In a queueing system, if the service time and inter-arrival time are both expoentially distributed, denoted by and respectively plus

- the system is in steady state
- the population is infinite

Let represent the probability the system has *i* customers
(including the ones in queue and ones in server).

Using the principle that the flow into a state is balanced by the flow out of the state, we have

= | ||

... |

solve this system of equations we get

Using the relation

and let

we have

thus

where

so

The meaning of the probability with no customer in the system is the
same as the server is idle. So the measure can be considered as
the probability that the server is busy, which is the utilization of
the server. With solved, all other items can be solved.

Use s, we can obtain all other measures of interest.

**Average number of customers in system :**-

**waiting time in system:**- use Little's Law

**waiting time in the queue:**- this can be calculated
using the fact that waiting time in the queue is the total waiting
time less the time spent in the server. The time spent in the server
is the average service time (note that is the
service rate here) so

**Average queue length:**- use the Little's Law again