IJMS, Vol., No. -4, (July-December 202), pp. 97-2 Serials Publications ISSN: 0972-754X A MATHEMATICAL MODEL FOR LINKAGE OF A QUEUING NETWORK WITH A FLOW SHOP SCHEDULING SYSTEM Deepak Gupta, Sameer Sharma & Seema Sharma Abstract: Waiting lines or queues are a common occurrence both in everyday life and in variety of business and industrial situations. Most queuing problems are centered about the question of finding levels of services that a system should provide. Flow shop scheduling with the objective of minimizing the makespan is also an important task in manufacturing system. This paper is an attempt to establish a linkage between a complex queuing network in which a common server is linked in series with each of two systems, one containing two Biserial servers and other with three parallel servers and a flow shop system consisting of two machines in series. In queue network, the arrival and service pattern both follows Poisson law. The completion time (waiting time + service time) of jobs (customers) will be the setup time for first machine in flow shop scheduling system. The objective of the paper is to develop an algorithm minimizing the total elapsed time with minimum completion time for this proposed queuing scheduling linkage model. A numerical illustration followed by a computer programme is also given to substantiate the proposed algorithm. Keywords: Biserial channels, Flow shop scheduling, Mean queue length, Makespan, Processing time, Waiting time.. INTRODUCTION Queues (waiting lines) help facilities or businesses provide service in an orderly fashion. It is used extensively to analyze production and service processes exhibiting random variability in market demand (arrival times) and service times. Queuing theory had its beginning in research work of a Danish engineer name A. K. Erlang. Jackson R. R. P. [6] studied the behaviour of a queuing system containing phase type service. Maggu [8] introduced the concept of bitendom in theory of queues which corresponds to a practical situation arise in production concern. Later on this idea was developed by various authors with different modifications and argumentations. Singh T. P. et al., [9, ] studied the transient behaviour of a queuing network with parallel biseries queue linked with a common channel. Gupta, Sharma and Gulati [2] studied steady state behaviour of a queue model comprised of two subsystem with biserial channel linked with a common channel. st International Conference on Mathematics and Mathematical Sciences (ICMMS), 7 July 202.
98 Deepak Gupta, Sameer Sharma & Seema Sharma The scheduling/sequencing problems are common occurrence in our daily life e.g., ordering of jobs for processing in a manufacturing plant, waiting air craft for landing clearance, programs to be run in a sequence at a computer center etc. Such problems exist whenever there is an alternative choice in which a number of jobs can be done. The selection of an appropriate order or sequence to receive waiting customer is called sequencing. In industries man, machines, material and money are involved for the production of them. The manager of an industry is interested to use men, machines, materials and money in an economic manner so that the cost associated with the production of an item is not increased or it is the minimum in the competitive market. The literature reveled that a lot of research work has already been done in the field of Queuing and Scheduling theory individually. Only some efforts have been done to establish a linkage between these two fields of optimization. Singh T.P. and Vinod Kumar [, 2,, 4, 5], Gupta, Sharma and Seema [2] have made efforts to optimized the total flow time, Waiting time