For obtaining the equilibrium state distributions of queues with Poisson arrivals and exponentially distributed service times (i.e. M/M/-/- Type Queues), either of the three methods given below may be used. (The two methods are actually equivalent and Methods 2 and 3 are usually much faster to apply than Method 1.)

Method 1 : Obtain the differential-difference equations describing the system state. Solve these under equilibrium conditions along with the normalization condition.

Method 2 : Directly write the flow balance equations for proper choice of closed boundaries as illustrated and solve these along with the normalization condition.

Method 3 : Identify the parameters of the birth-death Markov chain for the queue and directly use equations (2.2) and (2.3) of Module 2

Method 3 has been used in the examples analyzed in this lecture and the next one of this module.

However, we strongly recommend that the same analysis should be tried using Method 2 as that gives a better feel of how the system actually operates and can indeed be more useful for general analysis!

M/M/1/∞ Queue:

The M/M/1 (or M/M/1/∞ ) queue is a single server queue with an infinite number of waiting positions where the arrival process is Poisson and the service times are exponentially distributed.

As in Module 2, we assume that the state of the queue is represented by the number in the system, i.e. jobs waiting and possibly one in service.

We assume that the arrivals come to the queue at the average arrival rate of λ per second and that the mean service time provided by the server is 1/μ seconds whenever the server is working (i.e. for all states except state 0) .