Queuing Theory

Queuing theory was stated by A.K. Erlang. He started, in the year 1905, to explore the effect of fluctuating service demand on the utilization of automatic dial equipment.

A queue or a waiting line is something very common in everyday routine. There is a queue for bus, queue at ration shop, for cinema ticket and where not. Standing n queue wastes a lot of otherwise useful time. Besides everyday experience, queues can be seen on the shop floor, where in-process goods wait for next operation or inspection or wait for getting moved to another place. Such delays in the production lines naturally, inceases production cycle duration, add to the product cost, may upset the whole system and it may not be possible to meet the specified delivery dates thereby annoying the customers.

Waiting lines cannot be completely eliminated, they can attest be reduced by optimizing the number of service stations, or by adjusting the service times in one or more service station.

Queuing theory analyses the feasibility of adding facilities (equipment, manpower) and assess the amount and cost of waiting times. In general, this theory can be applied wherever combustion occurs  and waiting line or a queue is formed. The purpose in such situations is to determine the optimum amount of facilities. Queuing theory helps finding lack of balance between items coming and going. It gives information about peak loads. It mathematically relates the length of queue and wating time and the controllable factors, such as number of service station or the time taken to process each article. It can also tell, how often and how long the times or persons in queue will probably have to wait.

Queuing theory mathematically predicts the way in which the waiting line will develop over given periods and in turn helps allocating facilities to deal the situation efficiently. In order to build a mathematical model for a queue forming situation i.e., where items or personas arrive in a random manner, are processed or treated and then leave, following information is required:
(a)    Distribution of gaps between the arrivals of times. Say a telephone call comes after every ten minutes.
(b)    Distribution of service times. Say every time a telephone talk continues for three minutes
(c)    The priority system.
(d)    The processing facilities.

Assumptions
(a)There is only one type of queue discipline-that is first come, first served. A unit in queue will immediately go to the service station as soon as the station is free.

(B)There are steady state (stable) conditions i.e., the probability that n items are in queue at any time, remains the same with the passage of item. Another words at all times the number of items in the queue remain same, the queue does not length indefinitely with time.

(c) Both, the number of items in queue at any instant and the waiting time experience by a particular item are random variables. They are not functionally dependent on time. The problem thus reduces to estimate the average length of the queue at any instant, etc.

(d) Arrival rates follow Poisson distribution and service times follow exponential distribution. This situation referred as passion Arrivals and Exponential service times, holds goods in number of actual operating situations.

(e)Mean service rate n is grater than mean arrival rate m.

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