operating time u1 has an exponential distribution with the parameter λ1 + λ2, and the unavailable time u2 is the period of the system M |G|∞ being busy.
Since
then
It is not difficult to show that under the assumptions made, the results of section 1.6 are correct and the traffic rate ρ1 is determined by the expression
[1.23]
where
The necessary and sufficient condition for the stability of the process Q1 is the fulfillment of the inequality ρ1 < 1, and the capacity
If, for example H(t) = mνt, which corresponds to the assumption that each car crosses the pedestrian crossing during an exponentially distributed time with a parameter ν, then
When the real intensity
Their asymptotic analysis, as 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
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
[1.24]
So we have a queueing system with m unreliable servers. All servers break when a pedestrian of the first (second) type appears on lane 1 (the 2mth).This means that the available time
Assuming that
If
When m = 1, we get
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
Putting
After drawing the graphs for
Figure 1.2. Plots for α = 0.5, 1.5, 2
Currently there is no algorithm that estimates the number of cars before the crosswalk in model 2, however we can obtain asymptotic expressions for the average number of expected cars when ρ2 ↑ 1. It turns out that
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
