Читать книгу Queueing Theory 2 - Nikolaos Limnios - Страница 21

The study of traffic flows has a long history (Gideon and Pyke 1999; Grinbeerg 1959; Greenshields 1935; Inose and Hamada 1975; and references therein). Various methods such as cellular automate (Maerivoet and de Moor 2005), statistical mechanics and mathematical physics (Blank 2003; Chowdhury 1999; Fuks and Boccara 2001; Helbing 2001; Schadschneider 2000) or queueing theory (Afanasyeva and Bulinskaya 2009, 2010, 2011, 2013; Afanasyeva and Mihaylova 2015; Afanasyeva and Rudenko 2012; Baycal-Gursoy and Xiao 2004; Baycal-Gursoy et al. 2009; Caceres and Ferrari 2007) were used.

The purpose of the proposed study is an estimation of the carrying capacity of the automobile road, intersected by a crosswalk. Under the capacity, we mean the upper bound of the intensity of the flow of cars, when the queue of cars does not tend to infinity. This means that the stability condition for the process determining the number of these cars is satisfied, so our analysis will be based on the results obtained in section 1.6.

Let us move to the description of the models.

We consider an automobile road with two directions of traffic and m traffic lanes in each. The flow of cars Xi in the ith direction is a regenerative flow with intensity The road is intersected by a two-directional pedestrian crossing (Figure 1.1). We denote that pedestrians following from A to B have the first type, and from B to A have the second type. The flow of pedestrians of the ith type is a Poisson flow with intensity λi (i = 1, 2). Pedestrians cross the road independently of each other with a random (but constant during the entire time of being at the crosswalk) speed.

First, we assume that there is no traffic light at the crossing and pedestrians have an absolute priority over cars. In this case the number of pedestrians at the crosswalk is the number of customers in the infinite-channel service system of M |G|∞ type.

Figure 1.1. A road intersected by a pedestrian crossing

Let us assume that 2b is an average time of crossing the road by a pedestrian. Then the probability P₀ that there is no pedestrian at the crosswalk in a stationary regime is defined by the expression

[1.22]

Saaty (1961).

First assume that a car can cross a pedestrian crossing only if there are no pedestrians on it. Let us assume that the cars in the lanes 1, 2, ..., m are going in one direction and the cars in the lanes m + 1,..., 2m – in another. We consider the process Q₁(t) – the number of cars in the lanes 1,2,...,m at time t (the consideration of lanes m + 1,m + 2,..., 2m is analogous).

Denote Hj(t) the mathematical expectation of the number of cars that pass through the crosswalk at the lane j during time t under the condition that there are always cars at this lane and the crosswalk is free. Also denote In relation to the process Q₁(t), we have a single-channel service system with an unreliable server. The 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 is defined as

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 is less than, but close to large queues accumulate before the crosswalk.

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 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 then

[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 of the jth server is exponentially distributed with the parameter Let be a block time of the jth server and Then

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

[1.25]

If then [1.25] can be written as

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 we have

After drawing the graphs for for α = 0.5; 1.5; 2 (see Figure 1.2), we can see that the effect of the weakened rule (model 2) in comparison with the standard rule (model 1) is stronger, as the number of the lanes increases and the intensity of the number of pedestrians increases.

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 , where c is a constant.

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.