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Operations Research

MBA-024

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QUEUING (WAITING LINE) THEORY

UNITIV

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Basic Components of Queuing

Model The arrival pattern/arrival rate

Service mechanism/service rate

No. of service facilities Capacity of the system

Queue discipline

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Symbols

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Single Channel Queuing Model

Arrival rates follow Poisson distribution

Service time follows exponential distribution Single server

Capacity of system is infinite

Queue discipline: FIFO

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Formulae

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Formulae

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Formulae

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A TV repairman finds that the time spent on his

job has an exponential distribution with mean 30minutes. If he repairs sets in the order in which

they come in and if the arrival of sets is

approximately a Poisson with an average rate of 10

in an 8 hour day, what is the repairmans expected

idle time each day? How many jobs are ahead of

the average set just brought in?

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Customers arrive at a box office window being

manned by a single individual according to aPoisson input process with a mean rate of 30 per

hour. The time required to serve a customer has an

exponential distribution with a mean of 90

seconds. Find the average waiting time of the

customer.

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Arrival of machinists at a tool crip is considered to bePoisson distributed at an average rate of 6 per hour.The length of the time the machinists must remain atthe tool crip is exponentially distributed with anaverage time being 0.05 hours.

a. What is the probability that a machinist arriving at

the tool crip will have to wait?b. What is the average no. of machinists at the tool

crip?

c. The co. will install a 2nd tool crip when convincedthat a machinist would have to spend 6 minuteswaiting and being served at the tool crip. By howmuch the flow of machinists to the tool crip shouldincrease to justify the addition of a 2nd tool crip?

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Customers arrive at a window drive in a bank

according to Poisson distribution with mean 10 perhour. Service time per customer is exponential with

mean 5 minutes. The space in front of the window,

including that for the serviced car can accommodate a

maximum of 3 cars. Other cars can wait outside thisspace.

a. What is the probability that an arriving customer

car drives directly to the space in front of the

window?

b. What is the probability that an arriving customer

car will have to wait outside the indicated space?

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A repairman is to be hired to repair machines

which break down at an average rate of 3 per hour.The breakdown follows a Poisson distribution. Nonproductive time of a machine is considered to costRs. 10 per hour. Two repairmen have been

interviewed one is slow but cheap while theother is fast but expensive. The slow repairmancharges Rs. 5 per hour and he services brokendown machines at the rate of 4 per hour. The fast

repairman demands Rs. 7 per hour and he servicesat an average rate of 6 per hour. Which repairmanshould be hired?

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Applications of Queue Model

Scheduling of aircraft at landing and takeoff

from busy airports.

Scheduling of issue and return of tools byworkmen from tool cribs in factories.

Scheduling of mechanical transport fleets.

Scheduling distribution of scarce war material.

Scheduling of work and jobs in production

control.

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Applications of Queue Model

Minimisation of congestion due to traffic delay at

tool booths.

Scheduling of parts and components to assembly

lines.

Decisions regarding replacement of capital assets

taking into consideration mortality curves,

technological improvement and cost equations. Routing and scheduling of salesmen and sales

efforts.

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Other Benefits of Queuing Theory

Attempts to formulate, interpret and predict

for purposes of better understanding the

queues and for the scope to introduce

remedies such as adequate service with

tolerable waiting time.

Provides models that are capable of

influencing arrival pattern of customers.

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Other Benefits of Queuing Theory

Determines the most appropriate amount of

service or number of service stations.

Studies behaviours of waiting lines viamathematical techniques utilising concept of

stochastic process.