An Application of Queuing Theory to Rotary Intersection Traffic Model and Control Using Safe Petri-net for Deadlock Avoidance

International Journal of Manufacturing Science and Technology 5(2) December 2011; pp. 101-109 Serials Publications An Application of Queuing Theory to Rotary Intersection Traffic Model and Control Using Safe Petri-net for Deadlock Avoidance Makoto KATOH * Abstract: This paper focuses on the effect of Poisson arrival at a low-capacity entrance in a rotary intersection model. The model is constructed using Mark Flow Graph which is a kind of safe Petri-net. The purpose is deadlock avoidance. The arrival interval of car was found to change the deadlock time. This paper propose an Entrance passing and Exit passing control system using a waiting agent counter at each entrance. Keywords: Petri-net, Multi-agent, Poisson Arrival, Traffic and Logistic, Deadlock, Percolation. 1. INTRODUCTION There have been many studies on traffic models such as the percolation model [1] [2], cellular automata model [3], neural net work model [4], and Petri-net model [5] [6] [7]. In particular, there have been interesting attempts to obtain new information by comparing the cellular automata and multi-agent models, or to use a hybrid Petri-net model which allows continuous values for speed of agents, and discrete values for their existence. Recently, there has been a fusion of traffic studies and logistic ones in Japan. The present authors previously developed a simple, hybrid simulator using CAD (Simulink, Mathworks Co. Ltd.), and applied it to the analysis of an abnormal diagnosis warning display system and the causes of failure diagnosis and obtained excellent results. The purpose of the present study was to enhance this simple colored logic-mfg (Mark Flow Graph [8]: Safe Petri-Net) hybrid simulator by combining multi-valued logic specifying an agent s destination type with a conventional binary logic approach to the existence of an agent. A multi-agent flow model that was allowed to join and bifurcate was then applied to a rotary intersection model. It is well known that an economic loss and an environmental destruction due to congestion is very large in all of the world and the avoidance of congestion will be useful for both problems [8]. Though he presents the following various causes of congestion of reality, the modeling of them is very difficult and the useful results will not be obtained even if it could be done. (1) Obstacles caused by road works, accidents, break down cars or illegal parker etc.. (2) Natural congestion caused by speed down before slopes, tunnels, toll gates or railroad crossings etc.. (3) Forced congestion caused by suspension of traffic for a fixed time Then, we have simulated on the congestion rate characteristics based on cellular automata like ASEP and a percolation type complex system in a scientific manner. [1] * Department of Mechanical Engineering, Osaka Institute of Technology, 5-16-1, Ohmiya, Asahi-ku, Osaka, Osaka-fu 535-8585, Japan

102 / International Journal of Manufacturing Science and Technology Moreover, K. Nishinari has been studied precisely on congestion [9]. He presents some causes of congestion as follows: (1) Queuing Theory: When the average service rate of the toll gate becomes below the average arrival rate of the car, the queue is generated. The queue length is the average arrival rate of cars * average waiting time (2) Asymmetric Simple Execution Process (ASEP) Rule: All agents can advance if the cell forward is empty. (There is the same rule in MFG) (3) Matrix Products Ansatz (MPA) method in ASEP A strictly solution of ASEP. (4) ASEP with pheromone of ants As for the action that ants move in the state of the group generated naturally, similar happens to cars. (5) Stochastic Optimal Velocity Model (SOV) Assumption: Speed of a car in Optimal Velocity Model (OV) is advanced probability in ASEP. A study on the kinds, properties and judgment methods of deadlocks (stopping of mark flow) in a discrete production system described by MFG has been discussed [11] [12]. Deadlocks are caused by circular waiting due to an imbalance between the input and output. Two approaches for deadlock avoidance were proposed, which are the Input suppression method and States suppression method, and some compensation methods have also been presented [11]. The present paper focuses mainly on the effects of Poisson arrival at the entrance of the rotary intersection model proposed in the literature [1] on the occurrence of deadlock, which we define a criteria for judgment of it by becoming percolation rate as molecule of water in the rotary section to a constant. We assume the reason that cars move in the state of the group generated naturally as percolation of molecular of water not but pheromone of ants. The percolation system [13] is well known as a kind of complex systems. Then, we can introduce the critical probability which is the phenomena of the percolation system as one in general complex systems. In the next section, we will describe the specifications of the model used. 2.1. Basic Specifications 2. ROTARY INTERSECTION MODEL (1) In rotary boxes, the initial existence probability of a multi-agent can be set by initial marking (initially set of marks in rotary boxes). (2) In source boxes, the entry probability of multi-agents, can be set (entry probability of marks in source boxes). The color distribution of multi-agents in the entrance is uniform. The arrival distribution of multi-agents in the entrance hall is Poisson, and we refer to this situation as Poisson arrival [14]. The number of multi-agents in the entrance hall is computed from the arrival and departure rates.

