a study on m/m/c queueing model under monte carlo …€¦ · · 2016-11-05a study on m/m/c...

International Journal of Pure and Applied Mathematical Sciences.

ISSN 0972-9828 Volume 9, Number 2 (2016), pp. 109-121

© Research India Publications

http://www.ripublication.com

A Study on M/M/C Queueing Model under Monte

Carlo Simulation in a Hospital

P.Umarani1 and S.Shanmugasundaram2

1Department of Mathematics, AVS Engineering College,

Salem-3, Tamilnadu, India.

E-mail: [email protected].

2Department of Mathematics, Government Arts College,

Salem – 7, Tamilnadu, India.

E-mail: [email protected].

Abstract

In this paper, we analyze the performance of the multi-speciality hospital using Monte

Carlo Simulation method with different service distributions. Also we analyze the

future behaviour of the multi-speciality hospital both in simulation and analytical

method. Numerical examples illustrate that the feasibility of the system.

Keywords: Inter - arrival Time, Service time, Waiting time, M/M/C queueing model,

Monte Carlo Simulation, Queue length.

INTRODUCTION

A queue is a waiting line of people or things to be handled in a sequential order[1].

Queueing theory was introduced by A.K.Erlang in 1909. He published various articles

about the study of jamming in telephone traffic[2]. In a queueing model, customers

arrive from time to time and join a queue (waiting line), are eventually served, and

finally leave the system. The key elements of a queueing system are the customers

and servers. The term “customer” can refer to people, machines, trucks, patients,

pallets, airplanes, e-mail cases, orders or dirty clothes- anything that arrives at a

facility and requires service. The term “server” might refer to receptionists,

repairpersons, mechanics, tool-crib clerks, medical personal, automatic storage and

retrieval machines, runways at an airport, automatic packers, order pickers, CPU’s in

110 P.Umarani and S.Shanmugasundaram

a computer, or washing machines – any resource (person, machine, etc.) that provides

the required service[3].

Features of queueing systems: (i).The calling population: The population of potential

customers, referred to as the calling population, may be assumed to be finite or

infinite. For such systems, this assumption is usually innocuous and, furthermore, it

might simplify the model. (ii). System capacity: In many queueing systems, there is a

limit to the number of customers who may be in the waiting line or system. (iii).

Arrival process: Arrivals may occur at scheduled times or at random times. At random

times, the inter-arrival times are usually characterized by a probability distribution. In

addition, customers may arrive one at a time or in batches. The batch may be of

constant size or of random size. The most important model for random arrivals is the

Poisson Arrival Process. (iv). Service Pattern: It represents the pattern in which a

number of customers leaves the system. Departures may also be represented by the

service time, which is the time period between two successive services. The number

of customers served per unit of time is called service rate. This rate assumes the

service channel to be always busy. A general assumption used in most of the models

is that the service time is randomly distributed according to exponential distribution.

(v).Service Channels: The queueing system may have single service channel. The

system can also have a number of service channels where the customers may be

arranged in parallel or in series or complex combination of both[4]. (vi). Queue

Behaviour and Queue Discipline: Queue behaviour refers to the actions of customers

while in a queue waiting for service to begin. In some situations, there is a possibility

that incoming customers will balk, renege, or jockey. Queue discipline refers to the

logical ordering of customers in a queue and determines which customer will be

chosen for service when a server becomes free. Common queue disciplines include

first-in-first-out (FIFO); last-in first-out (LIFO); service in priority(PR)[5].

MODEL DESCRIPTION

For two servers, while the arrival distribution is the same, the service distributions

differs. In this paper, we discuss the application of simulation in M/M/C queueing

model in a hospital. Chi-Square test has been used to verify if the arrival is Poisson

distribution and if the services are 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 a patient in the queue, the waiting time of a patient

in the hospital, the idle time of the doctors, the queue length in M/M/C queueing

model and also to compare the simulation and analytical solutions. This paper

Arrival Departureeee

Queue Server1

Server2

A Study on M/M/C Queueing Model under Monte Carlo Simulation in a Hospital 111

composes: Section 1 gives introduction about basic simulation model, Section 2 gives

the Chi-square test, Section 3 gives calculation of simulation and analytical methods,

Section 4 describes the numerical study and Section 5 gives the conclusion.

