Application of Queuing Theory in Transportation

Application of Queuing Theory in Transportation

Vincy Verghese

Nivya Varghese V

PG Student Civil Engineering Department Jyothi Engineering College

Thrissur, India

Arun Chandran

Assistant Professor Civil Engineering Department Jyothi Engineering College

Thrissur, India

Scientist KSCSTE-

National Transportation Planning and Research Centre Thiruvananthapauram, India

Abstract Traffic congestions are always frustrating to the road users. Congestions are formed due to different reasons like reduced capacity of road stretch, accidents, overcrowding etc. The traffic congestions results in delays. Delay is a more subtle concept. Delays are defined as the difference between the particular time period on a given segment and a few ideal time period of that segment. These delays will surely end up in queues. Queues occur whenever immediate demand exceeds the capacity to provide a service. Here the need of application of queuing theory arises. Queuing theory was developed to supply models capable of predicting the behaviour of systems that provide service for randomly arising demands. Queuing theory deal with the study of queues (waiting lines). A queuing system is one during which customers arrive for service, await for service if it is not immediately available, and advance to subsequent server once they have been serviced. This paper mainly deals with the different application of queuing theory in transportation sector. Applications mainly include the application of queuing theory in material transportation system, in management of traffic intensity, for the performance analysis of toll plaza, to reduce traffic accidents, at Signalized Intersection etc.

KeywordsQueuing theory; road inventory; traffic volume; space mean speed; time headway; arrival rate; service rate; utilization rate

1. INTRODUCTION

   Traffic congestion occurs in busy and populated areas. It are often very frustrating due to the delay it causes on vehicular movement for commuters and item delivery. It is periodical and has several causative factors depending on the area. The causes of congestion include lack of internal route expansion, bad roads and many vehicles in transit, poor packing by commercial vehicles and the like. Traffic congestion is common in Lagos because it is heavily populated with a small land mass. Reference [1] describes traffic congestion as a situation on road network which occurs as its use increases. It is characterized by slower speeds, increased trip times and queuing of vehicles. It is therefore necessary to apply the principle of queuing theory to optimize the waiting time in queuing system as experienced in traffic congestion. The effect of traffic jam includes commuters frustration, vehicle collision and fuel wastage. Traffic congestion also has spill over effect from congested main routes to secondary roads and side-streets as alternative routes are sought. Such spill over effect results in delays which successively leads to late arrivals for meetings and business activities within locality. Reference [1] assert that traffic congestion occurs when a mass of traffic requires space

   greater than the available road capacity. According to [2] queuing theory is the mathematical study of waiting lines or queues. In queuing theory a model is constructed so that queue lengths and waiting times can be predicted. Generally, queuing theory may be a branch of research because the results are utilized in making business decisions about resources required within the provision of services. A queue is a waiting line but queuing is used broadly to cover variety of problems usually for economic balance and optimization involving waiting and delay in serving people or servicing machines and equipment.

2. QUEUING THEORY

   Queuing theory was developed to provide models capable of predicting the behaviour of systems that provide service for randomly arising demands. Queuing theory deals with the study of queues (waiting lines).The earliest use of queuing theory was in the design of a telephone system; randomly arising calls would arrive and need to be handled by the switchboard, which had a finite maximum capacity. Applications of queuing theory are found in fields such as; traffic control, hospital management, and timeshared computer system design. The following terms are commonly utilized in queuing theory;

   Customers: The persons or objects that require certain service are called customers.

   Server: The person or a machine that gives certain definite service is understood as server.

   Service: The activity between server and customer is called service, this consumes some time.

   Queue or Waiting line: A scientific arrangement of a gaggle of persons or objects that await for service.

   Arrival: The method of consumers coming towards service facility or server to receive a particular service.

