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QUEueING THEORY

Flow of customers from infinite/finite population

towards service facility form queue.Due to Lack of capability

Customers

any arriving unit(persons,machine,vehicle,parts,etc..)

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QUEueING system- Input Process

Input process : customer arriving pattern

Size of queue Finite , Infinite

Arrival distn for uncertain arrival

characterized by prob..distribution

Customers behavior

Balked -quitting initially

Renegedquitting after sometime

jockey -move from one queue to another

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QUEueING system- Queue descipline

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Capacity of the system ..

Finite source

Infinite source

Random arrivalPoisson dist..

Inter arrival - Exponential dist..

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LittLes Law

Most powerful , simple laws in queuing theory.

Flows in rate =1 ; Flow out = 2.

1 < 2 ( Tank empty )

1 > 2 ( Tank Overflows)

1 = 2 (steady state =1=2) ( Tank constant )

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Average residency time T = N/. The response time is T.

Example :

If a server processes 630,000 requests in a half hour and the average number of

requests in the server is 2. What's the average Response time?

= 630000/1800 = 350/second

N = T

T = N/ = 2/350 = 0.0058 seconds

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Line 1

Line 2

Line 3

Line 4

LINE 1

Line Interval 1 Sec ; Avg time b/w arrival 0.1 sec (100 ms )

Service time - 50 ms # No waiting

LINE 2

2 arrivals together #

1st transaction 50 ms ..

2nd transaction 100 ms ( 50 ms waiting + 50 ms service )

3 arrivals together #

1st transaction 50 ms ..

2nd transaction 100 ms ( 50 ms waiting + 50 ms service )

3rd transaction 100 ms ( 100 ms waiting + 50 ms service )

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Erlang dist

If one server is busy 50% of the time, probability of finding it busy on arrival

is 0.5

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If a system has 2 servers, for what utilization threshold, might

I expect for the probability of both servers being busy to be

less than 0.1?

Formula # C(2,T) = (2U2) / (1 + U)

0.1 = (2U2) / (1 + U)

20U2 -U - 1 = 0

U = -0.2, +.25

Ans #At 25% busy, the probability of finding both busy is 0.1.

Or

for 90% of the time a request will not wait.

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1) FIFO (First In First Out)

- FCFS (First Come First Serve)

- orderly queue.

2) LIFO (Last In First Out)

-LCFS (Last Come First Serve)- stack.

3) SIRO (Serve In Random Order)

4) Priority Queue

Pre emptive

Non pre emptive

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service MechaNisM

Single queueone server

Single queueseveral server

Several queueone server

Several servers

Parallel channels (barber shop, super market)

series channels (must pass thro all service channel)

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Queue models

Arrival Model

Poisson probability

Mean arrival rate; timet ; Meant

Interarrival time Exponential distnmean 1 /

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Problems

1) A T.V repairman finds that time spent on his job has an exponential distribution

with mean 30mins. If he repairs sets in the order in which they came in , and if the

arrival of sets is approx poison with an average rate of 10 per 8 hr day, what is

repairmans expected idle time each day ? How many jobs are ahead of the

average set just brought in ?

1) in the prodn shop of a company ,breakdown of machines is found to be poison

wth avg rate of 3 machine per hr. Breakdown time are one machine cost Rs.40 per

hr to the company.There are 2 choices bfor the company for hiring the repairmen

.one of the repairmen is slow but cheap, other fast but expensive.the slow-cheap

repairmen demands Rs.20 per hr and wil repair the broken down machines

exponentialy at the rate of 4 per hour.The fast expensive repairmen demand Rs.30per hour and wil repair machines exponentially at an average rate of 6 per

hour.Which repairmen should be hired ??