MONTE CARLO SIMULATION STUDY ON M/M/1 AND M/M/2 QUEUEING MODELS IN A MULTI SPECIALITY HOSPITAL

Transcription

1 Volume 117 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu MONTE CARLO SIMULATION STUDY ON M/M/1 AND M/M/2 QUEUEING MODELS IN A MULTI SPECIALITY HOSPITAL S.Shanmugasundaram 1, P.Umarani 2 1 Department of Mathematics, Government Arts College,Salem , Tamil Nadu,India. sundaramsss@hotmail.com 2 Department of Mathematics, AVS Engineering College,Salem - 3, Tamil Nadu, India. umasaravana2000@gmail.com Abstract This paper tries to check and analyze the waiting time for the new patients as well as the previous patients through Monte Carlo simulation and provides M/M/1(single server) or M/M/2 (multi server). Additionally, Chi-Square test verifies the Poisson distribution which is suitable to the arrival and the uniform distribution of the service time of patients. The behaviour of the system has been analyzed both in simulation and analytical methods. The numerical study illustrates the originality of the model. The results have been recommended to the hospital management for implementation, which will definitely reduce the waiting time of the patients. AMS Subject Classification: 90B22; 60K25. Key Words and Phrases: Inter - arrival Time, Service time, Waiting time, M/M/1 and M/M/2 queueing models, Monte Carlo Simulation, Queue length. 1 INTRODUCTION In all the activities in our day to day life, we have to wait in a queue. The same holds good for the patients in a hospital, which will increase the physical as well as psychological discomfort during the waiting. When compared with the queues in other activities, the patients waiting in the hospital becomes an emergency and that is the reason for this topic has been chosen. Waiting time in the queue is analyzed through a stochastic process which comprises the simulation method and analytical method. Further Chi - Square test verifies the Poisson distribution is apt to the arrival and the uniform distribution of the service. Brief explanation about the topics viz: i) Queuing Theory, ii) Simulation Method, iii) Analytical Method and iv) Chi - Square Test are given below. Queueing theory was introduced by Agner Krarup Erlang ( ), a Danish Engineer cum mathematician. It is a quantitative technique which consists of constructing mathematical models of various types of queueing systems and deals with the analysis of queues and queueing behaviour. It is the mathematics of waiting lines. A queuing system can be described by the flow of units for service forming, or joining the queue 271 1

2 2 S.Shanmugasundaram, P.Umarani when service is not available soon, and leaving the system after being served.this theory is well explained by A.K.Erlang (1909) in his articles related to the study of congestion in telephone traffic[1]. Six characteristics of queueing processes are explored in this section. These are (i) Arrival pattern of customers. (ii) Service pattern of servers (iii) Queueing discipline (iv) System capacity (v) Number of service channels and (vi) Number of service phases. All the six characteristics decide the strength of the system. The arrival pattern is the number of arrivals per unit time. This is expressed in terms of the mean number or mean inter-arrival time. The arrival may be single, bulk or batches of fixed or varied sizes. Service pattern may also be described by the number of customers served per unit time or the time required to serve a customer. Queue discipline refers to the rule based on which customers are selected for service. There are many queue disciplines and the commonly observed one is First In First Out (FIFO). The capacity refers to the limitation on the number of customers allowed for that service. Service channels refer to the parallel counter offering the same type of service and Service in phases refers to a series of service counters. In queueing theory, a discipline within the mathematical theory of probability, an M/M/1 queue represents the queue length in a system having a single server, where arrivals are determined by a Poisson process and job service times have an exponential distribution. An extension of this model with more than one server is the M/M/C queue. The simulation was introduced by Mr. John von Neumann and Mr.Stanishlaw ulam. It is a method of solving decision-making problems by designing, constructing and manipulating a model of the real system. It is defined to be the action of performing experiments on a model of a given system. Mr. J.V Neumann and Mr. Stanishlaw gave the first important application in the behaviour of Neutrons in a Nuclear Shielding Problem with remarkable success. Banks.J (2001) has described that in a Monte Carlo simulation, a random value is selected for each of the task, based on the range of estimates[2]. The model is calculated based on this random value. The result of the model is recorded, and the process is repeated. A typical Monte Carlo simulation calculates the model hundreds or thousands of times, each time using different randomly-selected values. When the simulation is completed, there are a large number of results from the model, each is based on random input values. These results are used to describe the likelihood, or probability, of reaching various results in the model. The analytical method calculates the waiting time in the Queue using Little s Formula, which says that, under steady state conditions, the average number of items in a queuing system equals the average rate at which items arrive multiplied by the average time that an item spends in the system. L = λw, where L is the average number of items in the queuing system, W is the average waiting time in the system for an item, and λ is the average number of items arriving per unit time. Chi-square test is used to test the suitability of a distribution and the independence of the attributes. It is used to test the significance of the difference between the observed frequencies in a sample and expected frequencies obtained from the theoretical distribution. A set of observed frequencies obtained under some experiment has been given to test if the experimental results support a particular hypothesis. Karl Pearson developed a test for testing the significance of discrepancy between experimental values and the theoretical values obtained under some hypothesis. This is χ 2 test fitness. The test statistic is given (O i E i ) 2 i=1 E i by χ 2 = k, where O i is the observed frequency in the i th class interval and E i is the expected frequency in that class interval. The expected frequency for each class interval is computed as E i is the np i where p i is the theoretical, hypothesized probability associated with the i th class interval. It can be shown that χ 2 approximately follow the Chi-square distribution with k s 1 degrees of freedom, where s represents the number of parameters of the hypothesized distribution estimated by the sample statistics. The 272

