62560818 Queueing MMI Model

7/27/2019 62560818 Queueing MMI Model

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Customers or Units

(Population)

Size of

Population

Size of

Arrivals

Arrival

ControlArrival

Distribution

Attitude of

Customers

Finite Infinite

Single Group

Controllable Uncontrollable

Uniform Random

Patient Impatient

Poisson Exponential Others

Customers arrivals

at a departmental

store (i.e hrs of

business etc) issubject to varying

degrees of

influence

If the arrivals cannot be predicted with certainty, then

the pattern of distribution is random. The random

distribution can be Poisson, Exponential or othersArrival of patients in an accident

& Emergency ward of a hospital


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Impatient Customers

Customers may not join the waiting line when they

find it is too long; this attitude is called balkingSome Customers after waiting in the queue for

sometime may leave it; this attitude is called reneging

Patient Customers

Those who remain in the queue either voluntarily or

due to the force of circumstances are classified aspatient customers.


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Single- channel Poisson Arrivals with Exponential Service Rate

A Single channel facility : Arrivals from an Infinite Population

THE FIRST MODEL (M/M/1)

M / M / 1 : : (FIFO ) :

[ Fixed Arrival rate and Fixed Service ]

PoissonArrival

Exponential

Service

Number of

Service

Station

First in

First Out

Queue

Length


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Assumptions:

1. Arrivals are described by Poisson probabilitydistribution.

2. Single waiting line and each arrival waits to beserved regardless of the length of the queue.(i.e.infinite capacity) and there is no balking or reneging.

3. Queue discipline is first-come, first-served.

4. Single server and service times follow exponential

distribution.5. Customer arrival is independent but the arrival rate

(avenge number of arrivals) does not change over time.

6. The average service rate is more than the average arrivalrate.


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The following events (possibilities) may occur

during a small interval of time, tjust before time t.

It is assumed that the system is in state n (number ofcustomers) at time t.

1. The system is in state n (number of customers) and no

arrival and no departure, leaving the total to ncustomers.\

2. The system is in state n+1 (number of customers) and no

arrival and one departure, reducing the total to n

customers.

3. The system is in state n-1 (number of customers) and one

arrival and no departure, bringing the total to ncustomers.


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The probability Pnof n customers in the system at

time t and the value of its various operating

characteristics is summarized as follows:

The figure below illustrates the probability Pnby

considering each possible number of customers either

waiting or receiving service at each state that may be

entered by the arrival of a new customer or left by the

completion of the leading customers service.


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0 1 2 n

-

n n+1 s.

Number of customers in the system

On arrival

One Service

completion


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/= Traffic Intensity ()

if, /> 1, in that case the queue will grow without endif, /= 1, then, no change in queue length will be noticed

if, /< 1, then the length of the queue will go on diminishing

gradually.

The first model is applicable under the condition that : /< 1.


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Let

(or Average Arrival rate) = Average number of customers

arriving in one unit of time.

( or Average Service rate) = Average number of customer being

serviced in one unit of time, assumingno shortage of customers.

1/= Average inter- arrival time or is the length of the time interval

between two consecutive arrivals (t & t +t)

1/= inter service time


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Ls= Expected Average number of units being serviced and/ or waiting

in the system.

Lq = Expected Average number in the queue (the number in the queue

does not include the unit being serviced)

pn = Probability of having n units in the system.

Wq = Expected Average time an arrival must wait in the queue.

Ws

= Expected Average time an arrival spends in the system (both in

queue and service)


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There are several relationships of interest:-

p0= (1- /) = (-)/Probability of no customer in

the system or systembeing empty or

expected idle time of

the system.

1- p0 = 1- (1-) = = /Probability of servicefacility being busy.

pn= (/)n

.p0 or (/)n

(1- /)or

pn= (/)pn-1Probability of n customers in the

system


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Since p0is the probability of the system being empty, it is the expected

idle time of the system.

Also, (1-p0)= /is the expected busy time of the system, or the

expected utilization of the system.

Or

= / (Probabilty of service facility being busy)


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We are concerned with the determination of the following:-

1. Expected number of units (or customers) in the waiting lineand or being serviced (i.e the expected/ Average number ofunits or customers in the system):-

L = /(- )or Ls

2. Expected or Average number of units (or customers) in the

queue is :-( or Average queue length) or The average number ofcustomers waiting to be served:

Lq = 2/ (- )


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3. The average waiting time (in the queue) of an arrival (Average waiting ofan arrival) or the average time a customer waits before being served is:

Wq= L

q/ = 2/ (- ) = /(- )

4. Also, since the mean service rate is , the mean service time is 1/;

hence the average time an arrival spends in the system (both waiting andin service) or Average time a customer spends in the system is :

W = Wq+ 1/= 1/ (-)

or Ws


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The probability that the number in the queue and being serviced is greaterthan K is :-

P (n>K) = (/)k+1

Above equations apply only if (/)

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62560818 Queueing MMI Model