A Study on M x /G/1 Queuing System with Essential, Optional Service, Modified Vacation and Setup time
|
|
- Kerrie McDowell
- 5 years ago
- Views:
Transcription
1 A Study on M x /G/1 Queuing System with Essential, Optional Service, Modified Vacation and Setup time E. Ramesh Kumar 1, L. Poornima 2 1 Associate Professor, Department of Mathematics, CMS College of Science & Commerce, Coimbatore, Tamil Nadu 2 Department of Mathematics, CMS College of Science & Commerce, Coimbatore, Tamil Nadu Abstract:-A study on M x /G/1 queuing system where the arrival follows Poisson Process and the server provides service in two stages. The first stage service is essential and the second stage of service is optional. If the system is empty then the server moves for a vacation of random duration and after returning from vacation the server decides to take a second vacation which is optional. Such a customer behavior is considered in both busy time and vacation time of the system. The vacation time follows general (arbitrary) distribution. Before providing service to a new customer or a batch of customers that joins the system in the renewed busy period, the server enters into a random setup time process such that setup time follows exponential distribution. We discuss the transient behavior and the corresponding steady state results with the performance measures of the model. For this model, we obtain the time dependent solution and the corresponding steady state solutions. Also we derive the performance measure the mean queue size and the average waiting time explicitly. Keywords: Batch Arrival, Essential Service, Optional Service, Setup Time, Bernoulli Vacation. I I. INTRODUCTION n this paper we consider queuing system such that the customers are arriving in batches according to Poisson stream. The server provide a first essential service to all incoming customers and a second optional service will be provided to only some of them those who demand it. Both the essential and optional service times are assumed to follow general distribution. Queues with server vacations have emerged as an essential area of queuing theory and have been studied widely and fruitfully due to their various applications in communication system, manufacturing systems, textile processing industries, food industries, and chemical industries etc. In this models server s setup time corresponds to the preparatory work of the server before starting his service. Hur. S and Park. J [1] and Ke. J.C [2] is some of the authors who analyzed the N policy of M x /G/1 queuing models with server s setup time. The batch arrival queuing system with double threshold policy, setup time and vacation are analyzed by Lee. S.S [3] is among the most general queuing system with threshold policies. In everyday life there are queuing situations where all the arriving customers require the first essential service and some may require the second optional service provided by the same server. Madan K.C [4] has introduced the concept of second optional service, where the customers may depart from the system either with probability (1-r) or may immediately opt for second optional service with probability r. The classical single server vacation model was generalized by Seri. L.D and Finn S.G [5] by considering working vacation. Simple explicit formula for the mean, variance of the number of customers in the system was provided. Arumuganathan. R and Jeyakumar S [6] considered M x /G/1 queuing system with multiple vacations, setup times and closedown times under N- policy. In many real cases, Almasi B and Roszik J and Sztrik. J [7] examined a single server retrial queue with finite number of homogenous sources of calls and a single removable server. Stability conditions are provided by Sherman. N.P and Kharoufeh. J.P [8] for an M/M/1 retrial queue with infinite capacity orbit. Alfa. A.S and Yang. X [9] studied a multiserver queuing system with identical unreliable server with phase type distributed service time. Madan [4] who first introduced the concept of second optional service while studying the time dependent as well as steady behavior of an M/G/1 queuing system with no waiting capacity, using supplementary variable technique. Supplementary variable technique was used to develop the time dependent probability generating function in terms of their Laplace transform for M/G/1 queue by Al-Jararha. J and Madan K.C [10]. It concentrates on such queuing system with setup time. In this paper probability generating function of the steady state queue size at an arbitrary time, expected queue length, expected busy period, expected idle period and numerical illustrations are presents. II. MATHEMATICAL DESCRIPTION OF THE QUEUING MODEL Let λdt be the first order probability of arrival of customers in batches in the system during a short period of time (t, t+dt). The single server provides the first essential service to all arriving customers. Let B 1 (v) and b 1 (v) be the distribution function and the density function of first service times respectively. There is a single server which provides service following a general (arbitrary) distribution with distribution function B i (v) and density function b (v). Let µ i (x) dx be the conditional probability density function of service completion of i th Page 72
