Distribution Of Arrival And Service Time

 Introduction:

In queuing model, two basic constituents are considered i.e. arrival rate and service rate; these two are the main problems of waiting line. The common basic waiting line models have been developed on the assumption that arrival rate follows the Poisson distribution and that service times follow the negative exponential distribution.

Distribution of Arrivals

•          This situation is commonly referred to as the Poisson arrival and Exponential holding time case.
•          These assumptions are often quite valid in operating situations. Unless it is mentioned that arrival and service follow different distribution, it is understood always that arrival follows Poisson distribution and service time follows negative exponential distribution.
•          Research scholars working on queuing models have conducted careful study about various operating conditions like - arrivals of customers at grocery shops, Arrival pattern of customers at ticket windows, Arrival of breakdown machines to maintenance etc. and confirmed almost all arrival pattern follows nearly Poisson distribution.
•          The commonly used symbol for average arrival rate in waiting line models is the Greek letter Lamda ( λ ), arrivals per time unit. It can be shown that when the arrival rates follow a Poisson processes with mean arrival rate λ, the time between arrivals follow a negative exponential distribution with mean time between arrivals of 1/ λ .
•          This relationship between mean arrival rate and mean time between arrivals does not necessarily hold process, but describes the time between arrivals and specifies that these time intervals are completely random.
•          The distribution of arrivals in a queuing system can be considered as a pure birth process. The term birth refers to the arrival of new calling units in the system the objective is to study the number of customers that enter the system, i.e. only arrivals are counted and no departures takes place.
•         Such process is known as pure birth process.

Exponential Service Times

•         The commonly used symbol for average service rate in waiting line models is the Greek letter .mu. .μ.,the number of services completed per time unit.
•        As with arrivals it can be shown that when service rates follow a Poison process with mean service rate μ, the distribution of serviced times follow the negative exponential distribution with mean service time 1 / μ.
•        The reason for the common reference to rates in the discussion of arrivals and to times in the discussion of service is simply a matter of practice.
•          The Poison and Negative exponential distributions are single parameters distributions; that is, they are completely described by one parameter, the mean.
•          For the Poisson distribution the standard deviation is the square root of the mean, and for the negative exponential distribution the standard deviation is equal to the mean.
•       The result is that the mathematical derivations and resulting formulas are not complex. Where the assumptions do not hold, the mathematical development may be rather complex or we may resort to other techniques for solution, such as simulation.