3. APPLICATION OF QUEUING THEORY

   1. Queuing Theory in Material Transportation System using Truck

      Surface mining is that the commonest mining method worldwide; open pit mining accounts for quite 60% of all surface output. Reference [3] uses queuing technique to optimize the transportation at Lafarge WAPCO (Sagamu plant) Nigeria. Queuing theory was developed to model systems that

      provide service for randomly arising demands and predict the behaviour of such systems. Time of arrival at excavator area (hr/min/sec), time of first load by excavator into the truck (hr/min/sec), number of loads, time of departure from the excavator (hr/min/sec) and time taken to load trucks gotten from the Lafarge (Sagamu plant) was analysed to develop a model M/M/1: FCFS//, based on the assumption of single channel and single server with infinite number of queues. The model was used to calculate the arrival rate, service rate and number of server which at the end of it gives 7turcks/hour, 21trucks/hour and 1loader respectively. @risk software was wont to fit both service and inter-arrival into exponential distribution. The result shows that because the size of the haulage truck getting used increases, shovel productivity increases and truck productivity decreases. An effective number of trucks must be chosen that will effectively utilize idle time, increase productivity and reduce cost of production to the barest minimum. The idle time gotten is 66.6%; this indicates that an additional 8 to 9 trucks can be added to the company truck fleet to make use of the idle time; since time translate to cost.

      Reference [4] proposes a completely unique queue-based modeling approach for aircraft arrivals at an airport and analyzes the strategies for air traffic tactical control which will mitigate arrival delays while allowing future increasing air traffic volumes. The modeling approach is motivated by a data- driven analysis of actual flight plans and radar data corresponding to the arrival traffic in 2016 and 2017 at Tokyo International Airport. This analysis estimates the arrival delays under increasing arrival rates and indicates that the most delay- efficient arrival tactical control strategy is to increase airspace capacity close to the arrival airport. In particular, allowing one or two additional aircraft in the airspace within approximately

      60 NM around the airport significantly reduces the arrival delays even in the case when the arrival rate increased up to 20% more than the current operations. The proposed approach provides support to the decision-making process of prioritizing tactical control strategies under various traffic conditions, i.e., providing support for introducing new operational procedures, designig route structures, introducing automation support for controllers. As future work, they plan to develop arrival optimization models for aircraft arrivals at an airport under increased traffic volumes such that arrival delays are minimized.

   2. Queuing Theory in the Management of Traffic Intensity

      Reference [5] apply the queuing theory in minimizing vehicular traffic congestion using four routes/channels in Victoria Island as case study. This study is focused on the management of vehicular traffic based on queuing theory. It is assumed that the time interval between successive arrival and service times is independent and normally distributed. The system is also assumed to reach a stable state with constant arrival and service rates. The study also adopts the FCFS (first come first serve) approach where the vehicles are made to line up or queue according to their time of arrival as customers waiting to be served by a signal of functioning traffic light in a given channel or location to minimize traffic congestion. The number of vehicles in each service station for every channel is counted and the time in minutes noted when waiting to be served and after being served. These values are used to derive the arrival and service rates of the vehicles. This

      situation was observed for 5 days in peak hours of morning, afternoon and evening (7-10am, 12-3pm and 5-8pm respectively). This study reveals that traffic intensity is highest in the morning session when commuters are reporting for work/business and in the evening session at the close of work/business. It is therefore necessary to allot more time at intersections for traffic into such routes in the morning and evening sessions. The increase of traffic light time will reduce traffic intensity which in turn minimizes delays on such routes/channels at peak periods of morning and evening sessions.

      Reference [6] analyzes the importance of queuing theory in the field of traffic management system for this Bhopal, Indore, Ujjain city which were located in the India is chosen. The paper review the range of queuing theory results in the area of waiting time, utilization analysis and design of system the traffic crowd follows a repeatable pattern during the day and the proper people accepts it as a daily routine. The work is based on the actual survey of traffic flow at various times at different locations of Bhopal, Ujjain and Indore city. The application of the queuing theory is exploited to minimized the traffic congestion at a particular time. By this work they found out different steps to avoid the congestion. Like, the traffic can be reduced by increasing road capacity, provide separate lane for specific user group, variable massage sign can be installed along the roadway to avoid road users, increasing width of channel of congested route, applying parking restrictions for the motor vehicle.