2 service during the interval (x, x+dx) given that the elapsed time is x, so that, and There is a single server which provides setup time following a general (arbitrary) distribution with distribution function Y i (v) and density function y (v). Let ξ i (x) dx be the conditional probability density function of service completion of i th service during the interval (x, x+dx) given that the elapsed time is x, so that, and The vacation time of the server follows a general(arbitrary)distribution with distribution function V 1 (s) and the density function v 1 (s).let v 1k (x) dx be the conditional probability of a completion of a vacation during the interval (x,x+dx) given that the elapsed vacation time is x so that ; k=1, 2, 3..M and Figure: 1. Schematic Representation of the Queuing Model 2.1 Notations λ The following notations are follows: = Bulk arrival = k th vacation in n th customer in the queue. = i th service in n th customer in the queue. P 0 (x) = probability at time x there are no customers in the system and the server is idle. Y n (x), v(x), s1(x), s 2 (x), r(x) and g(x) = Probability density function of setup time, V, S 1, S 2, R and G. = Laplace-Stieltjes transform of v(x), s 1 (x), s 2 (x), r(x) and g(x) III. EQUATIONS GOVERNING THE SYSTEM We let, probability that at time t the server is active providing i th service and there are n (n 1) customers in the queue including the one being served and the elapsed service time for this customers is x. Consequently denotes the probability that at time t there are n customers in the queue excluding the one customer in i th irrespective of the value of x. service Probability that at time t the server is on k th vacation with elapsed vacation time x, and there are n (n 0) customers waiting in the queue for service. Consequently denotes the probability that at time t there are n customers in the queue and the server is on k th vacation irrespective of the value of x. Probability that at time t the server is on k th vacation with setup time x, and there are n (n 0) customers waiting in the queue for service. Consequently denotes the probability that at time t there are n customers in the queue and the server is on k th vacation irrespective of the value of x. (t) = Probability that at time t there are no customers in the system and the server is idle but available in the system. 3.1 Steady State System Size Distribution: The following equations are obtained for the queuing system, Page 73
3 The steady state equations are, The Laplace-Stieltjes transforms of P n (1) (x), P n (2) (x), (x), V n (k) (x) and y n (x) are defined as: Page 74
4 (15) Taking Laplace-Stieltjes transform on both sides, we get [ ] [ ] θ [ ] [ ] 3.2. The system size distribution: ( θ) (θ) ( θ) (θ) ( θ) (θ) ( θ) (θ) Page 75
5 After simplification we get, ( ( ) ) [ ] ( ) ( ( ) ) 0 ( ) 1 a ( ( ) ) [ ( ) ( ( ) ) [ ] ( ) ] ( ( ) ) [ ] ( ) Let us define the following, Put θ=0, we get the PGF of a queue size P (z) at an arbitrary time epoch as { ( ( ) ) ( ( ) ) ( ( ) ) ( ( ) ). ( ) ( )/ ( ) } IV. THE STEADY STATE ANALYSIS Since P (z) satisfies the steady state condition P (1) =1. Since P (z) is analytic within and on the unit circle. The Numerator must vanish at the point of circles the equations can be solved by any suitable numerical technique. V. PERFORMANCE MEASURES Some important performance measures using the PGF of the queue size P (z) of (29) are derived Expected Busy Period Let T be the residence time that the server is rendering service or under second optional service. Therefore T=S with probability Π and T=S+G with probability 1-Π. The expected length of busy period is given by E (B) = E (B/J=0) P (J=0) +E (B/J=1) P (J=1) = *, - +, -, 5.2. Expected Idle Period: (30) Let I be the Idle Period random variable, then the expected idle period is, - (31) Where is the probability that there are n customers arriving during a vacation. The expected length of idle period due to multiple vacation E (I) is given by E (I) = E (I/U=0) P (U=0) +E (I/U=1) P (U=1) = E (V) P (U=0) + [E (V) +E (I)] P (U=1). Page 76
6 E (I) = Expected Queue Size: where, P (U=0) =, Let L q denote the mean number of customers in the queue under the steady state, then form. Mean waiting time of a customer could be found, as follows (37) By using little s Formula. L q = Since this formula gives, then we write P (z) = Where N (z) and D (z) are the numerator and denominator of the right hand side of equation (29) respectively, then we use, - Where {, - } { ( ), - ( ), - ( ( ), -, - ( ), ( -, - ( ) ( ( ) (, - } VI. NUMERICAL ILLUSTRATION: The unknown probabilities of the queue size distribution are computed using numerical techniques. Using Mat lab, the zeros of the function ( ) ( ) ( )are obtained and simultaneous equations are solved. A Numerical example is analyzed with the following assumptions: Table 1: Threshold value vs. performance measures with μ=7. λ E(Q) E(B) E(I) E(W) Where E (V 2 ) is the second moment of the vacation time and we substitute the values of from equations (31), (32), (33) and (34) in to (30) equation we obtain L q in a closed Figure 2 : Arrival rate vs Expected length of queue and expected waiting time Table 2: Threshold value vs. performance measures with μ=8. Page 77
