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 M/M/N Queuing

 M/M/N Queuing A more general formulation of the M/M/1 queue is the M/M/N queue, where N is the total number of departure channels. M/M/N queuing is a reasonable assumption at toll booths on turnpikes or at toll bridges, where there is often more than one departure channel available (more than one toll booth open). A parking lot is another example, with N being the number of parking stalls in the lot and the departure rate, μ, being the exponentially distributed times of parking duration. M/M/N queuing is also frequently encountered in non- transportation applications such as checkout lines at retail stores, security checks at airports, and so on.

The following equations describe the operational characteristics of M/M/N queuing. Note that unlike the equations for M/D/1 and M/M/1, which require that the traffic intensity, ρ, be less than 1, the following equations allow ρ to be greater than 1 but apply only when ρ/N (which is called the utilization factor) is less than 1.

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( ) ρ ρρ

−== +

− 0 101

! ! 1cc

N n N


n N N


0 for !

nnP P n N

n ρ

= ≤ (5.35)

5.5 Queuing Theory and Traffic Flow Analysis 179

ρ−= ≥0 for !

nn n N

P P n N

N N (5.36)

( ) 10! 1


N N N ρ


+> = −(5.37)


P0 = probability of having no vehicles in the system, Pn = probability of having n vehicles in the system,

Pn>N = probability of waiting in a queue (the probability that the number of vehicles in the system is greater than the number of departure channels),

n = number of vehicles in the system, N = number of departure channels, nc = departure channel number, and

ρ = traffic intensity (λ/μ).

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