CHAPTER 3 STOCHASTIC MODEL OF A GENERAL FEED BACK QUEUE NETWORK

Transcription

1 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 models in the study of feedback and cyclic queue network. Assuming various hypothetical situations different authors have introduced various models to promote the theory to apply in practical situations. It may be observed that the mathematical tools applied in the analysis are really elegant, in spite of the fact that many such models lack practical applicability in real life situations. Here a brief outline of some chosen work is given in a series by which they have developed to meet real life situations. Feedback queues relate to those queues in which a customer served once, when his service becomes unsuccessful and are served again and again till his service becomes successful (Takacs [1963], Kleinrock [1975], Takagi [1987], Singh T.P. [1996]).. Section 3.1 of chapter 3 based on the paper Transient Analysis of Three Tandem Feedback Queue System with Service Parameter Constraints, Yamuna Journal of Technology & Business Research [2011]. Vol. 1, No. 1-2, pp Section 3.2 of chapter 3 based on the paper Mathematical Analysis of a General Feedback Queue Tandem Network, Presented and Published in the Proceeding of 6 th International Multi Conference on Intelligent System, Sustainable, New and Renewable Energy Technology & Nanotechnology [ISSN 2012], ISTK March 16-18, [2012]. pp

2 Many real life situations could be modeled as a feedback queue for example, in data transmission a packet transmitted from the source to the destination may be returned and it may go on like until the packet is finally transmitted. A vast amount of research in the field of feedback and cyclic queue model has been conducted by the academic community during the past several decades (Burke s [1956], Jackson [1957], Finch P.D. [1959], Buchan & Koiengsberg [1958], Singh T.P. [1994]). Jackson did a remarkable work in the field of queue network theory. Let us assume that units (customers/message/packets) arrive singly in a Poisson stream with rate λ and that the service time is general. As soon as the service is completed, a unit whose service is successful apart from the system with probability p or if service is unsuccessful the unit is cycled back into the system with probability (1-p). This is known as Bernoulli feedback. The unit or customers may enter the system at some node, traverse from node to node in the system, and depart from some node, not all customers necessarily entering and leaving at the same node, or taking the same path once having entered the system customer may return to nodes previously visited, skip some nodes entirely and even choose to remain in the system forever. In contrast to Jackson s open network, Gordan and Newell [1967] consider a close network of Markovian queues in which a fixed and finite number of customers say (K) circulate through the network, there being no external input or departure from the network. If we consider a close network of k nodes such that the output of nodes i goes to the next node i+1 ( 1 i i-1) where as the output the last node k feedback the node i and so on. Such a queue is called a 76

3 cyclic queue. A cyclic queue system consists of several service channels arranged in series, the end s of which are joined to form a closed circle. Singh T.P. [1994] extended the study made by various researchers in network of cyclic queue. Further Singh T.P., Kusum [2010] made the analytical study of feedback queue model in detail assuming service rate proportional to queue numbers and heterogeneous channels with feedback. This work is further a generalized work in the field feedback queue made by earlier researchers. The chapter is divided into two sections Section Transient behavior of three stage feedback queue system: A specific study Section Feedback queue network with applications: A Generalized model SECTION 3.1 TRANSIENT BEHAVIOR OF THREE STAGE FEEDBACK QUEUE SYSTEM This section studies a network of three queues in series with feedback from the third server under poison assumption of both arrival and departure. We assume the service rate proportional to their respective queue number. The differential difference equation has been explored in transient form. The mean queue size and other parameters for the queue model have been derived by using statistical tools. The model finds its application in banking service system, administrative set up and in decision making process. The transient behavior of the system has been studied and the differential difference equation so formed in the model is being solved by using generating 77