7 λ E(Q) E(B) E(I) E(W) VII. CONCLUSION This paper analyze the steady state solution of batch arrival single server with second optional service such that first essential service for all incoming customer whereas few of them require a second optional service and multi optional vacation. The queue with working vacation may be applicable in modeling of many practical situations related to computers, communications and productions systems, etc., wherein the server works at different service rates rather than completely stopping the service during a vacation. REFERENCES [1]. Hur. S and Park. J (1999). The effect of different arrival rates on N-policy of M/G/1 with server setup. Applied Mathematical modeling [2]. Ke. J.C. Operating characteristics analysis of M x /G /1 system with variant vacation policy and balking, Applied Mathematical Modeling, Vol.31, (2007), [3]. Lee. H.W, Lee. S.S and Park. J.O (1994). Analysis of the M x /G/1 queue with N policy and multiple threshold control with early setup. International Journal of system science. [4]. Madan. K.C (2000). An M/G/1 queue with second optional service. Queuing systems. [5]. Seri. L.D and Finn. S.G. M/M/1 queues with working vacations (M/M/1/WV) Performance Evaluation, Vol.50, (2002), [6]. Arumuganathan. R and Jeyakumar. S. Steady state analysis of bulk queue with multiple vacations, setup times with N-policy and closedown times. Applied Mathematical Modeling. Vol.29, (2005) [7]. Almasi. B, Roszik. J.and Sztrik. J. Homogeneous finite source retrial queues with server subject to breakdowns and repair, Mathematical and Computer Modeling, Vol.42, (2005), [8]. Sherman. N.P and Kharoufeh. J.P. An M / M / 1 retrial queue with unreliable server, Operations Research Letters, Vol.34, (2006), [9]. Yang. X and Alfa. A.S. A class of multi server queuing system with server failure, Computers & Industrial Engineering, Vol. 56, (2009). [10]. Al-Jararha. J and Madan. K.C. An M/G/1 queue with second optional service with general service time distribution, Information Management Science, Vol. 14, (2003). Page 78
A Two Phase Service M/G/1 Vacation Queue With General Retrial Times and Non-persistent Customers
Int. J. Open Problems Compt. Math., Vol. 3, No. 2, June 21 ISSN 1998-6262; Copyright c ICSRS Publication, 21 www.i-csrs.org A Two Phase Service M/G/1 Vacation Queue With General Retrial Times and Non-persistent
More informationTime Dependent Solution Of M [X] /G/1 Queueing Model With second optional service, Bernoulli k-optional Vacation And Balking
International Journal of Scientific and Research Publications,Volume 3,Issue 9, September 213 ISSN 225-3153 1 Time Dependent Solution Of M [X] /G/1 Queueing Model With second optional service, Bernoulli
More informationBatch Arrival Queueing System. with Two Stages of Service
Int. Journal of Math. Analysis, Vol. 8, 2014, no. 6, 247-258 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.411 Batch Arrival Queueing System with Two Stages of Service S. Maragathasundari
More informationM [X] /G/1 with Second Optional Service, Multiple Vacation, Breakdown and Repair
International Journal of Research in Engineering and Science (IJRES) ISSN (Online): 3-9364, ISSN (Print): 3-9356 Volume Issue ǁ November. 4 ǁ PP.7-77 M [X] /G/ with Second Optional Service, Multiple Vacation,
More informationPriority Queueing System with a Single Server Serving Two Queues M [X 1], M [X 2] /G 1, G 2 /1 with Balking and Optional Server Vacation
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 11, Issue 1 (June 216), pp. 61 82 Applications and Applied Mathematics: An International Journal (AAM) Priority Queueing System
More informationOptimal Strategy Analysis of an N-policy Twophase. Server Startup and Breakdowns
Int. J. Open Problems Compt. Math., Vol. 3, No. 4, December 200 ISSN 998-6262; Copyright ICSRS Publication, 200 www.i-csrs.org Optimal Strategy Analysis of an N-policy Twophase M X /M/ Gated Queueing System
More informationAn M/M/1 Retrial Queue with Unreliable Server 1
An M/M/1 Retrial Queue with Unreliable Server 1 Nathan P. Sherman 2 and Jeffrey P. Kharoufeh 3 Department of Operational Sciences Air Force Institute of Technology Abstract We analyze an unreliable M/M/1
More informationSystem with a Server Subject to Breakdowns
Applied Mathematical Sciences Vol. 7 213 no. 11 539 55 On Two Modifications of E 2 /E 2 /1/m Queueing System with a Server Subject to Breakdowns Michal Dorda VSB - Technical University of Ostrava Faculty
More informationMultiserver Queueing Model subject to Single Exponential Vacation
Journal of Physics: Conference Series PAPER OPEN ACCESS Multiserver Queueing Model subject to Single Exponential Vacation To cite this article: K V Vijayashree B Janani 2018 J. Phys.: Conf. Ser. 1000 012129
More informationInternational Journal of Mathematical Archive-5(1), 2014, Available online through ISSN
International Journal of Mathematical Archive-5(1), 2014, 125-133 Available online through www.ijma.info ISSN 2229 5046 MAXIMUM ENTROPY ANALYSIS OF UNRELIABLE SERVER M X /G/1 QUEUE WITH ESSENTIAL AND MULTIPLE
More informationStationary Analysis of a Multiserver queue with multiple working vacation and impatient customers
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 932-9466 Vol. 2, Issue 2 (December 207), pp. 658 670 Applications and Applied Mathematics: An International Journal (AAM) Stationary Analysis of
More informationSECOND OPTIONAL SERVICE QUEUING MODEL WITH FUZZY PARAMETERS
International Journal of Applied Engineering and Technolog ISSN: 2277-212X (Online) 2014 Vol. 4 (1) Januar-March, pp.46-53/mar and Jenitta SECOND OPTIONA SERVICE QEING MODE WITH FZZY PARAMETERS *K. Julia
More informationNon-Persistent Retrial Queueing System with Two Types of Heterogeneous Service
