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- 1. Prepared by Dr Ajay Parulekar
- 2. Topic Objectives Prepared by Dr Ajay Parulekar To get acquainted with the basic terminology of queuing theory To understand its use in day today business To learn applications of queuing theory through real life business problems
- 3. Introduction Prepared by Dr Ajay Parulekar A queuing model is a suitable model to represent a service oriented problem where customers arrive randomly to receive some service, the service time being also a random variable The objective of a queuing model is to find out the optimum service rate and the number of servers (resources) so that the average cost of being in queuing system and the cost of service are minimized Elements of a Queuing System: I. Arrival Pattern: It is characterized by, a. Number of customers arriving can be finite or infinite
- 4. Introduction Prepared by Dr Ajay Parulekar b. Time period between the arriving customers can be constant or random. Most commonly used is Poisson Exponential Distribution with mean arrival rate (average number of arrivals per unit period – sec, min, hour etc) denoted by λ. Hence mean time between successive arrivals is (1 / λ) II. Customer Behaviour III. Queue (Waiting Line): It refers to customers who are waiting to be served a. Waiting time is the time spent by a customer in the queue before being served. b. Service time is the time spent by server in providing service to a customer
- 5. Introduction Prepared by Dr Ajay Parulekar c. Waiting time in the system is the total time (waiting time in the queue + service time) d. Queue length is number of customers waiting in the queue e. System length is total customers in the system (customers in the queue + customers being served)
- 6. Introduction Prepared by Dr Ajay Parulekar IV. Servers: Servers can be a single one or can be multiple ones. Arrangement of servers can be done in series or in parallel. The service time is the actual time for which service is rendered to the customer. It can be constant or random. Most of the times the service time follows exponential continuous probability distribution with mean service time as (1 / μ) where μ is the mean service rate (average number of customers served per unit period of time) Service discipline is usually FIFO or FCFS provided server is busy
- 7. Introduction Prepared by Dr Ajay Parulekar Service Discipline: Service discipline is usually FIFO (First In First Out) or FCFS (First Come First Served) provided server is busy. It can also be LCFS (Last Come First Served) under certain scenarios. Under SIRO (Service In Random Order), customers are selected at random and time of arrival of customer is inconsequential. Under ‘Priority Service’, customers in a queue are selected on a priority basis (for e.g. tenure of service, VIPs etc.) We shall be restricting ourselves to FCFS
- 8. Introduction Prepared by Dr Ajay Parulekar Customer Behaviour: Customers who are patient by nature, will join an existing queue and will wait for service. Whereas certain ‘impatient’ customers might defect from queue. These will not choose a queue randomly (in case of multiple queues) and will keep on switching to fast moving or shortest queue. This behaviour is called ‘jockeying’. Under extreme condition, customer might wait in the queue for sometime and then leave the system because he feels system is working too slowly. This is called as reneging. A certain type of customer might, upon arrival not join the queue for some reason and decide to return after sometime. Such behaviour is known as‘balking’. We shall be assuming there is no jockeying or balking or reneging and customers leave the system only after receiving service and not before.
- 9. Kendall’s Notation Prepared by Dr Ajay Parulekar A queuing model is symbolically represented in terms of some important characteristics known as Kendall’s notation given by (a/b/c) : (d/e) where, a: Probability distribution of the number of arrivals per unit time. b: Probability distribution of service time c: Number of servers d: Capacity of the system e: Service discipline
- 10. Single server (Channel) Queuing Model (M/M/1) : (∞ / FIFO) Prepared by Dr Ajay Parulekar Customers arrive through a single queue & are served by single server on FIFO basis. Other assumptions, i)Arrivals - as well departures follow Poisson Distribution ii) Queue discipline is FIFO or FCFS iii) Only one queue & one server iv) Infinite potential customers, infinite system capacity v) Mean arrival rate (λ) < Mean Service rate (μ)
- 11. IMP FORMULAE - Single server (Channel) Queuing Model (M/M/1) : (∞ / FIFO) Prepared by Dr Ajay Parulekar MeanArrival Rate = λ Mean Service Rate = μ Utilisation Parameter /Traffic Intensity = (λ / μ) Probability of having exactly‘n’ customers in the system, Pn = ρn (1 – ρ) Expected number of customers in the system Ls = ρ / (1 – ρ) Expected number of customers in the queue Lq = ρ2 / (1 – ρ) Mean waiting time in the systemWs = 1 / (µ – λ) Mean waiting time in the queueWq = ρ / (µ – λ)
- 12. Multiple Servers (Channel) Queuing Model (M/M/k) : (∞ / FIFO) Prepared by Dr Ajay Parulekar Customers arrive through a single queue & are served by two or more servers which are arranged in parallel, on FIFO basis. Other assumptions, i)Arrivals as well departures follow Poisson Distribution ii) Queue discipline is FIFO or FCFS iii) Parallel servers (say k in numbers) serve a single queue iv) All service facilities /servers provide the same service at the same rate (μ) v) Infinite potential customers vi) Mean arrival rate (λ) < Combined Mean Service rate of all the servers(kμ)
- 13. Example 1 Prepared by Dr Ajay Parulekar A repairman is to be hired to repair machines which breakdown at an average rate of 6 per hour. The breakdown follow Poisson distribution. The productive time of a machine is considered to cost INR 20 per hour. Two repairman Mr. X & Mr. Y have been interviewed for this purpose Mr. X charges INR 100 per hour and he services breakdown machines at rate of 8 per hour. Mr. Y demands INR 140 per hour and he services at an average rate of 12 per hour. Which repairmen should be hired? Why?(Assumes 8 hours shift per day).
- 14. Example 2 Prepared by Dr Ajay Parulekar The mechanic at CarPoint is able to install new mufflers at an average of three per hour while customers arrive at an average rate of 2 per hour. Assuming that the conditions for a single server infinite population model are all satisfied, find out a)The utilisation parameter b)Average number of customers in the system c)Average time a customer spends in the system d)Average time a customer spends in the queue e) Probability that there are more than 3 customers in the system
- 15. Example 3 Prepared by Dr Ajay Parulekar Customers arrive to a typist, known for quality typing, according to Poisson probability distribution with an average inter arrival time of 20 minutes. If typist is not free, the customer will wait. The typist completes the typing jobs at an average time of 15 minutes (with exponential distribution). Find out, a)What fraction of time is the typist busy? b) Probability of having less than 3 customers at any time c) Expected number of customers with typist d) Expected number of customers waiting in the queue e)Time spent by a customer in the queue f) Average elapsed time between customer reaching and leaving the typist g) Expected time customer spends in the system h) Probability that a customer has to wait for more than 10 min in queue i) Probability that a customer shall be in system for more than 10 min