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In this paper, we consider the number of both arrivals and departures seen bya tagged customer while in service in a classical $M/M/1$ processor sharingqueue. By exploiting the underlying orthogonal structure of this queuing systemrevealed in an earlier study, we compute the distributions of these twoquantities and prove that they are equal in distribution. We moreover derivethe asymptotic behavior of this common distribution. The knowledge of thenumber of departures seen by a tagged customer allows us to test the validityof an approximation, which consists of assuming that the tagged customer israndomly served among those customers in the residual busy period of the queuefollowing the arrival of the tagged customer. A numerical evidence shows thatthis approximation is reasonable for moderate values of the number ofdepartures, given that the asymptotic behaviors of the distributions are verydifferent even if the exponential decay rates are equal.

On the sojourn of an arbitrary customer in an $M/M/1$ Processor Sharing Queue