Excluded Volume Effect in Queueing Theory

We have introduced excluded volume effect, which is an important factor to model a realistic pedestrian queue, into queueing theory. The probability distributions of pedestrian number and pedestrian waiting time in a queue have been calculated exactly. Due to time needed to close up the queue, the mean number of pedestrians increases as pedestrian arrival probability ($λ$) and leaving probability ($μ$) increase even if the ratio between them (i.e., $ρ=λ/μ$) remains constant. Furthermore, at a given $ρ$, the mean waiting time does not increase monotonically with the service time (which is inverse to $μ$), a minimum could be reached instead.

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