3 MONTE CARLO SIMULATION STUDY ON M/M/1 AND M/M/2 QUEUEING MODELS IN A MULTI SPECIALITY HOSPITAL... 3 hypotheses are H 0 : The random variable conforms to the distributional assumption with the parameter given by the parameter estimates. H 1 : The random variable does not conform. The aims of this paper are i) to check the goodness-of-fit of arrival and service distribution, ii) to determine the waiting time of patients in the queue, iii) to determine the total time spent by the patients in the hospital, and iv) to determine the length of the queue. Shanmugasundaram,S and Punitha,S (2014) have analyzed the application of simulation in queueing model in tollgate[3]. Ishan P Lade, Sandeep A Chowriwar and Pranay B Sawaitul (2013) have described the use of queueing systems to decrease the waiting time of patients in the queueing system using simulation method[4]. Shanmugasundaram,S and Banumathi.P (2016) has aimed to reduce the queue length, system length, queue time and system time using simulation in Southern railway using simulation in queueing analysis. Simulation is a mimic of reality that exists or which is contemplated. Simulation is most effectively used as a queueing analysis[5]. Syed Shujauddin Sameer (2014) has helped in understanding the behaviour of a queueing system using simulation and also obtains certain very useful parameters. In this paper, simulation provides a good strategy to analyze the client - server systems and to solve the complicated problems. One of the important applications of the simulation is the analysis of the waiting line problem and it is classified into Deterministic Model, Probabilistic Model, Static Model and Dynamic Model[6]. Shanmugasundaram,S and Umarani,P (2016) have discussed the applications of simulation in queueing model at a medical center and how to calculate the queue time, system time, queue length and system length in the simulation table[7]. Soemon Takakuwa and Athula Wijewickrama (2008) have described the application of the simulation for patients coming to the hospital, the pertinent parameters like waiting time, service time, waiting time and service time ratio[8]. Shanmugasundaram,S and Umarani,P (2016) have analyzed the queueing system in a simulation model of a medical center in order to develop an efficient procedure for reducing the waiting time of the patients in the queue[9]. The researchers have dealt with the single classification of patients. They do not deal with new patients and previous patients served by a single server. They have analyzed the problem only in simulation and the analytical method was not explained. Moreover, the simulation table was also not used by them. All the above-mentioned issues are described in detail in this paper. 2 STATEMENT OF THE PROBLEM Generally, it is very tiresome for the patients to wait at a hospital as they may be both physically and psychologically weak. Further, patients have to wait for a long time in the queue for their turn which is undesirable. The problem of patients waiting in the queue is solved in this paper. 3 DESCRIPTION OF THE MODEL Initially, the patients arrival data in a multi specialty hospital has been collected for half an hour forenoon during the month of January The patients are classified into two categories like new patients and previous patients. Every department in a hospital has been assigned with two doctors the specialist and super specialist. The system is in such a way that a new patient has to meet the specialist and undergo the medical procedure; whereas the previous patients are directed to the super specialist (i.e. after the initial diag- 273