4 function technique and Lagrange s method of solution for partial differential equation. The aim of the section is to investigate various queue characteristics like mean queue length, variance etc. of the queue system which are useful in decision making process, manufacturing concern, banking service system and administrative set up etc PROBLEM FORMULATION: Consider a system of three service nodes say S 1, S 2, S 3. Customers or items arrive in Poisson process at S 1, and then go though S 2 and S 3 for required service, the service pattern at both the nodes are exponentially distributed. Let be mean arrival rate at s 1 and denotes exponential service parameters at S 1, S 2, S 3. Customer after service at S 1 go though mode S 2 and then S 3 in a tandem queue while customer at S 3 is either depart with probability p 3 or feed back to server s 1 with probability p 31. Such that p 3 +p 31 =1.It is further assumed that service parameters are directly proportional to their respective queue numbers defined by µ =, µ =, µ = MATHEMATICAL MODELING: Define the probability at time (t), there are n 1, n 2, n 3 items in system in front of nodes S 1, S 2, S 3 (n 1 n 2, n 3 >0), λ p 3 S 1 S 2 S 3 p 31 Figure Connecting the various state probabilities at time t+ time dependent differential difference equations for the model are obtained as: 78

5 (3.1.1) (3.1.2) (3.1.3) (3.1.4) (3.1.5) (3.1.6) (3.1.7) 79

6 (3.1.8) Taking initial condition at t=0 as, = 0 (3.1.9) SOLUTION METHODOLOGY: To solve the system of equations (3.1.1) to (3.1.8) we apply generating function technique. For this define g.f.t as: F(x, y,z, t) = (3.2.10) (3.2.11) (3.2.12) = (3.2.13) F(x, y, z, t)= (3.2.14) Solving (3.1.1) to (3.1.8) on the line of Maggu [1981] & Singh T.P. [1984],Using equations (3.1.10),(3.1.11), (3.1.12),(3.1.13) in (3.1.1) to (3.1.8) and applying initial condition (3.1.9), after simplifying, ultimate solution in transient form can be expressed as :- (3.1.15) 80

7 As the equation (3.1.15) is linear partial differential equation of first order in four independent variable x, y, z & t the Lagrange s Auxiliary Equations are given by (3.1.16) Now we proceed to obtain four independent solutions of (3.1.16). For this applying the solution technique of Lagrange s Auxiliary Equation, we have On integration we get, C 1 = (3.1.17) And + y (3.1.18) Similarly, C 2 = (3.1.19) + z (3.1.20) And C 3 = (3.1.21) (3.1.22) From (3.1.18), (3.1.20), (3.1.22), values of x, y, z are, x = (3.1.23) y (3.1.24) z (3.1.25) 81

8 using equation (3.1.16), C 4 =F (3.1.26) The general solution Ø(C 1, C 2,C 3,C 4 )=0 becomes, Ø =0 (3.1.27) Where Ø is an arbitrary function. The general solution of (3.1.27) is given by F = Using feedback initial condition F(x, y, z, 0) =1 in (3.1.28) we get, at t=0, (3.1.28) = (3.1.29) Taking C 1, C 2, C 3 at t=0,, (3.1.30) Using (3.1.30) in (3.1.29) we get (3.1.31) = 82

9 Now put the values of C 1, C 2 and C 3 from equations (3.1.17), (3.1.19), (3.1.21) in (3.1.31) then we get = (3.1.32) Putting the value from (3.1.32) to (3.1.28), we have F(x,y,z,t)= F(x,y,z,t)=exp (3.1.33) exp exp exp Which further written as F(x,y,z,t)= (3.1.34) Where A= B= 83

10 C= -(A+B+C)= (3.1.35) In order to find out we proceed as follows Since, F(x,y,z,t) = powers of x,y,z. represents as coefficient of in the expansion of (3.1.34) in i.e = (3.1.36) Using values of A, B and C in (3.1.36), we have = (3.1.37) which gives the required transient solution of the problem (a) MEAN QUEUE SIZE : Let the mean queue size be denoted by L. Then L as per statistical formula is defined as L= Now putting from (3.1.36), we have 84

