2 It holds that x 1 when x 6 0 134 123 Congestion M M 1 versus D D 1 5 The

2 it holds that x 1 when x 6 0 134 123 congestion m m

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2 It holds that x 0 = 1 when x 6 = 0.
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134 12.3. Congestion: M / M / 1 versus D / D / 1 5. The expected total sojourn time (queueing time + time in service), W , and the ex- pected waiting time, W q , can be found from application of Little’s formula as W = L λ = λ / μ 1 - λ / μ λ = 1 / μ 1 - λ / μ = 1 μ - λ and W q = L q λ = ( λ / μ ) 2 1 - λ / μ λ = λ / μ 2 1 - λ / μ = λ μ ( μ - λ ) . Note that W = W q + 1 μ , see the expression ( 9.1 ). 12.3. Congestion: M/M/1 versus D/D/1 In an M / M / 1 model the expected number of customers in the system equals L = ρ / ( 1 - ρ ) , where ρ = λ / μ is the utilization of the server. If the value of ρ tends to the value of one then L is getting larger than whatever value considered. For ρ 1 the expected number of customers in the system (in the long run) is even infinite. The behavior of L as a function of ρ has been sketched in Figure 12.9 . The situation for an M / M / 1 model, or any model infinity 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 L ρ = λ μ Figure 12.9.: Congestion in an M / M / 1 model. with variation in inter-arrival times and service times, contrasts with a situation in which inter-arrival and service times are fixed. A model with fixed lengths of inter-arrival and service times, and a single server, is denoted as an D / D / 1 model. If the server has ample capacity (‘speed’) then there is no waiting line in a D / D / 1 model. For example, under fixed inter-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 holds one 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. So L = ρ and this latter equality does not depend on the values chosen for the inter-arrival and service times. The behavior of L as a function of ρ implied by L = ρ is totally different from the behavior
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135 infinity 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 L ρ = λ μ Figure 12.10.: (Absence of) Congestion in a D / D / 1 model. of L as a function of ρ in the M / M / 1 model. The function L = ρ has been sketched in Figure 12.10 . For values of ρ greater than one the values for L are infinite in both the M / M / 1 and the D / D / 1 model. If the utilization of the server is slightly less than one then the number of customers in the system is very large in the M / M / 1 model whereas the number of customers in the system is still very small (less than one) in the D / D / 1 model. In this respect the predictions of the two models differ greatly. An M / M / 1 predicts congestion long before the server becomes fully utilized. 12.4. M/M/2 The M / M / 2 models is a system with two identical parallel servers that have exponentially distributed service times, a single waiting room with infinite capacity (while customers are not 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 sketch of the system is in Figure 12.11 . It can serve as a model for a post office with two identical tellers or a hairdresser’s shop with two identical hairdressers. arrivals uncapacitated waiting room servers Figure 12.11.: Sketch of a queueing system with two parallel servers.
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