4 4 S.Shanmugasundaram, P.Umarani nosis and treatment methods, for the first visit and from the next arrival onwards the same patient is categorised as the previous patient and the treatment procedures are as per the super specialist s advice). This system generates two different parallel queues for each department. The actual waiting time of the patients in the queue at the hospital is calculated through Monte Carlo Simulation and Analytical methods. The results of both the models were identical and have the waiting time in the queue and they are illustrated in Table 15(a). Chi - Square test is used to check whether the arrival follows the Poisson distribution and the service time of the patients are uniformly distributed. As the objective of this paper is to minimize or nullify the waiting time of the patients, the multi server system is recommended and again the calculations were repeated to compare with the original results. This recommendation of M/M/2 queuing model resulted in Zero waiting time of the patients in the queue and are illustrated in Table 15(b). The frequency distributions are given below. TABLE: 1 ARRIVAL DISTRIBUTION FOR NEW PATIENTS TABLE: 2 CHI-SQUARE TEST FOR NEW PATIENTS S.No Time(min) No of Patients Probability S.NO X No of Patients f fx P(x) Ei χ Total Total From Table 2. Null Hypothesis H 0 : The Poisson distribution fits well with arrival. Alternative hypothesis: The Poisson distribution does not fit well into the data. The level of significance: α = 0.01(1%).λ = X = 15.45; Degrees of freedom = 17. Calculated value of χ 2 = Tabulated value of χ 2 for 17 degrees of freedom at 1% level of significance is Since χ 2 < χ , H 0 is accepted and concluded that the arrival follows the Poisson distribution and it is suitable for the given data. From Table 4. Null Hypothesis H 0 : The Poisson distribution fits well with arrival. Alternative hypothesis: The Poisson distribution does not fit well into the data. The level of significance: α = 0.01.λ = X = 14.67; Degrees of freedom = 17. Calculated value of χ 2 = Tabulated value of χ 2 for 17 degrees of freedom at 1% level of significance is Sinceχ 2 < χ , it is accepted as H 0 and concluded that the arrival follows the 274

5 MONTE CARLO SIMULATION STUDY ON M/M/1 AND M/M/2 QUEUEING MODELS IN A MULTI SPECIALITY HOSPITAL... 5 TABLE: 3 ARRIVAL DISTRIBUTIONS FOR PREVIOUS PATIENTS TABLE: 4 CHI-SQUARE TEST FOR PREVIOUS PATIENTS S.No Time(min) No of Patients Probability S.NO X No of Patients f fx P(x) Ei χ Total Total Poisson distribution and it is suitable for the given data. TABLE: 5 TAG NUMBER TABLE FOR NEW PATIENTS TABLE: 6 TAG NUMBER TABLE FOR PREVIOUS PATIENTS S.No Time (min) No of Patients Probability Cumulative Probability Tag numbers S.No Time (min) No of Patients Probability Cumulative Probability Tag numbers Total Total TABLE: 7 SERVICE DISTRIBUTION FOR SPECIALIST TABLE: 8 CHI-SQUARE TEST FOR SPECIALIST S.No Time(min) No of Patients Probability S.NO X No of Patients Ei χ Total Total From Table 8. Null Hypothesis H 0 : The service time of patients are uniformly distributed. The level of significance: α = Degrees of freedom = 7. Calculated value of χ 2 = Tabulated value of χ 2 for 7 degrees of freedom at 1% level of significance is Since χ 2 < χ , it is is accepted as H 0 and concluded that the arrival follows the Poisson distribution and it is suitable for the given time of interval. From Table 10. Null Hypothesis H 0 : The service time of patients are uniformly distributed. The level of significance: α = 0.01.Degrees of freedom = 7. Calculated value of χ 2 = Tabulated value of χ 2 for 7 degrees of freedom at 1% level of significance is 275

6 6 S.Shanmugasundaram, P.Umarani TABLE: 9 SERVICE DISTRIBUTION FOR SUPER SPECIALIST TABLE: 10 CHI-SQUARE TEST FOR SUPER SPECIALIST S.No Time (min) No of Patients Probability S.NO X No of Patients f Ei χ Total Total Since χ 2 < χ , it is accepted as H 0 and concluded that the arrival follows the Poisson distribution and it is fit for the given time of interval. TABLE 11: TAG NUMBER TABLE FOR SPECIALIST TABLE: 12 TAG NUMBER TABLE FOR SUPER SPECIALIST S.No Time(min) No of Patients Probability Cumulative Probability Tag numbers S.No Time(min) No of Patients Probability Cumulative Probability Tag numbers Total Total TABLE: 13 DISTRIBUTIONS FOR ARRIVAL CHOSEN S.No Arrival Probability Cumulative Probability 1 New patients Previous patients TABLE: 14 TAG NUMBER TABLE FOR ARRIVAL CHOSEN S.No Arrival Probability Cumulative Probability Tag numbers 1 New patients Previous patients CALCULATION SIMULATION METHOD [From the Table 15(a)] Expected Waiting Time of new Patient in Queue = 19.6 min Expected Waiting Time of the previous Patient in Queue = 32.75min Total Time spent by new Patient in Hospital = 36.6 min Total Time spent by the previous Patient in Hospital = 49.2 min Average number of new patients in the Queue = 0.88 Average number of previous patients in the Queue = ANALYTICAL METHOD Expected Waiting Time of new Patient in Queue = 15 min Expected Waiting Time of the previous Patient in Queue = 23.3 min Total Time spent by new Patient in Hospital = 25 min Total Time spent by the previous Patient in Hospital = 33.3 min Number of new patients in the Queue = 0.9 Number of previous patients in the Queue =