Global Journal of Theoretical and Applied Mathematics Sciences. ISSN 2248-9916 Volume 1, Number 2 (211), pp. 157-164 Research India Publications http://www.ripublication.com Non-Persistent Retrial Queueing
More informationTransient Solution of M [X 1] with Priority Services, Modified Bernoulli Vacation, Bernoulli Feedback, Breakdown, Delaying Repair and Reneging
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 12, Issue 2 (December 217), pp. 633 657 Applications and Applied Mathematics: An International Journal (AAM) Transient Solution
More informationTwo Heterogeneous Servers Queueing-Inventory System with Sharing Finite Buffer and a Flexible Server
Two Heterogeneous Servers Queueing-Inventory System with Sharing Finite Buffer and a Flexible Server S. Jehoashan Kingsly 1, S. Padmasekaran and K. Jeganathan 3 1 Department of Mathematics, Adhiyamaan
More informationAn M/G/1 Retrial Queue with Non-Persistent Customers, a Second Optional Service and Different Vacation Policies
Applied Mathematical Sciences, Vol. 4, 21, no. 4, 1967-1974 An M/G/1 Retrial Queue with Non-Persistent Customers, a Second Optional Service and Different Vacation Policies Kasturi Ramanath and K. Kalidass
More informationA Heterogeneous Two-Server Queueing System with Balking and Server Breakdowns
The Eighth International Symposium on Operations Research and Its Applications (ISORA 09) Zhangjiajie, China, September 20 22, 2009 Copyright 2009 ORSC & APORC, pp. 230 244 A Heterogeneous Two-Server Queueing
More informationIntroduction to Queuing Networks Solutions to Problem Sheet 3
Introduction to Queuing Networks Solutions to Problem Sheet 3 1. (a) The state space is the whole numbers {, 1, 2,...}. The transition rates are q i,i+1 λ for all i and q i, for all i 1 since, when a bus
More informationResearch Article Binomial Schedule for an M/G/1 Type Queueing System with an Unreliable Server under N-Policy
Advances in Decision Sciences, Article ID 819718, 6 pages http://dx.doi.org/10.1155/2014/819718 Research Article Binomial Schedule for an M/G/1 Type Queueing System with an Unreliable Server under N-Policy
More informationReview Paper Machine Repair Problem with Spares and N-Policy Vacation
Research Journal of Recent Sciences ISSN 2277-2502 Res.J.Recent Sci. Review Paper Machine Repair Problem with Spares and N-Policy Vacation Abstract Sharma D.C. School of Mathematics Statistics and Computational
More informationJ. MEDHI STOCHASTIC MODELS IN QUEUEING THEORY
J. MEDHI STOCHASTIC MODELS IN QUEUEING THEORY SECOND EDITION ACADEMIC PRESS An imprint of Elsevier Science Amsterdam Boston London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo Contents
More informationA Vacation Queue with Additional Optional Service in Batches
Applied Mathematical Sciences, Vol. 3, 2009, no. 24, 1203-1208 A Vacation Queue with Additional Optional Service in Batches S. Pazhani Bala Murugan Department of Mathematics, Annamalai University Annamalainagar-608
More informationComprehending Single Server Queueing System with Interruption, Resumption and Repeat
Comprehending ingle erver Queueing ystem with Interruption, Resumption and Repeat Dashrath 1, Dr. Arun Kumar ingh 1 Research cholor, hri Venkateshwara University, UP dyadav551@gmail.com, arunsinghgalaxy@gmail.com
More informationOutline. Finite source queue M/M/c//K Queues with impatience (balking, reneging, jockeying, retrial) Transient behavior Advanced Queue.
Outline Finite source queue M/M/c//K Queues with impatience (balking, reneging, jockeying, retrial) Transient behavior Advanced Queue Batch queue Bulk input queue M [X] /M/1 Bulk service queue M/M [Y]
More informationAnalysis of a Machine Repair System with Warm Spares and N-Policy Vacations
The 7th International Symposium on Operations Research and Its Applications (ISORA 08) ijiang, China, October 31 Novemver 3, 2008 Copyright 2008 ORSC & APORC, pp. 190 198 Analysis of a Machine Repair System
More informationAnalysis of a Two-Phase Queueing System with Impatient Customers and Multiple Vacations
The Tenth International Symposium on Operations Research and Its Applications (ISORA 211) Dunhuang, China, August 28 31, 211 Copyright 211 ORSC & APORC, pp. 292 298 Analysis of a Two-Phase Queueing System
More informationQueues and Queueing Networks
Queues and Queueing Networks Sanjay K. Bose Dept. of EEE, IITG Copyright 2015, Sanjay K. Bose 1 Introduction to Queueing Models and Queueing Analysis Copyright 2015, Sanjay K. Bose 2 Model of a Queue Arrivals
More informationCost Analysis of Two-Phase M/M/1 Queueing system in the Transient state with N-Policy and Server Breakdowns
IN (e): 2250 3005 Volume, 07 Issue, 9 eptember 2017 International Journal of Computational Engineering Research (IJCER) Cost Analysis of Two-Phase M/M/1 Queueing system in the Transient state with N-Policy
More informationThe discrete-time Geom/G/1 queue with multiple adaptive vacations and. setup/closedown times
ISSN 1750-9653, England, UK International Journal of Management Science and Engineering Management Vol. 2 (2007) No. 4, pp. 289-296 The discrete-time Geom/G/1 queue with multiple adaptive vacations and
More informationM/M/1 TWO-PHASE GATED QUEUEING SYSTEM WITH UNRELIABLE SERVER AND STATE DEPENDENT ARRIVALS
Int. J. Chem. Sci.: 14(3), 2016, 1742-1754 ISSN 0972-768X www.sadgurupublications.com M/M/1 TWO-PHASE GATED QUEUEING SYSTEM WITH UNRELIABLE SERVER AND STATE DEPENDENT ARRIVALS S. HANUMANTHA RAO a*, V.