and Service time in Queuing-Scheduling linkage network. Recently Gupta, Sharma and Seema [4] has established a linkage mode of network of queues consisting of a system of parallel biserial servers and a system of two parallel servers both linked in series with a common server to a flow shop scheduling model. This paper is an attempt to extend the work by establishing a linkage between a complex queuing network in which a common server is linked in series with each of two systems, one containing two Biserial servers and other having three parallel servers to a flow shop system consisting of two machines in series. Hence, this paper establishes a linkage between a network of queue model given by Gupta, Sharma and Gulati [2] and the two stage scheduling flowshop system given by Johnson [5]. 2. PRACTICAL SITUATION Many practical situation of the model arise in industries, administrative setups, banking system, computer networks, office management, super markets and shopping malls, scheduling of patients in a hospital etc. For example, in 24 hours drive through meal department consisting of three sections, one is for food, second is for drink and third as common to both for billing. The food section consists of two sub channels and the drink section consists of three parallel sub channels. Suppose in food section, one channel is for Indian food items and second channel is for Chinese food items. Similarly at drink section one channel is for Coffee / Tea, Second channel is for different vegetables soups and Third section is for cold drinks (juice/shakes etc). The customers who arrive to take Indian food items may also take some Chinese food items and then go to the bill counter or may directly go to billing counter. Similarly the customers coming to take Chinese food items may also take some Indian food items and then go to the billing counter or may directly go to the
A Mathematical Model for Linkage of a Queuing Network with a Flow Shop Scheduling... 99 billing counter. Also the customers coming to take some drink items at the second section will join Coffee / Tea sub counter or vegetables soups sub counter or cold drinks sub counter according to their choice and will finally go to the billing counter. After bill counter there are two machines in series, one for collecting tokens and finally for delivery of food items. Similarly, in manufacturing industry, we can consider two parallel biserial channels one for cutting and other for turning. Some jobs after cutting may go to turning and viceversa. The second system consisting of three parallel servers can be regarded as Grinding, Molding and Casting. Both these system of servers are commonly connected to the server for stamping. After that the jobs has to pass thought two machine in series taken as inspection of quality of goods produced and second machine for the packing.. MATHEMATICAL MODEL The entire queue model is comprised of three service servers S, S 2 and S. The subsystem S consist of two biserial service servers S and S 2. The subsystem S 2 contain three parallel servers S 2, S 22 and S 2. The service server S is commonly linked in series with each of two servers S and S 2 for competition of final phase service demanded either at a subsystem S or S 2. The service time at S i j (i =, 2 and j =, 2, ) are distributed exponentially. We assume the service mean rate, 2,, 2, at S i j (i =, 2 and j =, 2, ) and at S respectively. Queues Q, Q 2, Q, Q 4, Q 5 and Q 6 are said to formed in front of the service servers S, S 2, S 2, S 22, S 2 and S respectively, if they are busy. Customers coming at the rate after completion of phase service at S will join S 2 or S (that is they may either go to the network of servers S S 2 S or S S ) with the probabilities p 2 or p such that p 2 + p = and those coming at the rate 2 after completion of phase service at S 2 will join S or S (that is they may either go to the network of servers S 2 S S or Figure : Linkage Model