An Application of Queuing Theory to Rotary Intersection Traffic Model and / 103 (3) In sink boxes, the exit probability of multi-agents can be set. The color distribution of multi-agents in the exit is uniform. (4) For all transitions, 3 input terminals are available, i.e., the mark input terminal, the permission/inhibition terminal and the activate/ inactivate terminal, which are connected to a common pulse signal with a fixed cycle time and a fixed split rate. This terminal is an extension of the original MFG [10] [11] [12]. Only a single output terminal exists. 2.2. Configuration and Assumptions A low-capacity (Transition = Box = 12) rotary intersection multi-agent flow model using a colored MFG format is configured as shown in Figure 1. Logic sets for functions such as color decoding are omitted to avoid confusion. A high-capacity (Transition = Box = 16) model was not used here because deadlock hardly occurred with such a model. The assumptions of this model are as follows. (1) There is only one lane and overtaking is prohibited. (2) Assuming a mark in box describe an agent, that is, a car. The color number (of colored mark in the colored Petri-net) of each agent mark is decoded in logic sets. The number ijk of each digit i,j,k shows 3 terms set (i,j,k) = (bifurcation position, bifurcation direction, speed). Then, a total of 999 kinds of colored multi-agent are classified by logic sets. Then, 50% of the multi-agents are initially set to travel fast on a straight lane and 50% are set to make a slow exit turn? (3) Movement from the entrance to the rotary section is executed using two competition transition on the entrance and rotary intersection for the same output box connected from the transitions, that is, if the one is fired, the other can not be fired.. All entrance signals are assumed to synchronous. (4) Bifurcation from the rotary section to the exit is executed using two competition transition on the rotary section and the exit for the same input box connected to the transition. Coordination of exit and advance competition is executed at the discretion of each agent, which is setting statistically in the simulation. No change is assumed to occur along the lane. Figure 1: Configuration of Low-capacity (Transition = Box = 12) Rotary Intersection Multi-agent Flow Model [1]

104 / International Journal of Manufacturing Science and Technology 3.1. Poisson Arrival and Entry Rate 3. SIMULATION AND RESULTS It is well known that many phenomena related to physics, biology and production systems are governed by Poisson distributions. Here, we assume that the number of arriving agents at the entrance hall of the rotary intersection per unit time is determined by such a Poisson distribution. Moreover, we create an algorithm to calculate the number X (t, τ) of Poisson arrivals in unit time t using a 2-D table with a probability variable x and a mean value µ as in (1). x µ µ p() x = e. (1) x! If you assign an entry pulse with color number to y(t), subtracting the number of entries Y(t) from the number of arrivals, the number waiting in the entrance hall is obtained using a digital integrator as in (2). KT Z( t,) τ = min[ {( s X,)()},max t τ Y t ] capacity z 1 d 1() when 1 y t < where Y() t = dt 0 otherwise. (2) where max-capacity in the entrance hall is assumed to 10. Non-linear feedback of Z is carried out to control the entries at a given probability based on a production rule. 3.2. Deadlock Generation Deadlock is defined as stopping marks or agents, then makes percolation rate to a constant. Moreover, we define a parameter referred to as the Deadlock Time (DLT) as shown in (3). d DLT = max[ ti {() t PR t 0}]. > eps > i dt This is the most significant monitoring parameter as it is changed by the Exit Control Probability (ECP) based on random arrival. The entry control probability and the initial existence probability in the rotary box are set to be the same as the independent variable ECP. This method is used to change the triple probability (Initial existence probability, Entry probability, Exit probability) similarly and concurrently. If the entry control probability is changed based on Poisson arrival, the DLT becomes 1181 s and the critical probability of ECP and the Congestion Rate (CR) become 65% as shown in Figure 2, because the entry control probability is matched to ECP only when there are more than one agents in the entrance. Here, the activation signal has a cycle time of 40 s and a split rate of 50%. Assuming Poisson arrival at all entrances of the rotary intersection, the number of arriving agents is calculated based on the mean and the variation. Subtracting the entry number from (3)

An Application of Queuing Theory to Rotary Intersection Traffic Model and / 105 the arrival number and integrating, we can obtain the number waiting in each entrance hall. Assuming that the Poisson arrival may not change if it attains once the maximum number in entrance hall (assumed to be 10 here), we obtain the results shown in Figure 2 for arrival intervals are 2 s and 3 s. The solid line indicates the number of agents through entrance No. 1, the broken line through entrance No. 2, the solid line through entrance No. 3, and the broken line through entrance No. 4. (a) At arrival interval 2 s (b) At arrival interval 3 s Figure 2: Number of Waiting Agents in Entrance Hall by Changing Interval of Poisson Arrival 3.3. Deadlock Avoidance There are many approaches to deadlock avoidance [11] [12]. It is well known that input flow into the system must be suppressed for deadlock avoidance [1] [12]. Here, we show that it is sensitive to the interval of Poisson arrival at the entrance hall of the rotary intersection, as seen in Figure 3. That is, a longer leads to more effective deadlock avoidance by suppression of the entry rate. Figure 3: Deadlock Avoidance Due to Entry Rate Suppression by Interval of Poisson Arrival

