Queuing Models. Queue. System

Transcription

1 Queuing Models Introduction The goal of Queuing model is achievement of an economical balance between the cost of providing service and the cost associated with the wait required for that service This model is applied to service oriented organizations like machine repair shop, production shop, food chain, restaurant etc. Characteristics of Queuing model Arrival rate Server S Calling population Queue System Service rate 1. System: The place or facility where the customer arrives in order to get service is called system and its capacity may be finite and infinite. 2. Arrival rate or Arrival pattern: The average number of customers arriving per unit time within the system to get service is called arrival rate. Arrival is random and so it is assumed to follow Poisson distribution. 3. Service rate or Service pattern: The number of customers serviced per unit time is called service rate and it is assumed to follow exponential distribution.

2 4. Service rule or Service order: Service rule provides information about Queue discipline which means the order by which customer are picked up from the waiting line to provide them service. Various service rules available are: (i) FIFO or FCFS - First in First out / First come First serve (ii) LIFO or LCFS Last in First out / Last come First serve (iii) SIRO Service in random order (iv) Priority treatment 5. Calling population: The entire sample of customers from which only a few visit the service facility is called Calling population or Input source. Its capacity may be finite or infinite. It is infinite when the arrival of few customers does not have any effect on the arrival pattern of future customers. Customer s attitude Attitude Patient Impatient Jockey Balking Reneging Cheater 1. Jockey: Customer keep on changing queue in hope to get the service faster. 2. Balking: Customer does not join the queue as the queue is very long. 3. Reneging: Customer join the queue for a short period then leave the system as queue is moving very slowly. 4. Cheater: Customer takes illegal means like bribing, fighting etc. in hope to get service faster.

3 Representation of Queuing Models Queuing models are represented by Kendall and Lee notations whose general form is (a/b/c) : (d/e/f ) Where a= probability distribution for arrival pattern b= probability distribution for service pattern c = number of servers in the system d= service rule or service order e= size or capacity of system f= size or capacity of calling population Symbols for a and b M Markovian (Poisson) for arrival pattern or Exponential service pattern. E Erlangian (Gamma) distribution for arrival or service pattern. D Deterministic arrival or service pattern. Symbols for c 1, 2, 3, 4, 5. Symbols for d FIFO / FCFS LIFO / LCFS SIRO GD: General Service discipline. Example Senior citizen, VIP line etc.

4 Symbols for e and f N finite - infinite Example: (E/M /3) : (SIRO/N/ ) Single Server Queuing Model (M/M /1) : (FIFO/ / ) λ = Arrival rate (Poisson in nature) Unit Customers / hour = Inter-arrival rate (Exponential in nature) = time gap between two consecutive arrivals Unit hour / customers µ = Service rate (Exponential in nature) Unit Customers / hour = Inter-service rate (Poisson in nature) Unit hour/ Customer Different conditions in single server queuing model Case-1: If λ µ In this situation the queue length will kept on increasing and ultimately system fails. There is no solution to such problem. Case-2: If λ µ System Works

5 Traffic intensity The ration between arrival to service rate is called traffic intensity or system utilization. It is denoted by ρ. ρ = (i) It tells the percentage time server is busy. (ii) It shows system utilization or shows system is busy. (iii) It also tells the probability that a customer has to wait. Traffic intensity is also known as utilization factor, average utilization, channel efficiency and clearing ratio. Example: λ = 5 customer / hr µ = 20 customer / hr then ρ = = = 0.25 i.e. Server performs customer s work in 25 % of the total customer s time in system and rest 75 % of the total time customer has to wait. Some basic formulae s pertaining to Queuing model 1. Probability that the system is idle or probability of zero customers in the system or the probability that a new customer does not have to wait in the queue Po = 1 - ρ 2. Probability of having exactly n customers in the system Pn = ρ n. Po = ρ n (1 ρ) Po + P1 + P2 + P3 + P4 +.. = 1

6 3. Average number of customers in the system: It includes both the customer waiting in the queue along with those provided service. It is denoted by Ls. S Queue 1 System 3 Queue 2 L s = n= number of customer in system Pn = probability of having n customers in system Ls = Average number of customers in the Queue: It does not consider the customers which are provided service. It is denoted by Lq. L q = Lq =

7 5. Average length of non- empty queue or queue containing at least one customer or queue that is formed from time to time. = Little s Law For a stable system average number of customers in the queue or system is equal to average customer arrival rate multiplied by average waiting time of the customer in the queue or system respectively. Lq = Wq and Ls = Ws W q = Average waiting time of the customer in the queue W s = Average waiting time of the customer in the System Ls = ρ 1 ρ Lq = Ls ρ = Ls ρ Lq = ρ2 1 ρ Ws= Ls λ Wq = Lq λ = Ws - 1

8 References 1. Operations Research by Kanti Swarup 2. Operations Research by Hira and Gupta