11 L= L= = = [C + (A+B) ] =A+B+C (3.1.38) (3.1.38) with the help of (3.1.35) on simplification gives the value of L, L= ( ) (b) STEADY STATE SOLUTION: The steady state solution of the model can be found in the usual manner by letting t. Hence when t equation (3.1.37) gives the corresponding steady state probability, as = (3.1.40) 85

12 Also the mean queue size L of the system in the steady state case is obtained by letting t in (3.1.39) which is as L = (3.1.41) 3.1.3(c) PARTICULAR CASE: On letting & and no feedback from third server, then one can easily obtain the result of the model that corresponds to the M M 1 queueing model discussed by Gross and Harries. For this if we identify = P n (t) (with n 1 =n, ) Then the result (3.1.37) corresponds to P n (t) = n 0 (3.1.42) On the similar line one can also easily identify the study state results corresponds to (3.1.40) & (3.1.41) as P n = n 0 (3.1.43) There will be no queue length & a balance is maintained between arrival pattern and service pattern. Other parameters such as variance, busy period distribution etc. can be calculated by using standard statistical formulae. 86

13 SECTION 3.2 GENERALISED FEEDBACK QUEUE MODEL WITH APPLICATIONS This section studies the steady state behavior of a feedback queue network in which after service at a serial channel the customer/items has two choices either to proceed to next or to feedback to one preceding channel. The arrivals and departure both follow the Poisson law at each channel. The Generating function technique (g. f. t), L-Hospital rule, laws of calculus & statistical formulae of Marginal probability distribution has been applied as a solution methodology in order to find various queue parameters. The objective of the study is to find out the mean queue size of each channel and for the entire system. The model is supported by numerical illustration MODEL DESCRIPTION: The model comprises of three service channels in tandem such that a unit or customer after service at initial channel either depart from the system or enter into the next succeeding channel for further operation. After getting service in the second service channel the items found defective are sent back &in similar fashion the units proceed to third channel. The steady state behavior of the system has been studied by applying the Generating function technique, L-Hospital rule & statistical formulae of Marginal probability distribution in order to find various queue parameters. The arrival and departure are assumed to be Poisson distributed. The mean queue size for each channel has been obtained for a decision making process. 87

14 3.2.2 PROBLEM FORMULATION: The units/customers in a Poisson stream at mean rate λ, demanding three phase services first arrive at S 1. Phase one service is completed at S 1 and Phase two service is completed at S 2 and Phase three service is completed at S 3 in a serial order. After phase one, the units either depart from the system at S 1 with probability p 1 or it goes through second channel S 2 with probability P 12 such that P 1 +P 12 = 1.After completing service S 2, the units either depart with probability p 2 or go to next channel with probability p 23 or feedback to preceding channel S 1 with probability p 21 (if items are found defective), such that p 2 +p 21 +p 23 =1.Similarly at S 3 the units either depart with probability p 3 or feedback to preceding channel S 2 with probability p 32 if items are found defective or feedback to channel S 1 with probability p 31, such that p 3 +p 31 +p 32 =1 The service parameters are assumed to be μ 1, μ 2, μ 3. p 1 p 2 p 3 λ p 12 p 23 S 1 S 2 S 3 p 21 p 32 p 31 figure MATHEMATICAL ANALYSIS OF QUEUE MODEL:- Define the probability, there are n 1 units in system in front of S 1, n 2 units in front of S 2, and n 3 units in front of S 3,. The standard argument leads to the following differential difference equations in steady state 88

15 (3.2.1) (3.2.2) (3.2.3) (3.2.4) (3.2.5) (3.2.6) (3.2.7) 89

16 (3.2.8) SOLUTION METHODOLOGY: To solve the system of equations (3.2.1) to (3.2.8) we apply generating function technique. For this define g.f.t as: F(x, y,z) = (3.2.9) (3.2.10) (3.2.11) = (3.2.12) F(x,y,z)= (3.2.13) Multiply (3.2.1) by and summing over n 1 from 0 to using (3.1.2) and (3.2.10). + = (3.2.14) 90

