Volume 116 No. 1 017, 199-07 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.173/ijpam.v116i1.1 ijpam.eu A Study on M/M/C Queue Model under Monte Carlo simulation in Traffic Model 1S.Aruna devi and Vijeta Iyer 1Dept of Science and Humanities, Kumaraguru College of Technology, Coimbatore email: arunadevi.s.sci@kct.ac.in Dept of Science and Humanities, Kumaraguru College of Technology, Coimbatore email: vijetaiyer.sci@kct.ac.in Abstract In this paper we present a stochastic queuing model for road traffic which captures the stationary density flow relationships. We have analysed the performance of controlling the homogeneous road traffic using Monte Carlo simulation with different service distributions. The future behaviour of road traffic network both in simulation and analytic methods have been analysed. Examples to illustrate these methods have been discussed. Keywords: Inter arrival pattern, Service pattern, M/M/C Queue, Monte Carlo Simulation INTRODUCTION A queue is a line of people or things waiting in an order to get the service [1]. Queuing theory was introduced by A.K.Erlang in 1909.He published various articles in telephone traffic []. Queueing Theory is mainly a branch of applied probability theory. It has many applications in different fields, namely, communication networks, computer systems, machine plants etc. Consider a service providing center and a population of customers, which at some time intervals enter the service center in order to get the service. Mostly, the service provider can only serve a limited number of customers. If a new customer arrives and the service is exhausted, he needs to stand in waiting line or queue until the service provider becomes free. So main three elements of a service providing centre are: a 199
population of customers, the service facility and the waiting line. Also it can be considered that the several service providers are arranged in a network and a single customer can walk through this network at a specific path, visiting several service centres. With the help of Queueing Theory, many questions can be answered e.g. the mean waiting time in the queue, the mean system response time (waiting time in the queue plus service times), distribution of the number of customers in the queue, mean utilization of the service facility, distribution of the number of customers in the system etc. 1. SIMULATION Monte Carlo simulation is a computerized mathematical technique that allows people to account for risk in quantitative analysis and decision making. Monte Carlo simulation helps the decision-maker with a range of possible outcomes and the probabilities they will occur for any choice of action. The technique was first used by scientists working on the atom bomb; it was named for Monte Carlo, the Monaco resort town renowned for its casinos. Since its introduction in World War II, Monte Carlo simulation has been used to model a variety of physical and conceptual systems. Monte Carlo simulation performs risk analysis by building models of possible results by uncertainty. It then calculates results over and over, each time using a different set of random values from the probability functions. Depending upon the number of uncertainties and the ranges specified for them, a Monte Carlo simulation could involve thousands or tens of thousands of recalculations before it is complete. Monte Carlo simulation produces distributions of possible outcome values. By using probability distributions, variables can have different probabilities of different outcomes 00
occurring. Probability distributions are a much more realistic way of describing uncertainty in variables of a risk analysis.. MODEL DESCRIPTION Traffic assessment of the road abutting the project site was carried out to estimate the peak traffic load. The existing traffic load during the morning and evening peak hours were studied and the vehicle counts were categorized under different heads as arrivals. For those differently categorized vehicles services can be provided in two different parking areas. The vehicles can be parked in Basement parking and Surface parking (parking available outside the building footprint). As the L&T by pass road traffic depends on the local traffic of vehicles available outside the hospital, the analysis of parking service provided by the hospital becomes very significant. In this paper we discuss the application of simulation in M/M/C queuing model on the traffic flow of vehicles on a L&T Bypass Road. Chi square test has been used to verify if the arrival is a Poisson distribution and if the services are of exponential distribution. The simulation table provides a systematic method of tracking system state over time. The main aim of this paper is to find the waiting time of vehicles in the queue, waiting time of vehicles in the parking area, the idle time of the parking area, the queue length and also to compare the simulation and analytical solutions. In section 1, Introduction and model description is discussed. In Section, Chi-Square test is applied. In Section 3, calculations of simulation and analytical methods are given. Fourth Section describes numerical study and lastly, section 5 gives the conclusion. 01
