2.6.1) For an \(M/M/1\) queue with mean interarrival time 1.25 minutes and mean service time 1 minute, find all five of \(W_q\), \(W\), \(L_q\), \(L\), and \(\rho\). For each, interpret in words. Be sure to state all your units (always!), and the relevant time frame of operation.
\(\lambda = 1\ entity / 1.25\ minutes = 0.8\ entities / minute\)
\(\mu = 1/E(S) = 1\ entity / minute\)
\(\rho = \lambda/\mu = 0.8\ entities / minute / 1\ entity / minute = 0.8\)
\(L = \rho/(1 - \rho) = 0.8/(1 - 0.8) = 4\ entities\)
\(L = \lambda W\)
\(W = L / \lambda = 4\ entities / 0.8\ entities / minute = 5\ minutes\)
\(W = W_q + E(S)\)
\(5\ minutes = W_q + 1\ minute\)
\(W_q = 4\ minutes\)
\(L_q = \lambda W_q = (0.8\ entities / minute) \times 4\ minutes = 3.2\ entities\)
\(W_q\), the steady-state or long-run average time in queue is 4 minutes. \(W\), the steady-state average time in system, or the sum of the expected time in queue and the expected service time, is 5 minutes. \(L_q\), the steady-state average number of entities in queue, or expected number of entities waiting to be served, is 3.2. The steady-state average number of entities in the system, \(L\), is 5. \(\rho\), the steady-state utilization of the single server in the system, or the average rate of utilization of the server over an infinte time horizon, is 0.8.