A Basic Queueing System. Herr Cutter’s Barber Shop. Herr Cutter is a German barber who runs a one-man barber shop. Herr Cutter opens his shop at 8:00 A.M. The table shows his queueing system in action over a typical morning. Arrivals.
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l = Mean arrival rate for customers coming to the queueing system
where l is the Greek letter lambda.
1 / l = Mean interarrival time
To identify which probability distribution is being assumed for service times (and for interarrival times), a queueing model conventionally is labeled as follows: Distribution of service times — / — / — Number of Servers Distribution of interarrival times
The symbols used for the possible distributions areM = Exponential distribution (Markovian)D = Degenerate distribution (constant times)Ek = Erlang distribution (shape parameter = k)GI = General independent interarrival-time distribution (any distribution)G = General service-time distribution (any arbitrary distribution)
L = Expected number of customers in the system, including those being served (the symbol L comes from Line Length).
Lq = Expected number of customers in the queue, which excludes customers being served.
W = Expected waiting time in the system (including service time) for an individual customer (the symbol W comes from Waiting time).
Wq = Expected waiting time in the queue (excludes service time) for an individual customer.
These definitions assume that the queueing system is in a steady-state condition.
L = r / (1 –r) = l / (m– l)
W = (1 / l)L = 1 / (m – l)
Wq= W – 1/m = l / [m(m – l)]
Lq= lWq = l2 / [m (m – l)]
Pn = (1 – r)rnThus,P0 = 1 – rP1 = (1 – r)rP2 = (1 – r)r2 : :
P(W > t) = e–m (1–r)tfor t ≥ 0
P(Wq > t) = r e–m (1–r)tfor t ≥ 0
E(T) = 1/m , Var(T) = s2.
3. The queueing system has one server.
P0 = 1 – r
Lq= l2[Var(T)+ E(T)2] / [2(1 – lE(T))]
L = Lq + l/m
The expected waiting time in the queue is
Wq= Lq/ l
W = Wq+ 1/m
Lq= l2[Var(T)+ E(T)2] / [2(1 – lE(T))]=[l2s2 + r2] / [2(1 – r)]
Decision: Adopt the third proposal