   3. Queuing Theory to address Traffic problems at a Highway Toll Plaza

      Reference [7] analyze the current situation, of traffic congestion, at a highway toll plaza using queuing theory and suggest possible solutions to encourage greater efficiency, thus reducing waiting time of the customers and money wasted because of that. This study has been carried out in various phases, i.e. problem identification, data collection, data analysis and results at a specific Toll Plaza in North India. The data analysis in the study helps to find out the current operational effectiveness of the Toll Plaza through parameters like, Arrival Rate, Service Rate and Number of toll booths. On analyzing the clustered graph (Mean number of vehicles in an hour vs Time of day) it was concluded that on both Sunday and Monday the peak hour is 5-6 pm and non-peak hour 4-5 am. Therefore, analysis was done in the peak hours as obtained from the graph using WinQSB software. The results show that during peak hours the mean waiting time is around 10 minutes and 5 minutes during non-peak hours. Service rate per server per minute currently is 4.3. If service rate is increased than waiting time decreases at a good pace up to a certain limit which was concluded from the sensitivity analysis. There are various steps and policies that need to be taken up to bring down the waiting time. Increasing the number of toll booths will help in reducing the length of the queue, Technologies like RFID, Mobile Collectors, and Smart card system would help in increasing the service time drastically, in turn reducing the waiting time. Another option can be put up a red light 1 km before the toll so that there is smooth vehicular flow at the toll. An LED (Light-Emitting Diode) screen can also be installed few kilometers before the toll describing the waiting time, length of queue in each lane. This will help commuters in getting into the right lane with less waiting. Less waiting in the

      system saves money as less fuel will be wasted. In this way the system can be optimized decreasing the queue length of the customers and time a customer spends in the system. Finally, possible solutions are suggested which may be recommended and implemented on various Toll Plazas within the country.

      Reference [8] presents an improved model base on M/M/1 queue theory for designing the toll plaza by considering different ways of charging in the actual toll service system (human-staffed, automated and electronic toll collection), and applies the Non-dominated Sorting Genetic Algorithm II (NSGA-II) to optimize the design of lane number and charging mode layout.

      As per [9] toll should be designed and planned in such a way that minimum time would be wasted in the queuing area. The toll booths are planned on the basis of queuing area. Queuing theory involves parameters like arrival, number of lanes, service time, waiting time, merging area. In present study road inventory, traffic volume, space mean speed, arrival rate, time headway and service rate are analyzed. A detailed study was carried out to analyse the performance of a tool booth. The following conclusion was drawn from the observed data; it had been found that flow remained constant on both directions 375 and 345 (veh/hr), the inter time of vehicle arrival between two vehicle was found out be 10 seconds on both the directions, the waiting time in the queuing area was found out to be 10 seconds as general, as the density increases the effective speed decreases thereon road section.

   4. Queuing Theory to Vehicular Traffic to reduce accidents

      Reference [10] establish the queue model for the Nakuru Salgaa road Stretch and test the model with real data from the Case Study. Data was collected between the Soil- junction and the Total junction. They derive the arrival rate, service rate, utilization rate and the probability of Bulking using the M/M/1 queuing model. It was estimated that the arrival rate at the Soil- junction is 37 vehicles per minute and at total junction the service rate is 44 Vehicles per minute this does not march the dwindle service rates in section that are now black spots. The average number of vehicles on single road stretch was on average 15 per minute with some sections recording a high of 40 vehicles per minute and the utilization of the sections of stretch was on average 0.8. The benefit of performing the queue analysis for the road stretch is finally discussed and recommendations provided.

      Reference [11] sets up the M/M/1 queuing model, analyses the traffic intensity of Palasia intersection (Indore city, India) through analyzing the queuing theory deeply, and uses the model to research the settings of the lane and signal timing that's supported a particular degree of accuracy.