200 Deepak Gupta, Sameer Sharma & Seema Sharma S 2 S ) with the probabilities p 2 or p 2 such that p 2 + p 2 =. The customers coming at the rate go to the network of servers S 2 S and those coming at the rate 2 go to the network of servers S 22 S and those coming at the rate go to the network of servers S 2 S. Further the completion time (waiting time + service time) of customers / jobs through Q, Q 2, Q, Q 4, Q 5 & Q 6 form the setup times for machine M. After coming out from the server S i.e., Phase I, customers / jobs go to the machines M and M 2 (in Phase II) for processing with processing time A i and A i2 in second Phase service. Our objective is to develop a heuristic algorithm to find an optimal sequence of the jobs / customers to minimize the makespan in this Queue-Scheduling linkage system with minimum completion time. 4. MATHEMATICAL ANALYSIS Let P n, n 2, n, n 4, n 5, n 6 be the joint probability that there are n units waiting in queue Q in front of S, n 2 units waiting in queue Q 2 in front of S 2, n units waiting in queue Q in front of S 2, n 4 units waiting in queue Q 4 in front of S 22, n 5 units waiting in queue Q 5 in front of S 2 and n 6 units waiting in queue Q 6 in front of S (Figure ). In each case the waiting includes a unit in service, if any. Also, n, n 2, n, n 4, n 5, n 6 > 0. The standard arguments lead to the following differential difference equations in transient form as Pn, n2, n, n4, n5, n ( t ) = 6 2 2 2 2 Pn, n2, n, n4, n5, n6 ( ) ( t) P ( t) P ( t) n, n2, n, n4, n5, n6 2 n, n2, n, n4, n5, n6 ( n ) p P ( t) ( n ) p P ( t) n, n2, n, n4, n5, n6 2 n, n2, n, n4, n5, n6 ( n ) p P ( t) ( n ) p P ( t) 2 2 2 n, n2, n, n4, n5, n6 2 2 2 n, n2, n, n4, n5, n6 P ( t) P ( t) P ( t) n, n2, n, n4, n5, n6 2 n, n2, n, n4, n5, n6 n, n2, n, n4, n5, n6 ( n ) P ( t) ( n ) P ( t) 6 n, n2, n, n4, n5, n6 n, n2, n, n4, n5, n6 ( n ) P ( t) ( n ) P ( t). 2 4 n, n2, n, n4, n5, n6 5 n, n2, n, n4, n5, n6 The steady state equation (t ) governing the model are depicted as ( ) P n n n n n n 2 2 2 2, 2,, 4, 5, 6 P P ( n ) p P = n, n2, n, n4, n5, n6 2 n, n2, n, n4, n5, n6 n, n2, n, n4, n5, n6 ( n ) p P ( n ) p P 2 n, n2, n, n4, n5, n6 2 2 2 n, n2, n, n4, n5, n6
A Mathematical Model for Linkage of a Queuing Network with a Flow Shop Scheduling... 20 ( n ) p Pn n n n n n Pn n n n n n Pn n n n n n 2 2 2, 2,, 4, 5, 6, 2,, 4, 5, 6 2, 2,, 4, 5, 6 P ( n ) P ( n ) P n, n2, n, n4, n5, n6 6 n, n2, n, n4, n5, n6 n, n2, n, n4, n5, n6 ( n ) P ( n ) P. () 2 4 n, n2, n, n4, n5, n6 5 n, n2, n, n4, n5, n6 Let us define the generating function as F( X, Y, Z, R, S, T) P X y Z R S T where X Y Z R S T. n 0 n2 0 n 0 n4 0 n5 0 n6 0 Also we define partial generating functions as n, n2, n, n4, n5, n6 n n2 n n4 n5 n6 n n2 n2, n, n4, n5, n 6 n, n2, n, n4, n5, n6 n, n4, n5, n 6 n2, n, n4, n5, n6 n 0 n2 0 F ( X) P X, F ( X, Y) P ( X) Y, n n4 n4, n5, n 6 n, n4, n5, n6 n5, n 6 n4, n5, n6 n 0 n 0 4 F ( X, Y, Z) P ( X, Y) Z. F ( X, Y, Z, R) P ( X, Y, Z) R, n5 n6 n (,,,, ) 6 n5, n ( X, Y, Z, R) S, F( X, Y, Z, R, S, T) (,,,, ) 6 Pn X Y Z R S T 6 n5 0 n6 0 F X Y Z R S P Now, on taking n, n 2, n, n 4, n 5, n 6 equal to zero one by one and then taking two of them pairwise, three of them at a time, four of them at a time, five of them at a time and all of them; we get 6 equations. Now proceeding on the lines of Maggu and Singh T. P. et al., and following the standard technique, which after manipulation gives the final reduced result as T Y p p2 F Y, Z, R, S, T X X T X 2 p2 p2 F X, Z, R, S, T Y Y T T F X, Y, R, S, T 2 F X, Y, Z, S, T Z R T F X, Y, Z, S, T F X, Y, Z, R, S S T F ( X, Y, Z, R, S, T) Y T X 2 Y p2 p X X T X 2 p2 p2 ( Z) 2( R) ( S) Y Y T T T 2 Z R S T. (2)
