module 7- queue

7/28/2019 Module 7- Queue

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Queuing Theory


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Queuing Theory

Called as waiting line-----theory is applied tosituations where customers arrive at some servicestation for some service: wait and leave the systemafter getting the system

Waiting line is developed because ----

Service demand can be met by-----

Adding capacity is costly affair ----

Cost of offering the serviceAssociated with the service facilities and their

operation, and

Cost incurred due to delay in offering service.

Associated with the cost of customers waiting time.


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Basic characteristics of a queuing

phenomenon: Customers arrive at regular or irregular

intervals of time. This is called arrivals of

customers. One or more service channels or service

facilities are assembled at the service

center. If the service station is empty, ..

If not queue is formed.


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Genera structure o Queuingsystem

Customer-arrival process

Queue

Service system

Customers leave the system


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1. Arrival Process

According to source- can be finite or infinite---customers at a supermarket

According to numbers-individually or in groups

According to time-arrival times are known with

certainty or at random Use Poisson distribution


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2.Service System

a. Structure of the service system Single server facility

Multiple, parallel facilities with single queue

Multiple, parallel facilities with multiple queues

Service facilities in a series

b. Speed of service Expressed in 2 ways

Service rate no.of customers serviced during a particulartime period

Service time amount of time needed to service acustomer

Service times are exponentially distributed


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3. Queue Structure

First come first served

Last come first served

Service in random order

Priority service


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Behavior of the customer

Customers can be patient or impatient Jockeying among many queues, i,e customers

may switch to other queues which are movingfast

Reneging stands in the queue for sometimeand leave the system because it is working tooslow

Bribing- Balking-customers will not join the system for

some reason and decide to join at the later stages

Assumption customers after getting the systemleave the ueue


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Operating Characteristics

a) Queue length average number of customers in queue waiting to get serviceb) System length

average number of customers in the system

c) Waiting time in queue

average waiting time of a customer to get serviced) Total time in system

average time a customer spends in the system from entryinto the queue to completion of service

e) Server idle time

relative frequency with which system is idle, directly related tocost (The server utilization factor (or busy period) - is theproportion of the time that a server actually spends with thecustomers. Gives an idea of expected amount of idle time,which can be used for some other work.)


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Queuing models

Deterministic model-customer arrivalat regular interval and service time isknown

Eg: interval between the arrival of any 2successive customers is 5 minutes, and 5minutes to serve each customer, can serve12 customers/hour

Let arrival rate be customers/unit time

Service rate is customers /unit time


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If> - waiting line

- no waiting time

The proportion of time service facility is idle

1(/ )

Average utilization = (/ )= row

Or traffic utilization

If row >1,system would fail

row 1,system works and row is the proportion of

time it is busy


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Kendalls Notation for representingqueuing models

(a/b/c) : (d/e)

a = arrival distribution b = departure distribution.

c = number of parallel service channels in thesystem.

d = service discipline. e = max number of customers allowed in the

system.


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Characteristics of Queuingmodules:

1)Input or arrival distribution:

Represents the pattern in which the number ofcustomers arrives at the system.

Arrivals may be represented by the inter-arrival time, which is the period between twosuccessive arrivals.

The number of customers arriving per unit of

time is called arrival rate. When arrivals are random, we have to use

probability distribution Poisson distribution


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2) Service (departure) distribution:

Represents the pattern in which the number ofcustomers leaves the system.

Servicetime = time period between twosuccessive services.

Service times are randomly distributed,exponential distribution is used.

Service rate = number of customers served prunit time.

Mean value of service rate - .


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3)Service channels:

Queuing system may have a single service channel

System may have number of channels, arranged either inseries or in parallel.

Service channels customers must pass successivelythrough all the channels before service is completed.

Eg:- product undergoing different processes overdifferent machines.

A queuing model One server model

- Multi server model.


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4)Service discipline:

Service discipline or order of service isthe rule by which customers are

selected from the queue for service.

FIFO FCFS

Eg: - cinema halls LCFS - eg: - big godown

Priority


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5)Maximum number of customersallowed in the system:

Either finite or infinite.

6. Calling source or population: The arrival pattern of the customers

depends upon the source that generates

them. If there are only a few potentialcustomers, the calling source (population)is finite.


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Symbols for a & b: M = Markovian Ek = Erlangian Symbols for d

FCFS LCFS SIRO service in random. e represent finite (N) or infinite ().

Eg:- (M/Ek/1) : (FCFS/N) Poisson arrival Erlangian departure, Single server, first come first served discipline, max allowable

customer N


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Transient and steady states of thesystem

If operating characteristic (behavior of thesystem) varies with time, it is said to be intransient state - initial stages.

A system is said to be in steady statecondition if its behavior becomesindependent of time.


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Probabilistic Models

Model 1: Poisson-exponential singleserver model infinite population

Assumptions:

Arrivals are Poisson with a mean arrival rate of, say

Service time is exponential, rate being Source population is infinite

Customer service on first come first served basis

Single service station

For the system to be workable,

Model 2: Poisson-exponential singleserver model finite populationHas same assumptions as model 1, except that population is finite


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Queuing Models

Model 3: Poisson-exponential multipleserver model infinite population

Assumptions

Arrival of customers follows Poisson law, mean rate

Service time has exponential distribution, mean servicerate

There are Kservice stations

A single waiting line is formed

Source population is infinite

Service on a first-come-first-served basis

Arrival rate is smaller than combined service rate of allservice facilities


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Notations: n = no. of customers in the system (waiting & in service) Pn = prob. of n customers in the system. = Mean customer arrival rate or average no. of arrivals in the queuing

system/unit time. = Mean service rate or average no. of customers completing service/unit time. /= P = Average service completion time (1/) Avg. inter arrival time (1/) = traffic intensity or server utilization factor.

S = no. of service channels (service facilities) N = max no. of customers allowed in the system. Ls = Mean no. of customers in the system (waiting & in service). Lq = Mean no. of customers in the queue (queue length). Lb = Mean length of non empty queue. Ws = Mean waiting time in the system.

Wq = Mean waiting time in queue. Ws = Mean waiting time of an arrival who has to wait.


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Single channel queuing theory

Random arrivals Poisson distribution

Random Service exponentiallydistribution.

Poisson-exponential Single server model-

infinite population

module 7- queue

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