper centers on a discrete-time retrial queue where the server experiences breakdowns and repairs when arriving customers may opt to follow a discipline of a last-come, first-served (LCFS)-type or to join the orbit. We focused on the extensive analysis of the system, and we obtained the stationary distributions of the number of customers in the orbit and in the system by applying the generation function (GF). We provide the stochastic decomposition law and the application bounds for the proximity between the steady-state distributions for the queueing system under consideration and its corresponding standard system. We developed recursive formulae aimed at the calculation of the steady-state of the orbit and the system. We proved that our discrete-time system approximates M/G/1 with breakdowns and repairs. We analyzed the busy period of an auxiliary system, the objective of which was to study the customer’s delay. The stationary distribution of a customer’s sojourn in the orbit and in the system was the object of a thorough and complete study. Finally, we provide numerical examples that outline the effect of the parameters on several performance characteristics and a conclusions section resuming the main research contributions of the paper.

Optimal control of service rates in networks of queues

1987 ◽  
Vol 19 (1) ◽  
pp. 202-218 ◽  
Author(s):  
Richard R. Weber ◽  
Shaler Stidham
Keyword(s):  
Rate Control ◽  
Queueing Networks ◽  
Arrival Rate ◽  
Service Rate ◽  
Discounted Cost ◽  
Holding Costs ◽  
Optimal Service ◽  
Monotonicity Result ◽  
Service Rates ◽  
Number Of Customers

We prove a monotonicity result for the problem of optimal service rate control in certain queueing networks. Consider, as an illustrative example, a number of ·/M/1 queues which are arranged in a cycle with some number of customers moving around the cycle. A holding cost hi(xi) is charged for each unit of time that queue i contains xi customers, with hi being convex. As a function of the queue lengths the service rate at each queue i is to be chosen in the interval , where cost ci(μ) is charged for each unit of time that the service rate μis in effect at queue i. It is shown that the policy which minimizes the expected total discounted cost has a monotone structure: namely, that by moving one customer from queue i to the following queue, the optimal service rate in queue i is not increased and the optimal service rates elsewhere are not decreased. We prove a similar result for problems of optimal arrival rate and service rate control in general queueing networks. The results are extended to an average-cost measure, and an example is included to show that in general the assumption of convex holding costs may not be relaxed. A further example shows that the optimal policy may not be monotone unless the choice of possible service rates at each queue includes 0.

scholarly journals A Discrete-TimeGeo/G/1 Retrial Queue with Two Different Types of Vacations

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Feng Zhang ◽  
Zhifeng Zhu
Keyword(s):  
Discrete Time ◽  
Queueing System ◽  
Retrial Queue ◽  
System Size ◽  
Stochastic Decomposition ◽  
Continuous Time Model ◽  
Time Model ◽  
Different Types ◽  
Number Of Customers ◽  
The Relationship

We analyze a discrete-timeGeo/G/1 retrial queue with two different types of vacations and general retrial times. Two different types of vacation policies are investigated in this model, one of which is nonexhaustive urgent vacation during serving and the other is normal exhaustive vacation. For this model, we give the steady-state analysis for the considered queueing system. Firstly, we obtain the generating functions of the number of customers in our model. Then, we obtain the closed-form expressions of some performance measures and also give a stochastic decomposition result for the system size. Moreover, the relationship between this discrete-time model and the corresponding continuous-time model is also investigated. Finally, some numerical results are provided to illustrate the effect of nonexhaustive urgent vacation on some performance characteristics of the system.

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.

Analysis of a Discrete-Time Queueing System with a Markov-Modulated input Process and constant service Rate

1996 ◽  
Vol 10 (3) ◽  
pp. 429-441 ◽  
Author(s):  
Woo-Yong Choi ◽  
Chi-Hyuck Jun
Keyword(s):  
Discrete Time ◽  
Queueing System ◽  
Service Rate ◽  
System State ◽  
Input Process ◽  
New Approach ◽  
One Dimensional ◽  
Markov Modulated ◽  
Modulated Process ◽  
Renewal Cycle

We propose a new approach to the analysis of a discrete-time queueing system whose input is generated by a Markov-modulated process and whose service rate is constant. Renewal cycles are identified and the system state on each renewal cycle is modeled as a one-dimensional Markov chain.

scholarly journals Performance Characteristics of Discrete-Time Queue With Variant Working Vacations

2020 ◽  
Vol 11 (2) ◽  
pp. 1-21
Author(s):  
P. Vijaya Laxmi ◽  
Rajesh P.
Keyword(s):  
Discrete Time ◽  
Cost Model ◽  
Probability Generating Function ◽  
System Size ◽  
Service Rate ◽  
Single Server ◽  
Working Vacation ◽  
Optimal Service ◽  
Working Vacations ◽  
Vacation Period

This article analyzes an infinite buffer discrete-time single server queueing system with variant working vacations in which customers arrive according to a geometric process. As soon as the system becomes empty, the server takes working vacations. The server will take a maximum number K of working vacations until either he finds at least on customer in the queue or the server has exhaustively taken all the vacations. The service times during regular busy period, working vacation period and vacation times are assumed to be geometrically distributed. The probability generating function of the steady-state probabilities and the closed form expressions of the system size when the server is in different states have been derived. In addition, some other performance measures, their monotonicity with respect to K and a cost model are presented to determine the optimal service rate during working vacation.

ANALYSIS OF A BULK QUEUE WITH FAST AND SLOW SERVICE RATES AND MULTIPLE VACATIONS

2005 ◽  
Vol 22 (02) ◽  
pp. 239-260 ◽  
Author(s):  
R. ARUMUGANATHAN ◽  
K. S. RAMASWAMI
Keyword(s):  
Performance Measures ◽  
Queue Length ◽  
Queueing System ◽  
Cost Model ◽  
Probability Generating Function ◽  
Service Rate ◽  
Multiple Vacations ◽  
Bulk Queue ◽  
Service Rates ◽  
Number Of Customers

We analyze a Mx/G(a,b)/1 queueing system with fast and slow service rates and multiple vacations. The server does the service with a faster rate or a slower rate based on the queue length. At a service completion epoch (or) at a vacation completion epoch if the number of customers waiting in the queue is greater than or equal to N (N > b), then the service is rendered at a faster rate, otherwise with a slower service rate. After finishing a service, if the queue length is less than 'a' the server leaves for a vacation of random length. When he returns from the vacation, if the queue length is still less than 'a' he leaves for another vacation and so on until he finally finds atleast 'a' customers waiting for service. After a service (or) a vacation, if the server finds atleast 'a' customers waiting for service say ξ, then he serves a batch of min (ξ, b) customers, where b ≥ a. We derive the probability generating function of the queue size at an arbitrary time. Various performance measures are obtained. A cost model is discussed with a numerical solution.

scholarly journals Finite M/M/1 retrial model with changeable service rate

2021 ◽  
Vol 56 (1) ◽  
pp. 96-102
Author(s):  
M.S. Bratiichuk ◽  
A.A. Chechelnitsky ◽  
I.Ya. Usar
Keyword(s):  
Queueing System ◽  
Service Rate ◽  
Numerical Examples ◽  
Retrial Queueing System ◽  
Retrial Queueing ◽  
Explicit Formulae ◽  
Number Of Customers ◽  
Theoretical Results

The article deals with M/M/1 -type retrial queueing system with finite orbit. It is supposedthat service rate depends on the loading of the system. The explicit formulae for ergodicdistribution of the number of customers in the system are obtained. The theoretical results areillustrated by numerical examples.

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