M/G/1 Queue or not?

Question

This is a question from an old exam and there are no solutions provided.

Customers arrive at this queue according a PP($λ$) and form a single line. At time $nd (n = 0, 1, 2, 3, · · ·)$, the customer at the head of the line (if the line is not empty) is removed from the queue. Let $X(t)$ be the number of customers in the system at time $t$ and $X_n = X(nd^{-})$, the number of customers just before time $nd, n \geq 0$.

Is $\{X(t), t\geq 0\}$ a queue length process in an M/G/1 queue?

In general, I was wondering what one needs to take into consideration when deciding whether a queue is an M/G/1 or G/M/1 queue.

Answer 1

This is an $M/G/1$ queue.

• The M refers to a Markovian arrival process. The arrival process is indeed Markovian because arrivals come to the queue according to a Poisson process.
• The G refers to a service time following a general distribution. Here, we can think of the service time as deterministic, in that each service time always takes exactly $d$ time units. This is because at times $nd$ for $n = 1, 2, \dots$, the customer at the head of the line (if the line is not empty) is removed.
• Additionally, service times are independent of each other and identically distributed, and service times and arrival times are also independent.
• Finally, the 1 refers to the fact that there is 1 server, with service rate $1/d$ customers served per time unit.