Waiting Line Formulas

μ = service rate = 1/service time

λ = arrival rate

[Be sure that both of these parameters are stated as rates, not times, and that the units are the same in both]

λ/μ are in the left column of the tables

M = the number of channels, or servers, given along the top row of the tables

Po = the probability that no one is already in the system, that is, the new entrant can be served by any of the M servers

Pw = the probability that all of the servers are occupied. Thus, 1 – Pw is the probability that at least one server is available

Lq = the expected number of units waiting in the queue to be served. This does not include those currently being served

Wq = the expected time waiting to be served = Lq/λ

Ws = Wq + 1/μ = expected total time in the system, both in line and being served

Ls = λWs = the expected total number in the system

[As a check, be sure also Ls = Lq + λ/μ; λ/μ is the expected number of servers occupied at any time and, therefore, the expected number of people being served at any time]