The details of the traffic assessment done in a multi speciality hospital is presented below [3]: Time (Hrs) Two wheelers (Bicycles and Motor cycles) Three Wheelers (Auto rickshaws) Four Wheelers (Passenger cars and vans) Six wheelers (Trucks and Buses) Morning peak 07.00-07.30 19 15 96 81 07.30-08.00 3 1 100 76 08.00-08.30 19 17 97 70 08.30-09.00 4 14 98 71 09.00-09.30 16 9 67 09.30-10.00 15 15 95 65 10.00-10.30 0 19 101 73 10.30-11.00 17 1 91 77 Total 159 10 770 580 Total Number of vehicles 169 Tag Number for Arrival Distribution of Two wheelers x Time Number P(x) Cumulative Tag Number of two wheelers probability 1 00.00-00.01 19 0.1 0.1 0-11 00.01-00.0 3 0.14 0.6 1-5 3 00.0-00.03 19 0.1 0.38 6-37 4 00.03-00.04 4 0.15 0.53 38-5 5 00.04-00.05 0.14 0.67 53-66 6 00.05-00.06 15 0.09 0.76 67-75 7 00.06-00.07 0 0.13 0.89 76-88 8 00.07-00.08 17 0.11 1 89-100 Tag Number for Arrival Distribution of Three wheelers x Time Number P(x) Cumulative Tag Number of three wheelers probability 1 00.00-00.01 15 0.13 0.13 0-1 00.01-00.0 1 0.10 0.3 13-3 00.0-00.03 17 0.14 0.37 3-36 4 00.03-00.04 14 0.1 0.49 37-48 5 00.04-00.05 16 0.13 0.6 49-61 6 00.05-00.06 15 0.13 0.75 6-74 7 00.06-00.07 19 0.15 0.90 75-89 8 00.07-00.08 1 0.10 1 90-100 0
Tag Number for Arrival Distribution of Four wheelers x Time Number P(x) Cumulative Tag Number of four wheelers probability 1 00.00-00.01 96 0.1 0.1 0-11 00.01-00.0 100 0.13 0.5 1-4 3 00.0-00.03 97 0.13 0.38 5-37 4 00.03-00.04 98 0.13 0.51 38-50 5 00.04-00.05 9 0.1 0.63 51-6 6 00.05-00.06 95 0.1 0.75 63-74 7 00.06-00.07 101 0.13 0.88 75-87 8 00.07-00.08 91 0.1 1 88-100 Tag Number for Arrival Distribution of Six wheelers x Time Number P(x) Cumulative Tag Number of six wheelers probability 1 00.00-00.01 81 0.14 0.14 0-13 00.01-00.0 76 0.13 0.7 14-6 3 00.0-00.03 70 0.1 0.39 7-38 4 00.03-00.04 71 0.1 0.51 39-50 5 00.04-00.05 67 0.1 0.63 51-6 6 00.05-00.06 65 0.11 0.74 63-73 7 00.06-00.07 73 0.13 0.87 74-86 8 00.07-00.08 77 0.13 1 87-100 Vehicle Parking are available in basement and also as surface parking outside the building foot print for both two wheelers and four wheelers separately as given below. For three wheelers and six wheelers separate parking is not required. Description No.of Two wheeler parks No.of Four wheeler car parks Basement Parking 55 80 Surface Parking 137 60 (Outside the building foot print) Total parking available 19 140 03
Service Distribution for Vehicle Parking x Time (Hrs) Number P(x) of two wheelers 1 0-1 40 0.5 1-37 0.3 3-3 5 0.33 4 3-4 30 0.19 Chi-Square test for Vehicle Parking x Time (Hrs) Observed frequency f fx P(x) Expected frequency Chi Square Value 1 0-1 40 40 0.5 40 0 1-37 74 0.3 40 0.5 3-3 5 156 0.33 40 3.6 4 3-4 30 10 0.19 40.5 Total 159 390 6.35 Null Hypothesis H0: The Exponential distribution fits well into the data. Alternative Hypothesis H1: The Exponential distribution does not fit well into the data. Level of Significance at α = 0.01 Test statistic under H0 is k ( Oi Ei ) e E i1 i = 6.35 x = fx f.45 x P( x)e Degrees of freedom = 8-1 = 7 Tabulated value for 7 degrees of freedom at 1% level of significance is 18.475( 0 ). Hence e 0. We accept H0 and conclude that the exponential distribution is a good fit for the given data. Tag Number for Vehicle Parking x Time (Hrs) Number P(x) Cumulative Tag Number of two wheelers probability 1 0-1 40 0.5 0.5 0-4 1-37 0.3 0.48 5-47 3-3 5 0.33 0.81 48-80 04
Time (mins) Time (mins) Time (mins) Time (mins) International Journal of Pure and Applied Mathematics Numerical Study 6 Inter Arrival Time 5.5 5 4.5 4 3.5 3.5 0 4 6 8 10 1 14 16 18 0 Number of vehicles 4 Service Time 3.5 3.5 1.5 1 0 4 6 8 10 1 14 16 18 0 Number of vehicles 1.4 Idle Time 1. 1 0.8 0.6 0.4 0. 0 0 4 6 8 10 1 14 16 18 0 Number of vehicles 5.5 Idle Time for server 5 4.5 4 3.5 3.5 1.5 1 0 4 6 8 10 1 14 16 18 0 Number of vehicles 05
3. Conclusion In this paper, we have presented a simulation table for queuing system with a multiple service station. Here we have developed a model for vehicle parking facilities and to decide whether two parking services are needed for two wheelers or can it be reduced to one parking service. Numerical examples are given to show the similarity between numerical and analytical calculation. Hence the excess available space of two wheeler parking in future can be used for four wheelers parking or it can be utilised by three wheelers or six wheelers vehicles. REFERENCES [1] Syed Shujauddin Sameer (014). Simulation Analysis of Single server Queuing Model, International Journal of Information Theory,vol3.No.3,doi:10.511/ijit.014.330547 [] Erlang.A.K (1909), The theory of probabilities of Telephone Conversations Nyt.Jindsskriff Mathematic, B0, 33-39. [3] P.Umarani and S.Shanmugasundaram, A study on M/M/C Queuing Model under Monte Carlo Simulation in a Hospital. International Journal of Pure and Applied Mathematical Sciences, Vol 9,No: (016),PP 109-11. [4] Shanmugasundaram.S and Banumathi.P (015). A study on single server queue in southern railway using Monte Carlo Simulation, Global Journal of Pure and Applied Mathematics, Vol 11. [5] S.Shanmugasundaram and P.Umarani (016), A Simulation study on M/M/1 Queueing Model in a Medical centre, International Journal of Science and Research, PP 84 88, Volume 5,Issue 3. 06
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