Discussion
Let \(D_{n}\) be the waiting time of the \(n\)th customer in G|M|1 queue. Consider the following recurrence equation :
\[ D_{n+1} = \begin{cases} D_{n} + V_{n} - U_{n} & \text{if } D_{n} + V_{n} - U_{n} \ge 0 \\ 0 & \text{otherwise} \end{cases}. \]
Which of the following interpretations of the variables in the above equation are valid ?
(i) \(U_{n}\) is the interarrival time between \(n\)th and \((n + 1)\)th customer
(ii) \(D_{n} + V_{n}\) is the departure time of the \(n\)th customer from the system
Answer: B
The keyed answer is both are valid.
This is Lindley's recurrence. If the \(n\)th customer waits \(D_{n}\) and is served for \(V_{n}\), he leaves \(D_{n} + V_{n}\) time units after his own arrival. The next customer arrives \(U_{n}\) after him, the interarrival time between the \(n\)th and \((n + 1)\)th customers, so (i) is valid. He must wait \(D_{n} + V_{n} - U_{n}\) if that is positive, and 0 otherwise.
So \(D_{n} + V_{n}\) is the departure time of the \(n\)th customer measured from his arrival, which is the sense in which (ii) is valid. The options rejecting either interpretation do not fit the recurrence.
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