202 Deepak Gupta, Sameer Sharma & Seema Sharma For convenience, let us denote F( Y, Z, R, S, T ) F, F( X, Z, R, S, T) F, F( X, Y, R, S, T) F 2 F( X, Y, Z, S, T) F, F( X, Y, Z, R, T) F, F( X, Y, Z, R, S) F 4 5 6 Also F (,,,,, ) =, being the total probability. On taking X = as Y, Z, R, S, T, F (X, Y, Z, R, S, T) is of 0 0 indeterminate form. Now, on differentiating numerator and denominator of (2) separately w.r.t X, we have ( p p2 ) F 2( p2) F2 ( p p ) ( p ) 2 2 2 F 2 p 2 F 2 2 p 2 ( p2 p ) () Similarly, on Diff. numerator and denominator of (2) separately w.r.t Y, on taking Y = and X, Z, R, S, T we have ( p2 ) F 2( p2 p2) F2 ( p ) ( p p ) 2 2 2 2 2 p 2 F 2 F 2 2 p 2 2 ( p2 p2 ) (4) Again, on Diff. numerator and denominator of (2) separately w.r.t Z, on taking Z = and X, Y, R, S, T we have F F Again, on Diff. numerator and denominator of (2) separately w.r.t R, on taking R = and X, Y, Z, S, T we have F F 2 4 2 4 2 2 2 2 Again, on Diff. numerator and denominator of (2) separately w.r.t S, on taking S = and X, Y, Z, R, T we have F F 5 5 Again, on Diff. numerator and denominator of (2) separately w.r.t T, on taking T = and X, Y, Z, R, S we have (5) (6) (7)
A Mathematical Model for Linkage of a Queuing Network with a Flow Shop Scheduling... 20 p F p F ( F ) ( F ) ( F ) F ( p ) ( p ) ( ) ( ) ( ) 2 2 2 2 2 4 5 6 2 2 2 PF 2P2 F2 F 2F4 F5 F6 = p p2 2 2 (8) On multiplying (4) with p 2 and adding to (), we get F ( p p ) ( p p ) p 2 2 2 2 2 2 p F ( ) 2 2 p2 p2 F F4 F5 On multiplying () withp 2 and adding to (4), we get (9) (Using (5)) (0) 2 (Using (6)) () 2 (Using (7)) (2) ( p p ) F p ( p p ) ( p p ) 2 2 2 2 2 2 2 2 2 2 2 2 p2 F2 ( p p ) 2 2 2 Now on putting the values of F, F 2, F, F 4, F 5 in (8), we get ( p ) p ( p ) p F6 ( p2 p2) 2 2 2 2 2 2 On using the values of F, F 2, F, F 4, F 5 and F 6, the joint probability is given by P ( )( )( )( )( )( ) n n 2 n n 4 n5 n6 n, n2, n, n4, n5, n6 2 4 5 6 2 4 5 6 Where F, 2 F2, F, 4 F4, 5 F5, 6 F6. Further the solution in a steady state condition exist if, 2,, 4, 5, 6. () (4) 4. Mean Queue Length Average number of the customer (L) = n 0 n2 0 n 0 n4 0 n5 0 n6 0 ( n n n n n n ) P n n n n n n 2 4 5 6, 2,, 4, 5, 6
204 Deepak Gupta, Sameer Sharma & Seema Sharma = n Pn, n2, n, n4, n5, n6 n 0 n2 0 n 0 n4 0 n5 0 n6 0 n 0 n2 0 n 0 n4 0 n5 0 n6 0 n P 2 n, n2, n, n4, n5, n6... n 0 n2 0 n 0 n4 0 n5 0 n6 0 n P 6 n, n2, n, n4, n5, n6 Where L = L + L 2 + L + L 4 + L 5 + L 6 L = n Pn, n2, n, n4, n5, n6 n 0 n2 0 n 0 n4 0 n5 0 n6 0 2 n 4 n5 n6 = n n n 2 n 4 5 6 2 4 5 6 n 0n2 0n 0n4 0n50n6 0 ( )( )( )( )( )( ) n n2 n n4 n5 n6 = ( )( 2)( )( 4 )( 5)( 6) n 2 4 5 6 n 0 n2 0 n 0 n4 0 n5 0 n6 0. = 2 4 5 6 Similarly, L2, L, L4, L5, L 6. 2 4 5 6 4.2 Average Waiting Time The average waiting time and the average number of items waiting for a service in a service system are important measurements for a manager. Little s Law relates these two metrics via the average rate of arrivals to the system. This fundamental law has found numerous uses in operations management and managerial decision making. Little s Law says that, under steady state conditions, the average number of items in a queuing system equals the average rate at which items arrive multiplied by the average time that an item spends in the system. Let L = Average number of items in the queuing system, W = Average waiting time in the system for an item, and A = Average number of items arriving per unit time. L i.e. Little s Law states that L = W; or W.