   5. Queuing Theory to Vehicular Traffic at Signalized Intersection

   Reference [12] seeks to model the vehicular traffic flow and explore how vehicular traffic could be minimized using queuing theory in order to reduce the delays on roads in the Kumasi metropolis of Ghana. The Oforikrom traffic intersection within the Kumasi metropolis of Ghana is currently operating with one service channel ech from the varied routes to the intersection. This paper sets up the M/M/1 queuing model, analyses the traffic intensity of Palasia intersection (Indore city, India) through analyzing the queuing

   theory deeply, and uses the model to analyze the settings of the lane and signal timing timing that's supported a particular degree of accuracy. The results showed that traffic intensity, <1 for all sessions, a condition that means an ideal traffic system. Consequently, smooth flow of traffic was shown since the server at each channel was ready to serve quite the cars in waiting queue when servers resume work. Again, it was found that heavy traffic occurs in the evening. Stakeholders can task Motor Traffic Transport Unit (MTTU) to see that drivers desist from such practices in order that there'll be free flow of traffic within the evening and also promote the utilization of bikes, which aside from serving as a sort of exercise also helps to scale back fuel consumption thereby saving money for the government to tackle problem of other sectors of the economy. Finally, the government of Ghana could introduce a conveyance system in order that people don't travel with private cars to their places of labor to scale back congestion on the roads, which in turn boosts productivity.

   Reference [13] sets up the queuing model, analyses the traffic flow of Shenzhen intersection through analyzing the queuing theory deeply, and uses the model to analyze the settings of the lane that supported the certain degree of accuracy. From the paper, the theoretical data is according to the truth. Therefore, it's economic that the tactic of the system metrics in confirming the amount of the lanes of the intersection and it can provide references for similar design.

4. METHODOLOGY THAT CAN BE ADOPTED

   The methodology that can be formulated to analyze the performance of toll plaza is as follows; as a first step we have to select the study area. Then field studies like road inventory, traffic volume, space mean speed, time headway, arrival rate and service pattern have to be done. Road inventory reflects the pavement characteristic. Inventory data basically consists of data necessary to identify the project under evaluation. This consists of the geometric details of the project which are collected by visually walking along the complete stretch. All of these data will remain constant until the pavement undergoes maintenance or repair. Traffic volume or traffic flow is defined as the product of the typical traffic intensity and therefore the period of time of the study. It is measured by the units vehicle per hour. The space mean speed is measured using the average travel time and length for the roadway Segment. Time headway (H) is that the difference between the time the front of a vehicle arrives at some extent on the highway and therefore the time the front of subsequent vehicle arrives at the same point. The time headway is usually expressed in seconds. Time headway is important to known the inter arrival rate among the vehicles which is required to seek out the capacity of a transportation system. Arrival is usually defined as The simple model assumes that the amount of arrivals occurring within a given interval of time follows a Poisson distribution. This parameter is that the average number of arrivals in time which is additionally the variance of the distribution. Service rate denotes the speed at which vehicles are been served in a system. It is the reciprocal of the service time. Suitable queuing model can be formulated using the field data collected, also the utilization factor can be calculated. Questionnaire survey can be done to study the trend of toll users. From the results the performance of particular system can be identified. Thus

   according to the results further performance improvement measures can be suggested.

5. SUMMARY

A queuing system is one during which customers arrive for service, await for service if it is not immediately available, and advance to subsequent server once they have been serviced. Queuing models in transportation are more likely to concentrate on the non stationary characteristics of queuing, as well as on the optimization of system design and system control.The queuing theory is an effective mathematical technique for solving various traffic problems of any system as queuing theory focused on representation of traffic situation by using various mathematical terms and formulas. This paper mainly deals with the different application of queuing theory in transportation sector. Applications mainly include the application of queuing theory in material transportation system, in management of traffic intensity, at Signalized Intersection, for the performance analysis of toll plaza, to reduce traffic accidents, etc. Methodology that can be adopted for application of queuing theory in transportation is also discussed in this paper.

ACKNOWLEDGMENT

First of all I would like to thank God, the almighty for the divine grace bestowed on me to complete this seminar successfully on time. I am gratefully acknowledging the constant support and cooperation of my seminar guide Ms. Vincy Verghese, Assistant Professor, Department of Civil Engineering. I am grateful to her for the timely corrections and scholarly guidance, which made me confident enough to come out.

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