M/M/1 Queue Stability and Limiting Distribution

The $M/M/1$ queue model describes a system with one server, where arrivals follow a Poisson process with rate $\lambda$, and service times follow an exponential distribution with rate $\mu$.

• Arrival Rate: $\lambda$ (mean interarrival time = $\frac{1}{\lambda}$)
• Service Rate: $\mu$ (mean service time = $\frac{1}{\mu}$)
• Traffic Intensity: $\rho = \frac{\lambda}{\mu}$

Queue Length Limiting Distribution

A limiting (or steady-state) distribution for the queue length exists only when the system is stable. The stability condition for an $M/M/1$ queue is met when the arrival rate is less than the service rate, i.e., $0 < \lambda < \mu$.

When the stability condition $0 < \lambda < \mu$ holds, the traffic intensity $\rho = \frac{\lambda}{\mu}$ is less than 1 ($\rho < 1$). In this case, the number of customers in the system, $N$, follows a geometric distribution.

The probability mass function (PMF) for the number of customers ($n$) in the system (waiting or being served) is given by:

$ P(N=n) = (1 - \rho) \rho^n = \left(1 - \frac{\lambda}{\mu}\right) \left(\frac{\lambda}{\mu}\right)^n $

This can be rewritten using $p_0 = P(N=0) = 1 - \rho$:

$ P(N=n) = p_0 (1 - p_0)^n $, where $p_0 = \frac{\mu - \lambda}{\mu}$.

This confirms that the limiting distribution is geometric when $0 < \lambda < \mu$.

Analysis of Options

• Options 1 & 2 (Poisson Distribution): The limiting distribution of the queue length in an $M/M/1$ queue is geometric, not Poisson. Poisson distribution typically describes the number of events in a fixed interval, not the steady-state queue size. Therefore, options 1 and 2 are incorrect.
• Option 3 (Geometric Distribution, $\mathbf{0 < \lambda < \mu}$): This matches the condition for stability and the known result that the queue length follows a geometric distribution. This option is correct.
• Option 4 (Geometric Distribution, $\mathbf{0 < \mu < \lambda}$): The condition $0 < \mu < \lambda$ implies $\rho > 1$, meaning the system is unstable. An unstable system does not have a limiting distribution; the queue length tends to grow indefinitely. Therefore, option 4 is incorrect.