More informationP (L d k = n). P (L(t) = n),
4 M/G/1 queue In the M/G/1 queue customers arrive according to a Poisson process with rate λ and they are treated in order of arrival The service times are independent and identically distributed with
More informationQueueing Systems: Lecture 3. Amedeo R. Odoni October 18, Announcements
Queueing Systems: Lecture 3 Amedeo R. Odoni October 18, 006 Announcements PS #3 due tomorrow by 3 PM Office hours Odoni: Wed, 10/18, :30-4:30; next week: Tue, 10/4 Quiz #1: October 5, open book, in class;
More informationSTEADY-STATE BEHAVIOR OF AN M/M/1 QUEUE IN RANDOM ENVIRONMENT SUBJECT TO SYSTEM FAILURES AND REPAIRS. S. Sophia 1, B. Praba 2
International Journal of Pure and Applied Mathematics Volume 101 No. 2 2015, 267-279 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v101i2.11
More informationGeom/G 1,G 2 /1/1 REPAIRABLE ERLANG LOSS SYSTEM WITH CATASTROPHE AND SECOND OPTIONAL SERVICE
J Syst Sci Complex (2011) 24: 554 564 Geom/G 1,G 2 /1/1 REPAIRABLE ERLANG LOSS SYSTEM WITH CATASTROPHE AND SECOND OPTIONAL SERVICE Yinghui TANG Miaomiao YU Cailiang LI DOI: 10.1007/s11424-011-8339-2 Received:
More informationQueueing Theory I Summary! Little s Law! Queueing System Notation! Stationary Analysis of Elementary Queueing Systems " M/M/1 " M/M/m " M/M/1/K "
Queueing Theory I Summary Little s Law Queueing System Notation Stationary Analysis of Elementary Queueing Systems " M/M/1 " M/M/m " M/M/1/K " Little s Law a(t): the process that counts the number of arrivals
More informationPreemptive Resume Priority Retrial Queue with. Two Classes of MAP Arrivals
Applied Mathematical Sciences, Vol. 7, 2013, no. 52, 2569-2589 HIKARI Ltd, www.m-hikari.com Preemptive Resume Priority Retrial Queue with Two Classes of MAP Arrivals M. Senthil Kumar 1, S. R. Chakravarthy
More informationλ λ λ In-class problems
In-class problems 1. Customers arrive at a single-service facility at a Poisson rate of 40 per hour. When two or fewer customers are present, a single attendant operates the facility, and the service time
More informationEQUILIBRIUM STRATEGIES IN AN M/M/1 QUEUE WITH SETUP TIMES AND A SINGLE VACATION POLICY
EQUILIBRIUM STRATEGIES IN AN M/M/1 QUEUE WITH SETUP TIMES AND A SINGLE VACATION POLICY Dequan Yue 1, Ruiling Tian 1, Wuyi Yue 2, Yaling Qin 3 1 College of Sciences, Yanshan University, Qinhuangdao 066004,
More informationRetrial queue for cloud systems with separated processing and storage units
Retrial queue for cloud systems with separated processing and storage units Tuan Phung-Duc Department of Mathematical and Computing Sciences Tokyo Institute of Technology Ookayama, Meguro-ku, Tokyo, Japan
More informationBIRTH DEATH PROCESSES AND QUEUEING SYSTEMS
BIRTH DEATH PROCESSES AND QUEUEING SYSTEMS Andrea Bobbio Anno Accademico 999-2000 Queueing Systems 2 Notation for Queueing Systems /λ mean time between arrivals S = /µ ρ = λ/µ N mean service time traffic
More informationInternational Journal of Informative & Futuristic Research ISSN:
Research Paper Volume 3 Issue 2 August 206 International Journal of Informative & Futuristic Research ISSN: 2347-697 Analysis Of FM/M//N Queuing System With Reverse Balking And Reverse Reneging Paper ID
More informationThe Performance Impact of Delay Announcements
The Performance Impact of Delay Announcements Taking Account of Customer Response IEOR 4615, Service Engineering, Professor Whitt Supplement to Lecture 21, April 21, 2015 Review: The Purpose of Delay Announcements
More informationComputer Networks More general queuing systems
Computer Networks More general queuing systems Saad Mneimneh Computer Science Hunter College of CUNY New York M/G/ Introduction We now consider a queuing system where the customer service times have a
More informationBurst Arrival Queues with Server Vacations and Random Timers
Burst Arrival Queues with Server acations and Random Timers Merav Shomrony and Uri Yechiali Department of Statistics and Operations Research School of Mathematical Sciences Raymond and Beverly Sackler
More informationAn M/M/1 Queue in Random Environment with Disasters
An M/M/1 Queue in Random Environment with Disasters Noam Paz 1 and Uri Yechiali 1,2 1 Department of Statistics and Operations Research School of Mathematical Sciences Tel Aviv University, Tel Aviv 69978,
More informationChapter 8 Queuing Theory Roanna Gee. W = average number of time a customer spends in the system.
8. Preliminaries L, L Q, W, W Q L = average number of customers in the system. L Q = average number of customers waiting in queue. W = average number of time a customer spends in the system. W Q = average
More informationA Batch Arrival Retrial Queue with Two Phases of Service, Feedback and K Optional Vacations
Applied Mathematical Sciences, Vol. 6, 212, no. 22, 171-187 A Batch Arrival Retrial Queue with Two Phases of Service, Feedback and K Optional Vacations D. Arivudainambi and P. Godhandaraman Department
More informationQueuing Theory. The present section focuses on the standard vocabulary of Waiting Line Models.