A Mathematical Model for Linkage of a Queuing Network with a Flow Shop Scheduling... 205 5. ASSUMPTIONS. We assume that the arrival rate in the queue network follows Poisson distribution. 2. Each job / customer is processed on the machines M and M 2 in the same order and pre-emission is not allowed, i.e., once a job is started on a machine, the process on that machine can not be stopped unless job is completed.. For the existence of the steady state behaviour the following conditions hold good: ( p ) ( 2 p2 ), (ii) 2, ( ) ( p p ) 2 2 (i) p2 p2 2 (iii), (vi) 4 (v) 2 2 2 2, 2 p ( 2 p2 ) p2 ( 2 p2 ) 5, (vi) 6. ( p2 p2 ) 6. ALGORITHM The following algorithm gives the procedure to determine the optimal sequence of the jobs to minimize the flow time for the machines M and M 2 when the completion time (waiting time + service time) of the jobs coming out of Phase I is the setup times for the machine M. Step : Find the mean queue length on the lines of Singh & Kumar [5] using the formula Here; 2 4 5 6 L 2 4 5 6 ( 2 p2) ( 2 p2 ) 2, 2,, ( p2 p2) 2( p2 p2) 4 2 2 p ( 2 p2) p2 ( 2 p2 ) 6 ; ( p p ) 2 2, 5, i is the mean arrival rate, i is the mean service rate and p ij are the probabilities. Step 2: Find the average waiting time of the customers on the line of Little s [9] using L relation E ( w), where 2 2. Step : Find the completion time (C) of jobs/customers coming out of Phase I, i.e., when processed thought the network of queues Q, Q 2, Q, Q 4, Q 5 and Q 6 by using the formula
206 Deepak Gupta, Sameer Sharma & Seema Sharma C E ( W). p p p p 2 2 2 2 2 2 Step 4: The completion time C of the customers / jobs through the network of queues Q, Q 2, Q, Q 4, Q 5 and Q 6 will form the setup time for machine M. Define the two machines M and M 2 with processing time A i = A i + C and A i2. Step 5: Apply Johnson s [5] procedure to find the optimal sequence(s) with minimum elapsed time. Step 6: Prepare In-Out tables for the optimal sequence(s) obtained in step 5. The sequence {S} having minimum total elapsed time will be the optimal sequence for the given problem. 7. NUMERICAL ILLUSTRATION Consider fourteen customers / jobs are processed through the network of queues Q, Q 2, Q, Q 5 and Q 6 with the servers S, S 2 and S. The server S consists of two biserial service servers S and S 2. The server S 2 contains three parallel servers S 2, S 22 and S 2. Server S is commonly linked in series with each of two servers S and S 2. The number of the customers, mean arrival rate, mean service rate and associated probabilities are given as in Table. Table The Detail Classification of the Linkage Model S. No. No. of customers Mean arrival rate Mean service rate Probabilities n = = 4 = 0 p 2 = 0.4 2 n 2 = 2 2 = 5 2 = 9 p = 0.6 n = 4 = = 7 p 2 = 0.6 4 n 4 = 2 2 = 5 2 = 6 p 2 = 0.4 5 n 5 = = 4 = 5 6 n 6 = 4 = 28 After getting service at Phase I jobs / customers are to be served at the machines M and M 2 with processing time A i and A i2 respectively as given in Table 2. Table 2 The Machines M and M 2 with processing times Jobs 2 4 5 6 7 8 9 0 2 4 M (A i ) 2 2 7 8 2 4 5 6 7 2 M 2 (A i2 ) 4 4 5 6 4 7 2 4 2 4
