Queueing Theory 2. Nikolaos Limnios

Queueing Theory 2 - Nikolaos Limnios


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well as some results concerning characteristics of the process Q1 in a stationary regime, when ρ(1) < 1 can be found in the papers (Afanasyeva and Rudenko 2012; Afanasyeva and Mihaylova 2015).

      Now we will consider model 2, in which the rules for crossing the crosswalk by a car are weakened. We assume that the car can move along the jth lane images if there are no pedestrians of the first type (going from A to B) on the lanes 1, 2, . . . , j, and there are no pedestrians of the second type on the lanes j, j + 1, ..., 2m. Denote P0(j) as the probability of this event in a steady-state.

      Since the number of pedestrians of the first type on the lanes (1,2,...,j) is the number of customers in the system M |G|∞ with the intensity λ2 and with an average service time images then

      [1.24]image

image

      Assuming that images and using the results of section 1.6, we can find the traffic rate ρ2 for model 2.

image

      When m = 1, we get

image

      It is easy to show that for all m ≥ 1, the inequality ρ2 (m) < ρ1 holds. Weakening the rules of crossing the crosswalk increases the capacity of the route. To estimate this effect, we consider the ratio

image

      Putting images we have

image

      If the length of the queue of cars is unacceptably high, it is necessary to make organizational decisions. One of these decisions is to install a traffic light. Then, in relation to the cars, we again get a one-channel service system with an unreliable server, but now the server will not work if the red light is on (for cars) and will work if the green light is on. This model has been studied in papers (Afanasyeva and Bulinskaya 2013, 2010), in which the algorithms for estimating the queue length were proposed and the number of the asymptotic results were received. It can happen that, with the available traffic intensities of the cars and the pedestrians, the installation of a traffic light, even at the optimum interrelationship between switching intervals, does not provide an acceptable level of queues of pedestrians and cars. This may be used as the basis for the construction of an underground (or overground) pedestrian crossing, its elimination or transfer to another location.


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