17 Multiply (3.2.3) by and summing over n 1 from 0 to using (3.2.5) and (3.2.10). + = (3.2.15) Multiply (3.2.4) by and summing over n 1 from 0 to using (3.2.7) and (3.2.10). + = (3.2.16) Multiply (3.2.6) by and summing over n 1 from 0 to using (3.2.8) and ( ). + = (3.2.17) Multiply (3.2.14) by and summing over n 2 from 0 to using (3.1.15) and (3.2.12). + + = (3.2.18) 91

18 Multiply (3.2.16) by and summing over n 2 from 0 to using (3.2.17) and (3.2.12). + = (3.2.19) Multiply (3.2.18) by and summing over n 3 from 0 to using (3.2.19) and (3.2.13) (3.2.20) On further simplification, (3.2.20) can be written as- [ (1-x) + + ] = + + (3.2.21) 92

19 Identity, =F 0, =F 1, =F 2 Where Now we find the value of, using L-Hospital rule for limits in case of indeterminate form and using the relation F(1,1,1)=1,the following results are obtained- 93

20 3.2.4(a) MEAN QUEUE LENGTH: The mean queue length L Q of the system is given by- L = L Q1 +L Q2 +L Q3 Now differentiating partially w.r.t. x we get- =

21 + Now, we calculate- (3.2.22) Now putting the values in equation (3.2.22) we get, (3.2.23) (3.2.24) (3.2.25) Mean Queue length: L Q = L Q1 +L Q2 +L Q3 + (3.2.26) 3.2.4(b) WAITING TIME IN QUEUE (W q ) = 95

22 3.2.5 NUMERICAL ILLUSTRATION: For a particular system taking following parameters, =8, =10, =14, λ=6, p 1 =.7, p 12 =.3, p 2 =.5, p 21 =.1, p 23 =.4,p 3 =.5, p 32 =.3, p 31 =.2, then we can find mean queue length and waiting time. After putting these values in the marginal queue length we get the expected queue length: =10, =0.44, =0.14 Then the mean queue length becomes: L = L Q1 + L Q2 + L Q3 = =10.58 W q = L Q /λ =10.58/6 = APPLICATION: The model finds its applications in different situations arising in manufacturing concern, office management, computer networking, banking service system, administrative set up, industrial concern etc. in manufacturing areas e.g. Consider the items which have three stages production, after first stage machine operation, the defective part of production item is dropped at back stage and non defective items are passed to next stage machine for further operation. At second stage machine, either item are dropped if it is defective, or it is sent to first stage for 96

23 reprocessing or sent to next third successive stage machine for further operation. For example, In glass cum soft drink industry we find that the services are done in distinct phases, from phase two the bottles which have defect in shape and sizes are sent back as a raw material to initial phase, from next phase leaked bottles or having some non traceable defects are sent back to phase two or one. The filled bottles are further sent for corking at next phase or not suitable for corking are sent back to initial one. Similarly, in an administrative set up files are sent from one section to second section for remarks then to final authority for checking or the authority either pass the file or sent back to initial one for further clarification. The concept of Network has been found to be very useful in the formulation and modeling of computer, communication, banking and other such system. Queuing network serve as model for multi programmed computer system and communication networks and certain parts manufacturing system. In some multistage queuing process recycling or feedback may occur. Recycling is common in manufacturing processes, where quality control inspections are performed after certain stages, and parts that do not meet quality standards are sent back for reprocessing. Similarly, a telecommunications network may process massages through a randomly selected sequence of nodes with the probability that some messages will require rerouting on occasion through the same stage CONCLUSION: The Aims of this study was to develop the Mathematical model for feedback queue network that would reflect a valuable data in industry. Two models in this study have been proposed. The mean queue length of entire system helps the decision maker how to redesign the queue system so that congestion can be minimized and the service system may no longer be idle. 97