106 / International Journal of Manufacturing Science and Technology Three different areas are seen in the figure. The upper left area is a region of deadlock, the center area is deadlock-free, and the lower area is without cars. 3.4. Mean Waiting Time The entrance passing control system is constructed by feedback of the waiting agent number computed as the difference between the Poisson arrival agent number and the passing agent number using a non-linear entrance passing rate controller with a probability distribution to an entrance passing gate with a passing rate for colored agents only when there is at least one waiting agent, as shown in Figure 4. Figure 4: Configuration of Entrance Passing Control System There were 3 critical probability 30%(~1/3), 40%(~1/2.5), 66%(~/1/1.5), (theoretical critical probability is atom numbers/total bifurcation numbers [13]). That is, in these probabilities, there are the largest effect on the mean waiting time in entrance hall computed as shown in Figure 5. Figure 5: Computation of Mean Waiting Time for Each Agent in Entrance Hall after Simulation for 1200 s with Arrival Interval of 3 s by Changing 3 Terms Similarly

An Application of Queuing Theory to Rotary Intersection Traffic Model and / 107 However, adequate standard probability value of the 3 terms (Initial existence probability, Entry probability, Exit probability) may not be 40% in all entrance. Here, the mean waiting time in the entrance hall after a simulation time T, MWT(T), is computed using the mean entry time, MET(T), and the mean waiting length in entrance hall, MWL(T), as follows, where MWL(T) = Z(T, τ) where d t i = t when y() t < 1 dt where MWT ()() T *{() = MEPT 1} T MWL T + (4) KTs MEPT () T = {() /()} t t Y t z 1 T 0 i i 1 i The theoretical MWL for one gate is obtained from queuing theory, as follows. λ = mean_arrival_rate µ = mean_service_rate M/M/1 λ2 MWL = ( x 1)() p x = µ ( µ ) λ x= 0 Figure 6 shows an example of the dependence of MWL described as vertical axis for the mean service rate described as the horizontal axis when an arrival rate is 0.367 and a maximum MWL is 10. (5) (6) Figure 6: MWL Dependence for Service Rate

108 / International Journal of Manufacturing Science and Technology We can determine from this figure that a new critical probability occurred in the left-hand side of Figure 5(b) may be caused the service rate (entry control probability) and the arrival rate in the approach. It is not due to deadlock in the rotary section which is the cause of the critical probability on the right-hand side of the figure. When deadlock occur, the actual service rate is decreases to zero. Then, the MWL in Figure 5 will exhibit a valley shape as shown in Figure 5(b). 4. CONCLUSIONS We determined that setting a long interval of Poisson arrival of multi-agents in the entrance hall of a rotary intersection is very effective for deadlock avoidance by entry rate suppression. However, the arrival interval has made not to change almost critical probability by concurrently changing the triple probabilities (Initial existence probability, Entry probability, Exit probability) in a similar manner. The longer arrival interval is also effective to keep the deadlock time larger in the periods. Moreover, it can be increasing the time that it takes to fill the entrance and reducing the time that the entrance is full. This study on Poisson arrival at a low-capacity rotary intersection provided new insights into methods of avoiding deadlock, and long waiting times, even though the results are intuitively clear. That is, control of interval of multi-agent arrival is important based on control of the entry rate. In particular, queuing theory may provide reasons of generating the critical probability in entrance systems with deadlock, considering that arrival interval of queuing cars gives very influence to the values or timings of the critical probability in which the steepest changing of state happens. We assume the reason that cars move in the state of the group generated naturally as percolation of molecular of water not but pheromone of ants. Then, we can introduce the critical probability which is the phenomena of the percolation system as one in general complex systems. In future work, we hope to improve the queuing theory such that it can treat not only service rate of an entrance gate and arrival rate of the agents to the gates but also conjunction rate after the entrance gate and bifurcation rate before next exit gate. And the relation between atoms of percolation systems and boxes, transitions of MFG is expected to be clear by theoretical critical probability. Moreover, the communication among agents or between each agent and the control center are expected as it has been realized recently or will be done in the actual agent flow systems. Acknowledgments The first author cordially acknowledges the cooperation of his many students including Mr. Y. Araki, Mr. H. Watanabe and Mr. T. Inoue for the initial research, Mr. M. Kawaguchi for the traffic simulation using the multi-agent simulator, and Mr. Xue Li, Mr. Mitsugu Arimura for discussions on queuing theory and congestion. He wish to express his gratitude also to Mr. Masaki Ishitani who encouraged the issue of this paper by an association study.

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