Posted in Operations and Supply Chain Terms, Total Reads: 4844

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**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.

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Generally there are two important distributions for arrival rate:

- Poisson Distribution
- 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(x_{1}<X<x_{2}) = e^{-} λ^{ X}_{1} - e^{-} λ^{ X}_{2}

= e^{-} 5 - e^{-} 10

= 6.6925 X 10^{-3} = 0.0067 (approximately)

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IMPORTANT DEFINITIONS

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