A Mathematical Model for Linkage of a Queuing Network with a Flow Shop Scheduling... 207 The objective is to find an optimal sequence of the jobs / customers to minimize the makespan in this Queue-Scheduling linkage system by considering the first phase service into account. Solution: We have 2 p2 = ( p p ) 2 2 2 p2 0.9205, 2 = ( p p ) 2 2 2 0.96492, = 2 0.42857, 4 = 0.8, 5 = 2 0.8, 6 = 2 p ( 2 p2) p2( 2 p2 ) 0.75. ( p p ) 2 2 Mean Queue Length = Average number of Jobs / Customers = 2 4 5 6 L 2 4 5 6 = 5.967 units. L Average waiting time of the jobs / customers = E ( w) = 2.472222 units. The total completion time of Jobs / Customers when processed through queue network in Phase I C E ( W) p p p p 2 2 2 2 2 2 = 2.4876 units. On taking the completion time C = 2.4876 as the setup time, when jobs / customers came for processing with machine M. The new reduced problem with processing times A i = A i + C and A i 2 on machine M and M 2 is as shown in Table. Table The Processing Times A i2 and A i 2 on Machine M and M 2 is Job 2 4 5 6 7 8 9 A i 4.4876.4876.4876 2.4876 7.4876 8.4876 2.4876.4876 4.4876 A i 2 4 4 5 6 4 7 2 Job 0 2 4 A i 5.4876 6.4876 7.4876.4876 2.4876 A i 2 4 2 4
208 Deepak Gupta, Sameer Sharma & Seema Sharma Using Johnson's [5] algorithm, we get the optimal sequence S = 2 7 4 5 0 6 4 2 9 8. The In-Out flow table for the sequence {S} is as in Table 4. Table 4 The In-Out Flow Table for the Sequence S Jobs Machine M Machine M 2 2 0.00000.4876.4876 7.4876 7.4876 7.9752 7.9752 4.9752 4 7.9752 2.4628 4.9752 9.9752 5 2.4628 2.9504 2.9504 27.9504 0 2.9504 29.48 29.48.48 29.48 4.9256 4.9256 8.9256 6 4.9256 45.4 45.4 49.4 45.4 50.9009 50.9009 54.9009 4 50.9009 55.885 55.885 58.885 2 55.885 64.876 64.876 67.876 64.876 69.67 69.67 72.67 9 69.67 75.85 75.85 78.85 75.85 84.89 84.89 86.89 8 84.89 89.8265 89.8265 9.8265 Therefore, the total minimum elapsed time for sequence S is 9.8265 units, average awaiting time is 2.472222 units and mean queue length is 5.967 units. 8. CONCLUSION The present paper establishes linkage between the queue network comprised of three servers S, S 2, S with a two stage flowshop scheduling system consisting machines M and M 2. The server S consists of two biserial service servers S and S 2. The server S 2 contains two parallel servers S 2, S 22 and S 2. Server S is commonly linked in series with each of three servers S and S 2 for completion of first phase service demanded either at a subsystem S or S 2.The objective of the model is to minimizing the total elapsed time. A heuristic algorithm by considering the completion time of jobs in Phase I as setup time for the machine M in Phase II is discussed. The study may further be extended by introducing various types of queuing models and different parameters for two and three stage flowshop scheduling models with different constraints.
