For an M/M/1 queue with mean interarrival time 1.25 minutes and mean service time 1 minute, find all five \(W_q, W, L_q, L\), and \(p\). for each interpret in words. Be sure to state all your units, and the relevant time frame of operation.

• \(\mu = 1~minute=\frac{1}{E(S)}\)
• \(\lambda = \frac{1}{1.25}~minutes~=~0.8~minutes\).

As per our text book, “the steady-state average number in system is \(L=\frac{p}{(1-p)}\) and \(p\) is the mean arrival rate \(\lambda\) divded by the mean service rate \(\mu\) for a single server”. Therefore, \(p = \frac{\lambda}{\mu}\) or \(p = \frac{.8}{1}\) then substitute that to solve for \(L\), \(L = \frac{.8}{(1-0.8)} = 4\)

• \(p = .8\)

• \(L = 4\)

Little’s Law

We now use Little’s Law to solve for the remaining values.

• \(L = \lambda W\); \(4 = .8 W\); \(W = 5\)

• \(W = W_q + E(S)\); \(5 = W_q + 1\); \(W_q = 4\)

• \(L_q = \lambda W_q\); \(L_q = 0.8 * 4\); \(L_q = 3.2\)

Simio Simulation

Results