Queuing Models

Queuing ModelsQueuing Models

The single server waiting line systemThe single server waiting line system Undefined and constant service time.Undefined and constant service time. Finite Queue length.Finite Queue length. Finite calling populationFinite calling population The multiple server waiting line.The multiple server waiting line.

Single –Server waiting line System The queue discipline: In what order the customers are

served. First-cum first, in any market situation. Last-in-first-out : A machine operator might stack in process

parts beside a machine so that so that the last part is on the top of the stack and will be selected first.

Random : Selecting any body from the queue at random. Pre arranged Schedule: Doctor’s appointment.

The nature of the calling population Source of the customers in the market. In general it is infinite. Finite Calling population : Customers from a particular place.

The arrival rate (Poisson distribution) The service time. ( Exponential Distribution)

Birth-Death Process

• En(t) : Number of times the process enters the state “n” by time t.

• Ln(t): Number of times the process leaves the state “n” by time t.

• Because the above two types of event must alternate, these two numbers must always equal or differ by one

• En(t) – Ln(t) ≤ 1

A Bank counter is an example of a single A Bank counter is an example of a single server queuing system.server queuing system.

1. Infinite calling population1. Infinite calling population 2. First-come –first served queue 2. First-come –first served queue

disciplinediscipline 3. Poisson arrival rate3. Poisson arrival rate 4. Exponential service time.4. Exponential service time. Arrival : 24 customers per hour.Arrival : 24 customers per hour. Service : 30 customers per hour.Service : 30 customers per hour.

Given customer’s expectations , the store’s manager Given customer’s expectations , the store’s manager believe that it is unacceptable for a customer to wait 8 believe that it is unacceptable for a customer to wait 8 mins and spend a total of 10 mins in the queuing system. mins and spend a total of 10 mins in the queuing system. The manager wants to test several alternatives for The manager wants to test several alternatives for reducing customer waiting time. ( 1) addition of another reducing customer waiting time. ( 1) addition of another employee. (2) addition of another counter. employee. (2) addition of another counter.

Service rate : 40 customers served per hour.Service rate : 40 customers served per hour. The addition of extra employee will cost the bank Rs The addition of extra employee will cost the bank Rs

150/week . The bank has determined that for each 150/week . The bank has determined that for each minute the average waiting time reduced the bank minute the average waiting time reduced the bank avoids the loss in business equivalent to Rs 75/ week. avoids the loss in business equivalent to Rs 75/ week.

Addition of a new counter will cost Rs 6000+ Rs Addition of a new counter will cost Rs 6000+ Rs 200/week. 200/week.

Operating Characteristics of each alternative

Operating Characteristics

Present System

Addition of a person

Addition of a new counter

L 4 persons 1.5 persons 0.67 persons

Lq 3.2 customers

0.9 customers

0.27 customers

W 10 mins 3.75 mins 3.33 mins

Wq 8 mins 2.25 mins 1.33 mins

U 0.8 0.6 0.4

The new loan officer of a bank interviews all customers for new accounts. The customers desiring to open new accounts arrive at the rate of 4 per hour according to a Poisson distribution, and the accounts officer spends an average of 12 minutes with each customer setting up a new account.

Determine the operating characteristics for this system.

Add an additional accounts officer to this system and determine the operating characteristics.

The ticket booth on the tech campus The ticket booth on the tech campus is operated by one person who is is operated by one person who is selling the tickets. The ticket seller selling the tickets. The ticket seller can serve on the average of 12 can serve on the average of 12 persons/hour. On average 10 persons persons/hour. On average 10 persons arrive to purchase the ticket. arrive to purchase the ticket. Determine the average time a ticket Determine the average time a ticket buyer must wait and the portion of buyer must wait and the portion of time the ticket seller is busy. time the ticket seller is busy.

• A service station has one pump for A service station has one pump for unleaded gas which with an attendant unleaded gas which with an attendant can service 10 customers per hour. can service 10 customers per hour. Cars arrive at a rate of 6/hour. Cars arrive at a rate of 6/hour. Determine the average queue length, Determine the average queue length, the average time a car in the system, the average time a car in the system, and the average time the car must wait. and the average time the car must wait. If the arrival rate increases to 12 cars / If the arrival rate increases to 12 cars / hour what will be the effect on the hour what will be the effect on the average queue length. average queue length.

A bank has one outside teller counter. It takes an A bank has one outside teller counter. It takes an average of 4 minutes to serve a bank customer. average of 4 minutes to serve a bank customer. Customers arrive at the window at the rate of Customers arrive at the window at the rate of 12/hour. The bank operation officer is currently 12/hour. The bank operation officer is currently analyzing the possibility of adding a 2analyzing the possibility of adding a 2ndnd window at window at an annual cost of Rs20,000. It is assumed that an annual cost of Rs20,000. It is assumed that the arriving customers would be equally divided the arriving customers would be equally divided between both windows. The officer estimates that between both windows. The officer estimates that each minutes reduction in customers waiting time each minutes reduction in customers waiting time would increase the banks revenue by 2000 would increase the banks revenue by 2000 annually. Should the 2annually. Should the 2ndnd window needs to be window needs to be installed ?installed ?

