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Queuing Theory

Queuing Theory. Unit 7. A pioneer: Agner Krarup Erlang (1878-1929). Examples. Wait for a bus Wait for traffic lights to turn green Wait for lift in bldg. Cars waiting at petrol pumps for service Customers waiting at bank Telephone subscribers waiting for connections

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Queuing Theory

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  1. Queuing Theory Unit 7

  2. A pioneer: Agner Krarup Erlang (1878-1929)

  3. Examples • Wait for a bus • Wait for traffic lights to turn green • Wait for lift in bldg. • Cars waiting at petrol pumps for service • Customers waiting at bank • Telephone subscribers waiting for connections • Aircrafts get delayed for want of free runway. • Broken down of machines waiting for repairs • Workmen waiting for tools • Goods in production shops waiting for cranes.

  4. Example • These all queuing models represents common feature. Customers arrive at a service center and wait for service. • The arrival of customers is not necessarily regular and so is the time taken for service not uniform. • Queues build up during hours of demand and disappear during the idle period.

  5. Introduction • Lack of adequate service facility would cause waiting lines of customers to be formed . • The only way that the service demand can be met with ease is to increase the service capacity to a higher level . • But this may be a costly affair and also uneconomic after a stage . • Therefore a manager has to decide on an appropriate level of service which is neither too low nor too high .

  6. 1.Arrival Process A. According to source –The source of the customers may be finite or infinite .eg.B. According to numbers-The customers may arrive for service individually or in groups .C. According to time –Customers may arrive at known times or in a random way .

  7. The queuing models wherein customers arrive at known times are called deterministic models and are easier to handle . • On the other hand, a majority of queuing models are based on the premise that customer enter the system at random point of time .These arrivals are Poisson distributed .

  8. Service System • There are two aspects in the service system – • Structure of the service system and • The speed of service

  9. Structure of service system • A Single Server Facility –eg. A library counter • Multiple, Parallel Facilities with Single Queue eg. Booking at a service station having several mechanics , each handling one vehicle. c. Multiple , Parallel Facilities with Multiple Queues – eg. Different cash counters at the electricity office d. Service Facilities in a Queue –eg. Machining of a certain steel item

  10. Speed of the Services It can be expressed in two ways – • The service rate – It describes the number of customers serviced during a particular time period . • The service time – It describes the amount of time needed to service a customer. Service rates and service times are reciprocals of each other .eg. If a cashier can attend 10 customers in an hour , calculate service rate and time .

  11. Queue Structure • It is the order in which customers are picked up from the waiting line for service. • First-cum-first-served- eg. Queue at a bus stop, • Last-cum-first-served eg. Letters to be typed by a typist . • Service-at-a-random eg. • Priority Service eg. Treatment of VIPs

  12. Operating characteristics of a Queuing System • Queue Length – the average no. of customers in the queue to get the services . • System length – the average no. of customer in the system , those waiting to be and those being serviced . • Waiting time in the queue- the avg. time that a customer has wait in a queue to get the service , • Total time in the system- the avg. time that a customer spends in the system , from entry in the queue to completion of service . • Server idle time – the relative frequency with which the system is idle .

  13. Mathematical Analysis of Queuing Process • Present the various parameters of queue in the form of mathematical model. • When specifying the statistical distribution of arrivals or service times, we have to find that when equilibrium state came. • Another important parameter is to find the length of queue. The number of customers will be very much different 15 minutes after opening the counter in post office from that after one hour. After the initial rush, one might reasonably expect to find the system with the same type of probabilities of arrivals.

  14. Deterministic queuing model • The first case where the customer arrive in the queuing system at regular intervals and the service time for each customer is known and constant. • Suppose that customer come to bank’s teller counter every 5 minutes. Thus the interval between the arrival of any two successive customer is exactly 5 minutes. Assume that banker also takes exactly 5 minutes to serve a customer. So the arrival and service rates are each equal to 12 customer per hour. In this situation there shall never be a queue and the banker shall always be busy with work.

  15. Deterministic queuing model • But if the banker can serve 15 customer per hour which shows high service rate than arrival rate. So banker would be busy in 4/5th of time and idle in 1/5th of the time. He will take 4 minute to serve a customer and wait for 1 minute for the next customer to come. So there is no queue. • On the other hand if banker can serve 10 customer per hour than in that case he/she would be very busy and queue length will increase continuously without limit with the passage of time. • Symbolically let the arrival rate is λ per unit time and the service time is µ customers per unit time

  16. Deterministic queuing model • If λ > µ, the waiting line would form and will increase indefinitely, service facility would always be busy. • If λ ≤ µ, there shall be no waiting time and the proportion of time service facility would be idle is 1 - λ / µ. • λ / µ = ρ = average utilization or traffic intensity or clearing ratio. • ρ > 1, the system would fail • ρ ≤ 1, the system works and ρ proportion of time it is busy.

  17. Probabilistic Queuing Models • Three possible queuing models. • Poisson-exponential single server model – infinite population • Poisson-exponential multi server model – infinite population • Poisson-exponential single server model – finite population

  18. Poisson - exponential • Poisson distribution – event occur in specified time interval. • The word ‘ Poisson – exponential’ indicate that the customer arrivals follow Poisson distribution while service times are distributed exponentially. • It means that the arrivals are independent with the average arrival rate equal to λ per period of time

  19. Poisson-exponential single server model – infinite population • λ = Average number of arrivals per unit time • µ = Average number of customers served per unit time. • ρ= Traffic intensity =average utilization • Lq = no. of customers in the queue or the length of queue • Ls = number of customers in the system = number of customers in the queue + no of customers being served

  20. Probability of n customers in the system = Pn • The ratio of λ / µ = ρ shows the proportion of time the service station is busy. From this the probability that the system is idle, i.e. no customers in the system equals to Po = 1 – ρ. • From this prob of having exactly one customer in the system is P1 = ρ*Po • Now prob of having exactly two customer in the system is P2 = ρ*P1 = ρ2*Po • Similarly, prob of having n customer in the system is Pn = ρn*Po = ρn*(1 - ρ )

  21. Examples • A tailor specializes in ladies’ dresses . The number of customers approaching the tailor appear to be Poisson distributed with a mean of 6 customers per hour . The tailor attends the customers on FCFS basis . The tailor can attend the customers at an average rate of 10 customers per hour . You are required to find

  22. Contd. • The utilization parameter • The probability that the system is idle • The avg. time that the tailor is free on a 10 hour working day • The prob that there are 5 customers in the system • What is the expected no. of customers in the shop ? • What is the expected no. of customers waiting for tailor’s services ?

  23. Example 2. Arrivals at a telephone booth are considered to be Poisson , with an avg. time of 10 minutes between one arrival and the next . The length of phone call is assumed to be distributed exponentially , with mean 3 minutes . Find – • The probability that an arrival finds that 4 persons for their turn , • The average no. of people waiting and making calls

  24. Example • A TV repairman finds that the time spent on his job has an exponential distribution with mean 30 min. if he repairs sets in order in which they come and if the arrival of sets is approximately Poisson with an avg. rate of 10 per 8-hour day , what is his expected idle time each day ? How many jobs are ahead of the set just brought in ?

  25. Example • Customers arrive at the First Class ticket counter at a rate of 12 per hour . There is one clerk serving the customers at the rate of 30 per hour . • What is the prob. that there are no customer in counter( i.e. the system is idle) ? • What is the probability that there are no customers waiting to be served ? • What is the prob. that a customer is being served and nobody is waiting ?

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