M/D/1 Queuing The assumption of exponentially distributed times between the arrivals of successive vehicles (Poisson arrivals) will, in some cases, give a more realistic representation of traffic flow than the assumption of uniformly distributed arrival times. Therefore, the M/D/1 queue (exponentially distributed arrivals, deterministic departures, and one departure channel) has some important applications within the traffic analysis field. Although a graphical solution to an M/D/1 queue is difficult, a mathematical solution is straightforward. Defining a new term (traffic intensity) for the ratio of average arrival to departure rates as
ρ = traffic intensity, unitless,
λ = average arrival rate in vehicles per unit time, and
μ = average departure rate in vehicles per unit time,
and assuming that ρ is less than 1, it can be shown that for an M/D/1 queue the following queuing performance equations apply:
( ) 22 1 Qρ ρ
= −(5.28)( )2 1 w
ρ μ ρ= −
(5.29)( ) 2
2 1 tρ μ ρ−= −(5.30)
Q = average length of queue in vehicles,
w = average waiting time in the queue, in unit time per vehicle, t = average time spent in the system (the summation of average waiting time in the
queue and average departure time), in unit time per vehicle, and
other terms are as defined previously. It is important to note that under the assumption that the traffic intensity is
less than 1 (λ < μ), the D/D/1 queue will predict no queue formation. However, a queuing model that is derived based on random arrivals or departures, such as the M/D/1 queuing model, will predict queue formations under such conditions. Also, note that the M/D/1 queuing model presented here is based on steady-state conditions (constant average arrival and departure rates), with randomness arising from the assumed probability distribution of arrivals. This contrasts with
176 Chapter 5 Fundamentals of Traffic Flow and Queuing Theory
the time-varying deterministic queuing case, presented in Example 5.9, in which arrival and departure rates changed over time but randomness was not present.
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