A Mathematical Model for Linkage of a Queuing Network with a Flow Shop Scheduling... 209 Programme #include<iostream.h> #include<stdio.h> #include<conio.h> #include<process.h> #include<math.h> int n[6],u[6],l[5],j[50],j[50],m; float p[4],r[6], g[50],h[50],a[50],b[50]; APPENDIX float a,b,a2,b2,a,b,a4,b4,a5,b5,b6,c,c2,p,q,v,w,m,z,z2,z,x,q,q2,q,z,f,c; void main() { clrscr(); for( int i=;i<=5;i++) {cout<<"enter the number of customers and Mean Arrival Rate for Channel S"<<i<<":";cin>>n[i]>>L[i];} m=n[]+n[2]+n[]+n[4]+n[5]; for(int d=;d<=6;d++) {cout<<"\nenter the Mean Service Rate for the Channel S"<<d<<":";cin>>u[d];} for(int k=;k<=4;k++) { cout<<"\nenter the value of probability p"<<k<<":";cin>>p[k]; } for(i=;i<=m;i++) {j[i]=i; cout<<"\nenter the processing time of "<<i<<" job for machine A : ";cin>>a[i]; cout<<"\nenter the processing time of "<<i<<" job for machine B : ";cin>>b[i];} a=l[]+l[2]*p[];b=(-p[]*p[])*u[];r[]=a/b;a2=l[2] +L[]*p[];b2 =(-p[]*p[])*u[2];r[2]=a2/b2; a=l[];a4=l[4];b=u[];b4=u[4],a5=l[5],b5=u[5],b6=u[6];r[]=a/b;r[4]= a4/b4;r[5]=a5/b5; c2=(-p[]*p[])*b6;z=(a+a4+a5)/b6;z2= a*p[2]/c2;z= a2*p[4]/c2; r[6]=z+z2+z;m=l[]+l[2]+l[]+l[4]+l[5]; for(i=;i<=6;i++)
20 Deepak Gupta, Sameer Sharma & Seema Sharma {cout<<"r["<<i<<"]\t\t"<<r[i]<<"\n";} for(i=;i<=6;i++) { if(r[i]>) {cout<<"steady state condition does not holds good for"<<r[i]<<"...\ nexitting";getch();exit(0);}} Q=(r[]/(-r[]))+(r[2]/(-r[2]))+(r[]/(-r[]))+(r[4]/(-r[4]))+(r[5]/(-r[5])+ (r[6]/(-r[6]))); cout<<"\nthe mean queue length is :"<<Q<<"\n"; W=Q/M; cout<<"\naverage waiting time for the customer is:"<<w<<"\n"; z=u[]* p[]+u[]*p[ 2] +u[2]* p[]+u[2]*p[4]+u[]+u[ 4]+u[5]+u[6]; f=/z;c= W+f; cout<<"\n\ntotal completetion time of Jobs / Customers through Queue Network in Phase :"<<c; for(i=;i<=m;i++) for(i=;i<=m;i++) {g[i]=a[i]+c;h[i]=b[i];} {cout<<"\n\n"<<j[i]<<"\t"<<g[i]<<"\t"<<h[i];cout<<endl;} float mingh[6];char ch[6]; for(i=;i<=m;i++) { if(g[i]<h[i]) else } {mingh[i]=g[i];ch[i]='g';} {mingh[i]=h[i];ch[i]='h';} for(i=;i<=m;i++) {cout<<endl<<mingh[i]<<"\t"<<ch[i];} for(i=;i<=m;i++) { for(int k=;k<=m;k++) if(mingh[i]<mingh[k]) {float temp=mingh[i]; int temp=j[i]; char d=ch[i];
A Mathematical Model for Linkage of a Queuing Network with a Flow Shop Scheduling... 2 mingh[i]=mingh[k]; j[i]=j[k]; ch[i]=ch[k]; mingh[k]=temp; j[k]=temp; ch[k]=d;} } for(i=;i<=m;i++) {cout<<endl<<endl<<j[i]<<"\t"<<mingh[i]<<"\t"<<ch[i]<<"\n";} // calculate scheduling float sbeta[6];int t=,s=0; for(i=;i<=m;i++) { if(ch[i]=='h') {sbeta[(m-s)]=j[i];s++;} else if(ch[i]=='g') {sbeta[t]=j[i];t++;}} int arr[6], m=; cout<<endl<<endl<<"job Scheduling:"<<"\t"; for(i=;i<=m;i++) {cout<<sbeta[i]<<" ";arr[m]=sbeta[i];m++;} cout<<"\nscheduled job sequence"; for(i=;i<= m;i++) {cout<<endl<<arr[i]<<" ";} //calculating total computation sequence float macha[50], machb[50],maxv[50];float time=0.0; macha[]=time+g[arr[]]; for(i=2;i<=m;i++) {macha[i]=macha[i-]+g[arr[i]];} machb[]=macha[]+h[arr[]]; for(i=2;i<=m;i++) {if((machb[i-])>(macha[i])) maxv[i]=machb[i-]; else maxv[i]=macha[i]; machb[i]=maxv[i]+h[arr[i]]; }
