Queuing Theory

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  • Topic: Queueing theory, Poisson distribution, Queueing model
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TOPIC VII

QUEUING THEORY

Definition

A queue can be defined as items awaiting service. Queues may consist of people, cars, components awaiting machining, telephone calls, aeroplanes, indeed any discrete items. Queues form when the rate of arrival of items requiring service is greater than the rate of service.

Queuing theory can be defined as the construction of mathematical models of various types of queuing systems so that predictions may be made about how the system works with the demands made upon it.

Applications

Queuing theory may be applied in the following areas:
a) Shop counters. Customers arrive at varying intervals requiring service which takes a variable time. What is number of assistants that will maximize profit, or provide the best service? b) Telephone exchange. The smaller the exchange the lower the cost but the greater the congestion. The larger the exchange the higher the costs but with reduced congestion. c) Parts stores. Production workers waiting to draw out parts. What is the appropriate number of service points and staff to produce lowest overall cost? d) Airport runways. How many runways are needed to provide landing facilities after a reasonable queue time?

Terms used in Queuing Theory

Customer- persons or units arriving at a station or service
Service station- point where service is provided
Waiting time- time a customer spends in the queue before being served Time spent by a customer in the system- waiting time plus service time Number of customers in the system- number of customers in the queue plus those being served Queue length- number of customers waiting in the queue

Jockeying- joining the other queue and leaving the first one Reneging- joining the queue and leaving it afterwards
Balking- customer decides not to join the queue
Queuing system- system consisting arrival of customers, waiting in queue, picked up for service according to a certain discipline, before being serviced and the departure of customers.

Elements of queuing systems

A queue system can be divided into four elements as

Arrival[pic] queue[pic] service[pic] outlet

a) Arrivals. This is the element concerned with how items (people, components, cars etc) arrive in the system. b) Queue. This is concerned with what happens between the arrival of an item requiring service and the time when the service is carried out. This is known as queue discipline, which is generally assumed to be First Come, First Served. (First In First Out). c) Service. This is concerned with the time taken to service a customer. d) Outlet. The exit from the system.

e) Queuing. The whole situation being considered from arrival to exit. The time in the system is generally taken to be the queuing time plus the service time.

Models in queuing theory

The four most important classes of queuing system are:
Queue Service Point
a) Single queue-single service model [pic]. . b) Multiple queues-multiple service model[pic] .
c) Single queue-multiple serves model [pic] . d) Multiple queues-single service model[pic] .

Simple queuing model characteristics

A simple queue is a term, which is applied to queuing problems with the following characteristics: a) Single queue and a single service point.
b) The queue discipline is First Come First Served.
c) The queue has infinite capacity.
d) Arrivals are random and follow a Poisson distribution. e) No simultaneous arrivals in a small time interval.
f) Service times are random and follow a negative exponential distribution. g) Discrete customers from an infinite population of potential customers. h) Single, follow-on service discipline.

i) In order to study a queuing model allow the system to be in operation long enough to settle down...
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