Queuing Theory Introduction Waiting lines are the most frequently encountered problems in everyday life. For example, queue at a cafeteria, library, bank, etc. Common to all of these cases are the arrivals
More informationAnalysis of an M/M/1/N Queue with Balking, Reneging and Server Vacations
Analysis of an M/M/1/N Queue with Balking, Reneging and Server Vacations Yan Zhang 1 Dequan Yue 1 Wuyi Yue 2 1 College of Science, Yanshan University, Qinhuangdao 066004 PRChina 2 Department of Information
More informationQueuing Analysis. Chapter Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Queuing Analysis Chapter 13 13-1 Chapter Topics Elements of Waiting Line Analysis The Single-Server Waiting Line System Undefined and Constant Service Times Finite Queue Length Finite Calling Problem The
More informationCost Analysis of a vacation machine repair model
Available online at www.sciencedirect.com Procedia - Social and Behavioral Sciences 25 (2011) 246 256 International Conference on Asia Pacific Business Innovation & Technology Management Cost Analysis
More informationResearch Article On the Discrete-Time Geo X /G/1 Queues under N-Policy with Single and Multiple Vacations
Applied Mathematics Volume 203, Article ID 58763, 6 pages http://dx.doi.org/0.55/203/58763 Research Article On the Discrete-Time Geo X /G/ Queues under N-Policy with Single and Multiple Vacations Sung
More informationSlides 9: Queuing Models
Slides 9: Queuing Models Purpose Simulation is often used in the analysis of queuing models. A simple but typical queuing model is: Queuing models provide the analyst with a powerful tool for designing
More informationQueuing Analysis of Markovian Queue Having Two Heterogeneous Servers with Catastrophes using Matrix Geometric Technique
International Journal of Statistics and Systems ISSN 0973-2675 Volume 12, Number 2 (2017), pp. 205-212 Research India Publications http://www.ripublication.com Queuing Analysis of Markovian Queue Having
More information5/15/18. Operations Research: An Introduction Hamdy A. Taha. Copyright 2011, 2007 by Pearson Education, Inc. All rights reserved.
The objective of queuing analysis is to offer a reasonably satisfactory service to waiting customers. Unlike the other tools of OR, queuing theory is not an optimization technique. Rather, it determines
More informationChapter 5: Special Types of Queuing Models
Chapter 5: Special Types of Queuing Models Some General Queueing Models Discouraged Arrivals Impatient Arrivals Bulk Service and Bulk Arrivals OR37-Dr.Khalid Al-Nowibet 1 5.1 General Queueing Models 1.
More informationMarkov Chains. X(t) is a Markov Process if, for arbitrary times t 1 < t 2 <... < t k < t k+1. If X(t) is discrete-valued. If X(t) is continuous-valued
Markov Chains X(t) is a Markov Process if, for arbitrary times t 1 < t 2
More informationM/G/1 and M/G/1/K systems
M/G/1 and M/G/1/K systems Dmitri A. Moltchanov dmitri.moltchanov@tut.fi http://www.cs.tut.fi/kurssit/elt-53606/ OUTLINE: Description of M/G/1 system; Methods of analysis; Residual life approach; Imbedded
More informationM/M/1 Queueing System with Delayed Controlled Vacation
M/M/1 Queueing System with Delayed Controlled Vacation Yonglu Deng, Zhongshan University W. John Braun, University of Winnipeg Yiqiang Q. Zhao, University of Winnipeg Abstract An M/M/1 queue with delayed
More informationNon Markovian Queues (contd.)
MODULE 7: RENEWAL PROCESSES 29 Lecture 5 Non Markovian Queues (contd) For the case where the service time is constant, V ar(b) = 0, then the P-K formula for M/D/ queue reduces to L s = ρ + ρ 2 2( ρ) where
More information57:022 Principles of Design II Final Exam Solutions - Spring 1997
57:022 Principles of Design II Final Exam Solutions - Spring 1997 Part: I II III IV V VI Total Possible Pts: 52 10 12 16 13 12 115 PART ONE Indicate "+" if True and "o" if False: + a. If a component's
More informationAIR FORCE INSTITUTE OF TECHNOLOGY
APPROXIMATE ANALYSIS OF AN UNRELIABLE M/M/2 RETRIAL QUEUE THESIS Brian P. Crawford, 1 Lt, USAF AFIT/GOR/ENS/07-05 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY Wright-Patterson
More informationAn Analysis of the Preemptive Repeat Queueing Discipline
An Analysis of the Preemptive Repeat Queueing Discipline Tony Field August 3, 26 Abstract An analysis of two variants of preemptive repeat or preemptive repeat queueing discipline is presented: one in
More informationContents Preface The Exponential Distribution and the Poisson Process Introduction to Renewal Theory
Contents Preface... v 1 The Exponential Distribution and the Poisson Process... 1 1.1 Introduction... 1 1.2 The Density, the Distribution, the Tail, and the Hazard Functions... 2 1.2.1 The Hazard Function