22 Deepak Gupta, Sameer Sharma & Seema Sharma //displaying solution cout<<"\n\n\n\n\n\t\t\t #####THE SOLUTION##### "; cout<<"\n\n\t***************************************************************"; cout<<"\n\n\n\t Optimal Sequence is : "; for(i=;i<=m;i++) {cout<<" "<<arr[i];} cout<<endl<<endl<<"in-out Table is:"<<endl<<endl; cout<<"jobs"<<"\t"<<"machine M"<<"\t"<<"\t"<<"Machine M2" <<"\t"<<endl; cout<<arr[]<<"\t"<<time<<"--"<<macha[]<<" \t"<<"\t"<<m acha[]<<"-- "<<machb[]<<" \t"<<endl; for(i=2;i<=m;i++) { cout<<arr[i]<<"\t"<<macha[i-]<<"--"<<macha[i]<<" "<<"\t"<<maxv[i]<<"-- "<<machb[i]<<" "<<"\t"<<endl; } cout<<"\n\n\ntotal Elapsed Time (T) = "<<machb[m]; cout<<"\n\n\t***************************************************************"; getch(); } REFERENCES [] Gupta Deepak, T. P. Singh, and Rajinder Kumar, (2007), Analysis of a Network Queue Model Comprised of Biserial and Parallel Channel Linked with a Common Server, Ultra Science, 9(2)M: 407-48. [2] Gupta Deepak, Sharma Sameer, and Gulati Naveen, (20), On Steady State Behaviour of a Network Queuing Model with Biserial and Parallel Channels Linked with a Common Server, Computer Engineering and Intelligent Systems, 2(): 22. [] Gupta Deepak, Sharma Sameer, and Seema, (202), On Linkage of a Flow Shop Scheduling Model Including Job Block Criteria with a Parallel Biserial Queue Network, Computer Engineering and Intelligent Systems, (2): 7 28. [4] Gupta Deepak, Sharma Sameer, and Seema, (202), On Linkage of a Queue Network with Biserial and Parallel Channels Linked with a Common Server to a Two Stage Flow Shop Scheduling System, International Journal of Emerging Trends in Engineering and Development, 2(2): 265-275. [5] S. M. Johnson, (954), Optimal Two & Three Stage Production Schedules with Set Up Times Includes, Nav. Res. Log. Quart., : 6 68.
A Mathematical Model for Linkage of a Queuing Network with a Flow Shop Scheduling... 2 [6] R. R. P. Jackson, (954), Queuing System with Phase Type Service, O. R. Quat., 5:09 20. [7] D. C. Little John, (965), A Proof of Queuing Formula: L = W, Operation Research, : 400 42. [8] P. L. Maggu, (970), Phase Type Service Queue with Two Channels in Biserial, J. OP. Res. Soc. Japan, (). [9] Singh T. P., and Kumar Vinod, (2007), A Note on Serial Queuing & Scheduling Linkage, PAMS, LXV(), 7 8. [0] M. Nawaz, E. E. Enscore, and L. Ham, (98), A Heuristic Algorithm for the m Machine, n Job Flow Shop Sequencing Problem, OMEGA, (): 9 95. [] T. P. Singh, Kumar Vinod, and K. Rajinder, (2005), On Transient Behaviour of a Queuing Network with Parallel Biserial Queues, JMASS, (2): 68 75. [2] T. P. Singh, and Kumar Vinod, (2007), On Linkage of Queues in Semi-Series to a Flowshop Scheduling System, Int. Agrkult. Stat. Sci., (2): 57 580.. [] T. P. Singh, Kumar Vinod, and Kumar Rajinder, (2008), Linkage Scheduling System with a Serial Queue-Network, Lingaya's Journal of Professional Studies, 2(): 25 0. [4] T. P. Singh, and Kumar Vinod, (2009), On Linkage of a Scheduling System with Biserial Queue Network, Arya Bhatta Journal of Mathematics & Informatics, (): 7 76. [5] Kumar Vinod, T. P. Singh, and K. Rajinder, (2007), Steady State Behaviour of a Queue Model Comprised of Two Subsystems with Biserial Linked with Common Channel, Reflection des ERA., (2): 5 52. Deepak Gupta Prof. & Head, Department of Mathematics, M.M. University, Mullana, Ambala, India. E-mail: guptadeepak200@yahoo.co.in Sameer Sharma Research Scholar, Department of Mathematics, D.A.V. College, Jalandhar, Punjab, India. E-mail: samsharma@yahoo.com Seema Sharma Department of Mathematics, D.A.V. College, Jalandhar, Punjab, India. E-mail: seemasharma7788@yahoo.com