APPLICATION OF BINOMIAL DISTRIBUTION AND UNIFORM DISTRIBUTION TO STUDY THE FINITE QUEUE LENGTH MULTIPLE SERVER QUEUING MODEL

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1 Volume 120 No , ISSN: (on-line version) url: APPLICATION OF BINOMIAL DISTRIBUTION AND UNIFORM DISTRIBUTION TO STUDY THE FINITE QUEUE LENGTH MULTIPLE SERVER QUEUING MODEL Damodhar F Shastrakar 1, Sharad S Pokley 2 1,2 Department of Mathematics, 1 Smt Radhikatai Pandav College of Engineering, Nagpur 2 KITS, Ramtek dfshastrakar@yahoo.com sharadpokley@gmail.com July 25, 2018 Abstract This paper is focus on study of various performance measures of queuing model having limited queue and multiple servers using binomial and uniform probability distributions. It also explains the use of relation customers spending time in the queue to analyze various performance measures of the system. For the derivation of customers spending time in the queue, the relations of probability are used and further by using Littles formula other parameters will be determined. Keywords: Mean arrival rate, Mean service rate, Mean inter-arrival time, Mean service time, Traffic intensity

2 1 Introduction As the capacity of the system is finite binomial distribution follows arrival rate of the customers and servers service rate. The customers inter-arrival time and servers service time follows uniform distribution. By using binomial distribution customers probability in the queuing system is derived first. For the analysis of various performance measures of the queuing system a relation of expected customers spending time in the queue is derived and using it and Little s law the other related effective measures are determined. Figure 1. Basic Structure of Finite Queue Length Multiple Server Queuing Model 2 Methodology Let the queue length capacity of the system be M and there are n servers for service. Let α be the mean arrival rate of customers. Mean inter-arrival time of customer = 1/alpha Let β be the mean service rate of each server. Mean service time of a customer by each server = 1/beta Let x customers be there in the system. Since there are n multiple servers in the queuing model, the mean service rate of server is classified in to two possible cases. Case-1: If the servers are more in number than the arriving customers in the system (x<n) then, mean service rate is = xβ. Case-2: If the arriving customers exceeds or equal than the number of servers in the system (n x M) then, mean service rate is = nβ

3 To find traffic intensity of servers can be calculated when all the servers are busy. The queuing model is working on queue discipline FCFS. By using binomial distribution, probability of x customers arrived in timet is (1) From this relation it is clear that the time required for the arrival of M number of customers is 0 t M/α. For the probability of served customers Case-1: For (x<n) Probability of x customers served by the servers less than n in timet is (2) From this relation it is clear that the service time required to serve the customers which are less than the available servers is 0 t M/xβ. Case-2: For (n x M) Probability of x customers served by the n number of servers in timet is (3) From this relation it is clear that the service time required for M customers is 0 t M/nβ. (4)

4 3 Distribution of Time for Inter-arrival Time and Service Time of Finite Queue Length Multiple Server Queuing Model by using Uniform Distribution 3.1 For Arrival of Customers: By uniform distribution probability density function for continuous random variablet is The cumulative distribution function of uniform distribution is (5) For arrival time distribution, average inter-arrival time of customer is 1/α and mean of uniform distribution is (a+b)/2 (6) Hence probability distribution function is (7) (8)

5 Which gives the probability of another customer arriving in the next coming timet if a customer already arrived. The time limit indicates that the 100% arrival of customer in the next coming time 2/α. 3.2 For Service Time Distribution of Customers: Average service time a customer served by each server is 1/β And cumulative distribution function is (9) (10) Which gives the probability of another customer served in the next timet by each server if a customer already served. The time limit indicates that the 100% served another customer in the next time 2/βby each server. 4 Relation of Probability to Find the Expected Customers in the System System will have probability of containing x customers at time (t+ωt) is

6 (11)

7 (12) At steady state condition system is independent of time. Hence by using substitution in (11) for case-1 and in (12) for case-2 and simplifying to find the recurrence relation of probability as Probability of system having x customer is (13) (14)

8 5 Total Time Spent by a Customer in the Queue The arriving customer has to wait in the queue only if customers are more in number than the servers. Therefore to find the customers waiting time in the queue only case-2 is applicable. In steady state condition the waiting time distribution of every customer is similar and is a continuous random variable. Let T is a time that all customers are served by the servers. Let F T (t) be the cumulative probability distribution function of T Where, (15) If all the servers in the system are busy the arrival next customer will get service after the completion of service of all the customers in the system. As n servers are serving n number of customers Probability of serving n customers by n servers = Probability of serving one customer by one server

9 (16)

10 (17)

11 By using the relation (17) several others parameters of queuing model can be calculated. 6 Expected Time Spent by a Customer in the System Expected time spent by a customer in the system is given by T S = Expected time spent by a customer in the queue+ Time of service (18) 7 Expected Customers Waiting for Service in the Queue Expected customers waiting for service in the queue is possible only when the arrival customers are greater than the number of server. Therefore we use here probability of case-2. By using Littles law Expected customers waiting for service in the queue is given by (19) 8 Expected Customers Present in the System Expected customers present in the system are the customer under service and customers waiting for service in the queue. Again we use here probability of case-2 By using Littles law Expected customers present in the system is given by

12 (20) 9 Expected Time Spent By a Customer in the Queue for Busy System Expected time spent by a customer in the queue for busy system is given by (21) 10 Expected Customers Served Per Busy Period Expected customers served per busy period is given by (22) 11 Probability of a Customer has to Wait in the Queue for any Particular Time Customer has to wait in the queue when the condition is case-2. Binomial distribution gives the time to serve M number of customers is 0 t<m/nβ. By using equation (16) probability of a customer waiting for more than time a minutes in queue is

13 (23) 12 Conclusion Binomial and uniform distribution shows the time required to arrive and serve finite number of customers. Uniform distribution gives the 100% arrival of another customer and 100% served another customer by server in the next particular time limit. The study of different parameters of the model takes place when model works under case-2 condition. Relation of the expected time spent by a customer in the queue plays an important role in the determination of all others parameters. For the finite queue length multiple server queuing model binomial distribution and uniform distribution are the better option as they gives the particular time for the arrival and service. References [1] Anish Amin, Piyush Mehta, AbhilekhSahay, Pranesh Kumar And Arun Kumar (2014),Optimal Solution of Real Time Problems Using Queuing Theory, International Journal of Engineering and Innovative Technology,Vol. 3 Issue 10, pp [2] Babes M, Serma GV (1991), Out-patient Queues at the Ibn- Rochd Health Centre, Journal of the Operations Research 42(10), pp

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