More informationLink Models for Circuit Switching
Link Models for Circuit Switching The basis of traffic engineering for telecommunication networks is the Erlang loss function. It basically allows us to determine the amount of telephone traffic that can
More informationINDEX. production, see Applications, manufacturing
INDEX Absorbing barriers, 103 Ample service, see Service, ample Analyticity, of generating functions, 100, 127 Anderson Darling (AD) test, 411 Aperiodic state, 37 Applications, 2, 3 aircraft, 3 airline
More informationInventory Ordering Control for a Retrial Service Facility System Semi- MDP
International Journal of Engineering Science Invention (IJESI) ISS (Online): 239 6734, ISS (Print): 239 6726 Volume 7 Issue 6 Ver I June 208 PP 4-20 Inventory Ordering Control for a Retrial Service Facility
More informationRelating Polling Models with Zero and Nonzero Switchover Times
Relating Polling Models with Zero and Nonzero Switchover Times Mandyam M. Srinivasan Management Science Program College of Business Administration The University of Tennessee Knoxville, TN 37996-0562 Shun-Chen
More informationBalking and Re-service in a Vacation Queue with Batch Arrival and Two Types of Heterogeneous Service
Journal of Mathematics Research; Vol. 4, No. 4; 212 ISSN 1916-9795 E-ISSN 1916-989 Publishe by Canaian Center of Science an Eucation Balking an Re-service in a Vacation Queue with Batch Arrival an Two
More informationPerformance Analysis of an M/M/c/N Queueing System with Balking, Reneging and Synchronous Vacations of Partial Servers
The Sixth International Symposium on Operations Research and Its Applications (ISORA 06) Xinjiang, China, August 8 12, 2006 Copyright 2006 ORSC & APORC pp. 128 143 Performance Analysis of an M/M/c/ Queueing
More informationChapter 2. Poisson Processes. Prof. Shun-Ren Yang Department of Computer Science, National Tsing Hua University, Taiwan
Chapter 2. Poisson Processes Prof. Shun-Ren Yang Department of Computer Science, National Tsing Hua University, Taiwan Outline Introduction to Poisson Processes Definition of arrival process Definition
More informationCHAPTER 3 STOCHASTIC MODEL OF A GENERAL FEED BACK QUEUE NETWORK
CHAPTER 3 STOCHASTIC MODEL OF A GENERAL FEED BACK QUEUE NETWORK 3. INTRODUCTION: Considerable work has been turned out by Mathematicians and operation researchers in the development of stochastic and simulation
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.262 Discrete Stochastic Processes Midterm Quiz April 6, 2010 There are 5 questions, each with several parts.
More information4.7 Finite Population Source Model
Characteristics 1. Arrival Process R independent Source All sources are identical Interarrival time is exponential with rate for each source No arrivals if all sources are in the system. OR372-Dr.Khalid
More informationPerformance Analysis and Evaluation of Digital Connection Oriented Internet Service Systems
Performance Analysis and Evaluation of Digital Connection Oriented Internet Service Systems Shunfu Jin 1 and Wuyi Yue 2 1 College of Information Science and Engineering Yanshan University, Qinhuangdao
More informationAn M/M/1/N Queuing system with Encouraged Arrivals
Global Journal of Pure and Applied Mathematics. ISS 0973-1768 Volume 13, umber 7 (2017), pp. 3443-3453 Research India Publications http://www.ripublication.com An M/M/1/ Queuing system with Encouraged
More informationInternational Journal of Pure and Applied Mathematics Volume 28 No ,
International Journal of Pure and Applied Mathematics Volume 28 No. 1 2006, 101-115 OPTIMAL PERFORMANCE ANALYSIS OF AN M/M/1/N QUEUE SYSTEM WITH BALKING, RENEGING AND SERVER VACATION Dequan Yue 1, Yan
More informationPerformance Evaluation of Queuing Systems
Performance Evaluation of Queuing Systems Introduction to Queuing Systems System Performance Measures & Little s Law Equilibrium Solution of Birth-Death Processes Analysis of Single-Station Queuing Systems
More informationPBW 654 Applied Statistics - I Urban Operations Research
PBW 654 Applied Statistics - I Urban Operations Research Lecture 2.I Queuing Systems An Introduction Operations Research Models Deterministic Models Linear Programming Integer Programming Network Optimization
More information2905 Queueing Theory and Simulation PART III: HIGHER DIMENSIONAL AND NON-MARKOVIAN QUEUES
295 Queueing Theory and Simulation PART III: HIGHER DIMENSIONAL AND NON-MARKOVIAN QUEUES 16 Queueing Systems with Two Types of Customers In this section, we discuss queueing systems with two types of customers.