7 MONTE CARLO SIMULATION STUDY ON M/M/1 AND M/M/2 QUEUEING MODELS IN A MULTI SPECIALITY HOSPITAL NUMERICAL STUDY COMPARISION OF SIMULATION WITH ANALYTICAL IN M/M/1 Figure 1: Bar Chart comparing the waiting time (in minutes) of New & Previous patients in the queue for both Simulation and Analytical Methods (M/M/1 - model). In the chart, the first bar indicates the result through simulation method and the second bar indicates the result through Analytical method. Figure 2: Bar Chart comparing the total time (in minutes) spent by New & Previous patients in the Hospital for both Simulation and Analytical Methods (M/M/1 - model). In the chart, the first bar indicates the result through simulation method and the second bar indicates the result through Analytical method. Figure 3: Bar Chart comparing the Queue length of New & Previous patients for both Simulation and Analytical Methods (M/M/1 - model). In the chart, the first bar indicates the result through simulation method and the second bar indicates the result through Analytical method. 6 CALCULATION SIMULATION METHOD [From the Table 15(b)] Expected Waiting Time of new Patient in Queue = 0 min Expected Waiting Time of the previous Patient in Queue = 0.1 min Total Time spent by new Patient in Hospital = min Total Time spent by the previous Patient in Hospital = min Average number of new patients in the Queue = 0 Average number of previous patients in the Queue = ANALYTICAL METHOD Expected Waiting Time of new Patient in Queue = 1 min 277

8 8 S.Shanmugasundaram, P.Umarani Expected Waiting Time of the previous Patient in Queue = 1.28min Total Time spent by new Patient in Hospital = 11 min Total Time spent by the previous Patient in Hospital = min Number of new patients in the Queue = Number of previous patients in the Queue = NUMERICAL STUDY: COMPARISION OF SIMULATION WITH ANALYTICAL IN M/M/C Figure 4: Bar Chart comparing the total time (in minutes) spent by New and Previous patients in the Hospital for both Simulation and Analytical Methods (M/M/2 - model). In the chart, the first bar indicates the result through simulation method and the second bar indicates the result through Analytical method. 8 CONCLUSION This paper has analyzed the single server (M/M/1) and multi server (M/M/2) queueing models in both Monte Carlo Simulation and Analytical models. The queue formed by the arrival of new patients and the previous patients in the M/M/1 model has been resolved by recommending the M/M/2 model for the specialist doctor and the super specialist doctor. It brings out that the analytical model coincides with the simulation model. Though the study has been directed for a multi specialty hospital, the same technique can be adopted in all the sectors where queue formation is foreseen, which will be a remedial measure for reducing the waiting time of the society. References [1] A.K. Erlang, The Theory of Probabilities of Telephone Conversations, Nyt Jindsskriff Mathematic, B20, (1909), [2] J. Banks, J.S. Carson, B.L, Nelson, and D.M. Nicol, Discrete Event System Simulation, Prentice Hall International Series, Third Edition, (2001), [3] S. Shanmugasundaram and S.Punitha, A Simulation Study on Toll Gate System in M/M/1 Queueing Models, IOSR - Journal of Mathematics, Vol.10, (3), (2014). [4] Ishan P Lade, Sandeep A Chowriwar and Pranay B Sawaitul (2013), Simulation of Queueing Analysis in Hospital, International Journal of Mechanical Engineering and Robotics Research, (2013),Vol. 2, No. 3. [5] S. Shanmugasundaram, and P. Banumathi A Simulation Study on M/M/C Queueing Models, International Journal for Research in Mathematics and Mathematical Sciences, (2016), Vol.2 (2). 278

9 MONTE CARLO SIMULATION STUDY ON M/M/1 AND M/M/2 QUEUEING MODELS IN A MULTI SPECIALITY HOSPITAL... 9 [6] Syed Shujauddin Sameer, Simulation: Analysis of Single Server Queueing Model, International Journal of Information Theory, Vol.3, No.3, (2014). [7] S. Shanmugasundaram and P. Umarani, A Study on M/M/C Queueing Model under Monte Carlo Simulation in a Hospital, International Journal of Pure and Applied Mathematical Sciences. Vol. 9 (2), (2016), [8] Soemon Takakuwa and Athula Wijewickrama, Optimizing Staffing Schedule in Light of Patient Satisfaction for the whole Outpatient Hospital Ward, Proceedings of the 2008 Winter Simulation Conference (2008). [9] S. Shanmugasundaram and P. Umarani A Simulation Study on M/M/1 and M/M/C Queueing Models in a Medical Centre, Mathematical Sciences International Research Journal, Vol.5 (2), (2016). 279

10 10 S.Shanmugasundaram, P.Umarani Above Table continue

11 MONTE CARLO SIMULATION STUDY ON M/M/1 AND M/M/2 QUEUEING MODELS IN A MULTI SPECIALITY HOSPITAL Above Table continue

12 282