Question

Suppose 3 cashiers are working in a store, independently. Time to serve a customer by cashier i is Ti ∼ exp(λi ). What is the probability that none of the cashiers complete their current service (assuming that all are busy) in the next 2 minutes?

Answer 1

There are 3 cashiers in the store, working independently.

Here Time to serve a customer by cashier i is Ti ∼ exp(λi )

so, cashier 1 will not be able to complete his current service in next 2 minutes = 1 - [1 - e^-t] = e^-2

similarly,

cashier 2 will not be able to complete his current service in next 2 minutes = 1 - [1 - e^{- * 2} = e^-2

cashier 3 will not be able to complete his current service in next 2 minutes = 1 - [1 - e^{- * 2} = e^-2

so

Pr(None of the cashiers complete their current service in the next 2 minutes) = e^{-2( + + )}