More informationSingle Server Bulk Queueing System with Three Stage Heterogeneous Service, Compulsory Vacation and Balking
International Journal of Scientific and Research Publications, Volume 7, Issue 4, April 217 3 ISSN 225-3153 Single Server Bulk Queueing System with Three Stage Heterogeneous Service, Compulsory Vacation
More informationStochastic inventory system with two types of services
Int. J. Adv. Appl. Math. and Mech. 2() (204) 20-27 ISSN: 2347-2529 Available online at www.ijaamm.com International Journal of Advances in Applied Mathematics and Mechanics Stochastic inventory system
More informationFinite source retrial queues with two phase service. Jinting Wang* and Fang Wang
Int. J. Operational Research, Vol. 3, No. 4, 27 42 Finite source retrial queues with two phase service Jinting Wang* and Fang Wang Department of Mathematics, Beijing Jiaotong University, Beijing, 44, China
More informationChapter 6 Queueing Models. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation
Chapter 6 Queueing Models Banks, Carson, Nelson & Nicol Discrete-Event System Simulation Purpose Simulation is often used in the analysis of queueing models. A simple but typical queueing model: Queueing
More informationOperations Research II, IEOR161 University of California, Berkeley Spring 2007 Final Exam. Name: Student ID:
Operations Research II, IEOR161 University of California, Berkeley Spring 2007 Final Exam 1 2 3 4 5 6 7 8 9 10 7 questions. 1. [5+5] Let X and Y be independent exponential random variables where X has
More informationThe Israeli Queue with a general group-joining policy 60K25 90B22
DOI 0007/s0479-05-942-2 3 4 5 6 7 8 9 0 2 3 4 The Israeli Queue with a general group-joining policy Nir Perel,2 Uri Yechiali Springer Science+Business Media New York 205 Abstract We consider a single-server
More informationBatch Arrival Queuing Models with Periodic Review
Batch Arrival Queuing Models with Periodic Review R. Sivaraman Ph.D. Research Scholar in Mathematics Sri Satya Sai University of Technology and Medical Sciences Bhopal, Madhya Pradesh National Awardee
More informationYORK UNIVERSITY FACULTY OF ARTS DEPARTMENT OF MATHEMATICS AND STATISTICS MATH , YEAR APPLIED OPTIMIZATION (TEST #4 ) (SOLUTIONS)
YORK UNIVERSITY FACULTY OF ARTS DEPARTMENT OF MATHEMATICS AND STATISTICS Instructor : Dr. Igor Poliakov MATH 4570 6.0, YEAR 2006-07 APPLIED OPTIMIZATION (TEST #4 ) (SOLUTIONS) March 29, 2007 Name (print)
More informationName of the Student:
SUBJECT NAME : Probability & Queueing Theory SUBJECT CODE : MA 6453 MATERIAL NAME : Part A questions REGULATION : R2013 UPDATED ON : November 2017 (Upto N/D 2017 QP) (Scan the above QR code for the direct
More informationTRANSIENT ANALYSIS OF A DISCRETE-TIME PRIORITY QUEUE
RANSIEN ANALYSIS OF A DISCREE-IME PRIORIY QUEUE Joris Walraevens Dieter Fiems Herwig Bruneel SMACS Research Group Department of elecommunications and Information Processing (W7) Ghent University - UGent
More informationQueuing Theory. 3. Birth-Death Process. Law of Motion Flow balance equations Steady-state probabilities: , if
1 Queuing Theory 3. Birth-Death Process Law of Motion Flow balance equations Steady-state probabilities: c j = λ 0λ 1...λ j 1 µ 1 µ 2...µ j π 0 = 1 1+ j=1 c j, if j=1 c j is finite. π j = c j π 0 Example
More informationGlossary availability cellular manufacturing closed queueing network coefficient of variation (CV) conditional probability CONWIP
Glossary availability The long-run average fraction of time that the processor is available for processing jobs, denoted by a (p. 113). cellular manufacturing The concept of organizing the factory into
More informationKendall notation. PASTA theorem Basics of M/M/1 queue
Elementary queueing system Kendall notation Little s Law PASTA theorem Basics of M/M/1 queue 1 History of queueing theory An old research area Started in 1909, by Agner Erlang (to model the Copenhagen
More informationMaking Delay Announcements
Making Delay Announcements Performance Impact and Predicting Queueing Delays Ward Whitt With Mor Armony, Rouba Ibrahim and Nahum Shimkin March 7, 2012 Last Class 1 The Overloaded G/GI/s + GI Fluid Queue
More informationRETRIAL QUEUES IN THE PERFORMANCE MODELING OF CELLULAR MOBILE NETWORKS USING MOSEL
RETRIAL QUEUES IN THE PERFORMANCE MODELING OF CELLULAR MOBILE NETWORKS USING MOSEL JÁNOS ROSZIK, JÁNOS SZTRIK, CHE-SOONG KIM Department of Informatics Systems and Networks, University of Debrecen, P.O.
More informationAnalysis of Two-Heterogeneous Server Queueing System
UDC 519.872 Analysis of Two-Heterogeneous Server Queueing System H. Okan Isguder, U. U. Kocer Department of Statistics, Dokuz Eylul University, Tinaztepe Campus, Izmir, 3539, Turkey Abstract. This study
More informationQueueing Review. Christos Alexopoulos and Dave Goldsman 10/6/16. (mostly from BCNN) Georgia Institute of Technology, Atlanta, GA, USA
1 / 24 Queueing Review (mostly from BCNN) Christos Alexopoulos and Dave Goldsman Georgia Institute of Technology, Atlanta, GA, USA 10/6/16 2 / 24 Outline 1 Introduction 2 Queueing Notation 3 Transient
More informationA Queueing System with Queue Length Dependent Service Times, with Applications to Cell Discarding in ATM Networks
A Queueing System with Queue Length Dependent Service Times, with Applications to Cell Discarding in ATM Networks by Doo Il Choi, Charles Knessl and Charles Tier University of Illinois at Chicago 85 South
More informationThe Unreliable M/M/1 Retrial Queue in a Random Environment
The Unreliable M/M/1 Retrial Queue in a Random Environment James D. Cordeiro Department of Mathematics and Statistics Air Force Institute of Technology 2950 Hobson Way (AFIT/ENC) Wright Patterson AFB,
More informationReadings: Finish Section 5.2
LECTURE 19 Readings: Finish Section 5.2 Lecture outline Markov Processes I Checkout counter example. Markov process: definition. -step transition probabilities. Classification of states. Example: Checkout
More information