# Arrival Rate Distribution

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## Definition: Arrival Rate Distribution

Arrival Rate Distribution can be defined as the distribution of mean number of calling units arriving at a service center per unit time where service center maybe ticket booking counter, airport, post office etc. with number of ticket bookers, passengers, mails etc. respectively as calling units.

Generally there are two important distributions for arrival rate:

1. Poisson Distribution
2. Exponential Distribution

PoissonDistribution

Sequence of random events like arrival of passengers in airport, can be explained by counting function N(t), where t is the number of events that occur in time interval [0,t]. This process comes under poisson distribution only if

• Arrivals occur one at a time
• N(t) has stationary increments     (Distribution of arrivals does not depends on ‘t’)
• N(t) has independent increments (Number of arrivals are random variables)

Probability for N(t)=n is P{N(t)=n} = (e-λt)* (λt)*n/n! where α = λt

Exponential Distribution

Arrival rate distribution is said to be exponential when

• The average arrival rate is ‘λ’
• Arrivals occur independently

Exponential Distribution is linked with Poisson distribution. When arrival rate is governed by time interval of ‘λ’ , then for fixed time T, arrivals are governed by poisson distribution with mean value ‘λT’.

f(x) for exponential distribution is given by

f(x; λ) = λe-(λx) if x > 0 or x= 0

0 if x < 0

Example

Consider a telephone exchange where calls arrive exponentially with λ =5.Find probability thattime between one call and next call is greater than 1 minute but less than 2 minutes.

P(x1<X<x2) = e- λ X1 - e- λ X2

= e- 5 - e- 10

= 6.6925 X 10-3 = 0.0067 (approximately)

Hence, this concludes the definition of Arrival Rate Distribution along with its overview.

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