Suppose that you arrive at a single-teller bank to find five...

Suppose that you arrive at a single-teller bank to find five other customers in the bank, one being served and the other four waiting in line. You join the end of the line. If the service times are all exponential with rate $\mu$, what is the expected amount of time you will spend in the bank?

Key Concepts

Gamma (Erlang) Distribution

When summing independent exponential random variables, the resulting distribution is a Gamma distribution, and more specifically an Erlang distribution when the number of summed variables is an integer. In queuing theory, this property is used to determine the waiting (or service) time for several successive events, as the total time is the sum of individual service times.

Queuing Theory - Single-Server Queue

In queuing theory, a single-server queue such as an M/M/1 system models scenarios where customers arrive and are served one at a time. The arrival process typically follows a Poisson distribution, and the service times are exponential. This framework is utilized to estimate performance measures like expected wait times, total time in the system, and the probability of queue buildup.

Memoryless Property

A notable characteristic of the exponential distribution is its memoryless property, meaning that the probability of an event (such as service completion) occurring in the next interval is independent of how much time has already elapsed. This makes analytical treatment simpler in queuing and reliability problems, as the system does not 'accumulate' history.

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between events in a Poisson process. In the context of service times, it implies that each customer's service duration is random, with the probability of service completion in the next instant being constant in time, and it is uniquely characterized by a single parameter, the rate ?.

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