I am deeply puzzled by a question about the behaviour of an M/G/1 queue – i.e., a queue with a **M**arkovian distribution of arrival times, a **G**eneral distribution of service times and **1** server. I have asked about this on the Math Stackexchange (and there’s now a bounty on the question if you’d like to answer it there – but as I am getting nowhere with it, I thought I’d ask it here too.

(This is related to getting a more rigorous presentation on thrashing into my PhD thesis.)

Considering an M/G/1 queue with Poisson arrivals of rate λ – this comes from Cox and Miller’s (1965) “The Theory of Stochastic Processes” (pp 240 – 241) and also Cox and Isham’s 1986 paper “The Virtual Waiting-Time and Related Processes“.

My question is what is the difference between (using the authors’ notation) and ? The context is explained below…

In the 1965 book (the 1986 paper presents the differentials of the same equations), is the “virtual waiting time” of a process and the book writes of “a discrete probability that , i.e., that the system is empty, and a density for “.

The system consumes virtual waiting time in unit time, i.e., if and there are no arrivals in time then .

The distribution function of is then given by:

They then state:

I get all this – the first term on the RHS is a run-down of with no arrivals, the second is adding of service time when the system is empty at and the third, convolution-like, term is adding of service time from an arrival when it’s not empty at . (The fourth accounts for their being more than one arrival in but it tends to zero much faster than so drops out as approaches the limit.)

And … and this is where I have the problems …

The first term of the RHS seems clear – the probability that the system is empty at multiplied by the probability there will be no arrivals in , but the second is not clear to me at all.

I assume this term accounts for the probability of the system “emptying” during but I don’t see how that works, is anyone able to explain?

In other words, how does represent this draining? Presumably again represents the possibility of zero arrivals in , so how does represent the situation?

If we take the equilibrium situation where and then, if we differentiate and as , we get – so, again, what does represent?

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