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2It holds thatx0=1 whenx6=0.
13412.3. Congestion: M/M/1 versus D/D/15.The expected total sojourn time (queueing time+time in service),W, and the ex-pected waiting time, Wq, can be found from application of Little’s formula asW=Lλ=λ/μ1-λ/μλ=1/μ1-λ/μ=1μ-λandWq=Lqλ=(λ/μ)21-λ/μλ=λ/μ21-λ/μ=λμ(μ-λ).Note that W=Wq+1μ, see the expression (9.1).12.3. Congestion: M/M/1 versus D/D/1In an M/M/1 model the expected number of customers in the system equalsL=ρ/(1-ρ),whereρ=λ/μis the utilization of the server. If the value ofρtends to the value of onethenLis getting larger than whatever value considered. Forρ≥1 the expected number ofcustomers in the system (in the long run) is even infinite. The behavior ofLas a functionofρhas been sketched in Figure12.9. The situation for an M/M/1 model, or any modelinfinity0.10.20.30.126.96.36.199.80.91Lρ=λμFigure 12.9.: Congestion in an M/M/1 model.with variation in inter-arrival times and service times, contrasts with a situation in whichinter-arrival and service times are fixed. A model with fixed lengths of inter-arrival andservice times, and a single server, is denoted as an D/D/1 model. If the server has amplecapacity (‘speed’) then there is no waiting line in a D/D/1 model. For example, under fixedinter-arrival intervals of length 15 minutes and a fixed service-time length of 10 minutes,each customer has left before the next one arrives. Two thirds of the time the system holdsone customer, and this consequently is the average number of customers in the system.Two thirds is evidently also the fraction of time that the server is busy. SoL=ρand thislatter equality does not depend on the values chosen for the inter-arrival and service times.The behavior ofLas a function ofρimplied byL=ρis totally different from the behavior
135infinity0.10.20.30.188.8.131.52.80.91Lρ=λμFigure 12.10.: (Absence of) Congestion in a D/D/1 model.ofLas a function ofρin the M/M/1 model. The functionL=ρhas been sketched inFigure12.10.For values ofρgreater than one the values forLare infinite in both the M/M/1 and theD/D/1 model. If the utilization of the server is slightly less than one then the numberof customers in the system is very large in the M/M/1 model whereas the number ofcustomers in the system is still very small (less than one) in the D/D/1 model. In thisrespect the predictions of the two models differ greatly. An M/M/1 predicts congestionlong before the server becomes fully utilized.12.4. M/M/2The M/M/2 models is a system with two identical parallel servers that have exponentiallydistributed service times, a single waiting room with infinite capacity (while customers arenot balking), and that accommodates a single queue from which customers are served.The arrival process is Poisson and the service times are exponentially distributed. A sketchof the system is in Figure12.11. It can serve as a model for a post office with two identicaltellers or a hairdresser’s shop with two identical hairdressers.arrivalsuncapacitated waiting roomserversFigure 12.11.: Sketch of a queueing system with two parallel servers.