An Airport has a single runway and one air traffic An Airport has a single runway and one air traffic controller to land planes. It takes an airplane 12 mins controller to land planes. It takes an airplane 12 mins to land and clear the run way. Planes arrive at the to land and clear the run way. Planes arrive at the Airport at the rate of 4/hour.Airport at the rate of 4/hour.

Determine the average number of planes that will Determine the average number of planes that will stack up waiting to land.stack up waiting to land.

Find the average time a plane must wait in line before Find the average time a plane must wait in line before it can land.it can land.

Calculate the average time it takes a plane to clear Calculate the average time it takes a plane to clear the runway once it has notified the airport that it is in the runway once it has notified the airport that it is in the vicinity and wants to land.the vicinity and wants to land.

The airport has a rule that an air traffic controller can The airport has a rule that an air traffic controller can on the average land planes a maximum of 45 minutes on the average land planes a maximum of 45 minutes out of every hour. There must be 15 mins of idle time out of every hour. There must be 15 mins of idle time available to relieve the tension. Will this airport needs available to relieve the tension. Will this airport needs to hire an extra air traffic controller ?to hire an extra air traffic controller ?

A machine shop has 5 machines that periodically breakdown and require service. The average time between breakdowns is 4 days, distributed according to exponential distribution. The average time to repair a machine is one day, distributed according to exponential distribution. One mechanic repairs the machines in the order in which they are breakdown.

Determine that the probability that the mechanic being idle.

Determine the mean number of machines waiting to be repaired.

Determine the mean time machines wait to be repaired. Find the probability that 3 machines are not operating

• A motor garage has the space for only one vehicle in service and 3 vehicles lined up to wait for service. If the space is full( 3 cars in waiting line) , prospective customers should not enter for service.

• The mean time between arrivals of cars for lubrication is 3 Mins.

• The mean time required to perform the lube operation is 2 mins.

• Both the inter arrival and service time is expoential.

At a Railway station , only one train is handled at a time. The railway yard is sufficient for 2 trains to wait while the other is given signal to leave the station. Trains arrive at the station at an average rate of 6/hour and the railway station can handle them on an average of 12/hour. Assuming Poisson arrivals and exponential service distribution, find the service characteristics. Also find the average waiting time of a new train coming into the ward.

A manufacturing co. operates a shop that A manufacturing co. operates a shop that includes 20 M/C s. Due to the type of work includes 20 M/C s. Due to the type of work performed in the shop , they require performed in the shop , they require frequent repair. The co. has one senior frequent repair. The co. has one senior repair person with an assistant. They repair repair person with an assistant. They repair the M/C s in the same order as they break the M/C s in the same order as they break down. M/c break down are according to down. M/c break down are according to Poisson and service is exponential. Each M/c Poisson and service is exponential. Each M/c operates an average of 200 hours before operates an average of 200 hours before breaking down. The average time to repair a breaking down. The average time to repair a M/c is 3.6 Hours. The co. would like an M/c is 3.6 Hours. The co. would like an analysis of M/c idle time due to break downs analysis of M/c idle time due to break downs to determine if the present repair staff is to determine if the present repair staff is sufficient. sufficient.

• A survey of customer service dept. for a 12 month period shows that the arrival and service rates in a bank are 10 and 4 respectively. Let the bank has 3 counters. Calculate the Operational characteristics.

The Bank management has observed that customers are frustrated by the relatively long waiting time of 21 minutes and 0.703 probability of waiting. To try to improve matters , management has decided to consider addition of an extra counter. Find the new operational characteristics.

• The Dupit-corp is a long time leader in the office photo copier marketplace. One reason for this leadership position is the service the co. provides its customers.

• The service is provided at the customers’ site by the co.’s service tech reps.

• Each tech is given responsibility for a specified territory. Each tech rep’s territory should be assigned enough machines so that the tech reps will be busy 75% of the time. When working continuously each tech rep should able to repair an average of 4 machines per day. Since the co.’s machines now are averaging 50 work days between needing repairs, the target to assign approximately 150 machines to each tech rep’s territory.

• Under this policy the co. has nearly 10,000tech reps with a total payroll of approximately $600 million per year.

Suggestion by the Co.’s VP Modify the current policy by decreasing the

percentage of time that the tech reps are expected to be active in repairing machines. This involves simply decreasing the number of machines assigned to each tech rep and adding more tech reps.

By assigning 100 machines to each tech rep we can get a new set of performance characteristics.

= (Number of machines assigned to each tech rep)/ 50