Question

In an \(\mathrm{M} / \mathrm{G} / 1\) queue, (a) what proportion of departures leave behind 0 work? (b) what is the average work in the system as seen by a departure?

Short answer

In an M/G/1 queue, (a) the proportion of departures leaving behind 0 work is \(P0 = 1 - \frac{1}{\mu} \cdot \frac{\lambda}{\mu}\), and (b) the average work in the system as seen by a departure is \(E(W) = \frac{\rho^2}{1-\rho}\cdot \frac{1}{\mu}\), where \(\rho = \frac{\lambda}{\mu}\), \(\lambda\) is the mean arrival rate, and \(\mu\) is the mean service rate.

Step-by-step solution

Part (a): Proportion of Departures Leaving Behind 0 Work

In an M/G/1 queue, the proportion of departures leaving behind 0 work is equal to the proportion of the system being empty when a general arrival occurs. Let P0 be the probability that the system is empty when a general arrival occurs. We can write this probability as: \[P0 = 1 - E(S)\rho\] where \(E(S)\) is the expected service time and \(\rho\) is the utilization factor. We are given that this is an M/G/1 queue, so we can use the fact that \(E(S)=1/\mu\), where \(\mu\) is the mean service rate. From the definition of the utilization factor, we have: \[\rho = \frac{\lambda}{\mu}\] where \(\lambda\) is the mean arrival rate. Now, we can compute \(P0\): \[P0 = 1 - \frac{1}{\mu} \cdot \frac{\lambda}{\mu}\] Once we have the values of \(\lambda\) and \(\mu\), we can plug them into the above formula to find the proportion of departures leaving behind 0 work.

Part (b): Average Work in the System as Seen by a Departure

The average work in the system as seen by a departure in an M/G/1 queue can be found using Little's Law and the fact that the average work seen by a general arrival is equal to the average work seen by a departure: \[E(W) = E(Q) \cdot E(S)\] where \(E(W)\) is the average work in the system as seen by a departure, \(E(Q)\) is the average number of customers in the queue, and \(E(S)\) is the expected service time. We know that \(E(S) = 1/\mu\). To find the average number of customers in the queue, we can use Little's Law: \[E(Q) = \frac{\rho^2}{1-\rho}\] Now, we can find the average work in the system: \[E(W) = \frac{\rho^2}{1-\rho}\cdot \frac{1}{\mu}\] Once we have the values of \(\lambda\) and \(\mu\), we can plug them into the above formula to find the average work in the system as seen by a departure.