1. Simulation

A Simulation is the imitation of the operation of a real-world process or system over

time. Whether done by hand or on a computer, simulation involves the generation of

an artificial history of a system and the observation of that artificial history to draw

inferences concerning the operating characteristics of the real system. The behaviour

of a system as it evolves over time is studied by developing a simulation model. This

model usually takes the form of a set of assumptions concerning the operation of the

system. Simulation can also be used to study systems in the design stage, before such

systems are built. Thus, simulation modelling can be used both as an analysis tool for

predicting the effect of changes to existing systems and as a design tool to predict the

performance of new systems under varying sets of circumstances. Mr.John Von

Neumann and Mr. Stanislaw Ulam were given the first important application in the

behaviour of neutrons in the nuclear shielding problem [6].

Monte Carlo methods are used for simulating the behaviour of physical and or

mathematical systems, especially when analytical solutions are difficult to obtain.

These methods are non-deterministic or stochastic. Applications of Monte Carlo

methods are quite varied: these include physics, computer science, engineering,

environmental sciences, finance etc., and systems with uncertainties in addition to

pure mathematical systems not involving any uncertainty [7]. A random value of X is

known as a random number. In practice, values of X can be deterministically

generated and the random numbers so generated are known as pseudorandom

numbers: such a pseudorandom number contains a bounded (fixed) number of digits,

implying that continuous uniform distribution is approximated by a discrete one.

Pseudorandom numbers are made use of it in simulation studies. Monte Carlo

methods need large number of pseudorandom numbers and computers are in their

generation [8].

2. Chi-Square Test Goodness-of-fit tests provide helpful guidance for evaluating the suitability of a

potential input model. One procedure for testing the hypothesis that a random sample

of size n of the random variable 𝑋 follows a specific distributional form is the Chi-

square Goodness-of-fit test. The test procedure begins by assuming the n

observations into a set of 𝑘 class intervals or cells. The test statistic is given by 2 =

∑(𝑂𝑖−𝐸𝑖)2

𝐸𝑖

𝑘𝑖=1 where Oi is the observed frequency in the ith class interval and Ei is the

expected frequency in that class interval. The expected frequency for each class

interval is computed as E i= npi where pi is the theoretical, hypothesized probability

associated with the ith class interval. It can be shown that 2 approximately follows

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 hypotheses are H0: The random variable 𝑋, conforms to the


distributional assumption with the parameter given by theparameter estimates. H1:

The random variable 𝑋 does not conform.

3. Calculation

At a hospital, the patients’ arrival is a random phenomenon and the time between the

arrivals varies from 6 a.m. to 12 p.m. and the service time of doctor 1 varies from four

minutes to thirty two minutes and the service time of doctor 2 varies from five

minutes to forty minutes[9]. The frequency distributions are given below.

Table 1: ARRIVAL DISTRIBUTION

S.No Time No of Patients Probability

1 0-6 0 0

2 6-7 0 0

3 7-8 1 0.01

4 8-9 2 0.01

5 9-10 3 0.02

6 10-11 7 0.04

7 11-12 8 0.04

8 12-13 12 0.07

9 13-14 17 0.10

10 14-15 21 0.13

11 15-16 24 0.15

12 16-17 22 0.13

13 17-18 18 0.11

14 18-19 9 0.06

15 19-20 8 0.05

16 20-21 7 0.04

17 21-22 3 0.02

18 22-23 3 0.02

19 23-24 0 0

Total - 165


Table 2: CHI-SQUARE TEST FOR ARRIVAL

Null Hypothesis𝐻0: The Poisson distribution fits well into the data.

Alternative Hypothesis 𝐻1 : The Poisson distribution does not fit well into the data.

Level of significance: 𝛼 = 0.01 .Test statistic: Under𝐻0, the test statistic is

2 =∑(𝑂𝑖−𝐸𝑖)2

𝐸𝑖

𝑘𝑖=1 ; �̅� = 0.957 ; P(x) =

𝑒−𝜆𝜆𝑥

𝑥!

Degrees of freedom = 11 . Tabulated value of 2 for 11 degrees of freedom at 1%

level of significance is 24.725. Since 2 < 20.01, we accept 𝐻0 and conclude that

the Poisson distribution is a good fit to the given data.

X

No of Patients

f

fX

P(x)

Ei

2

0 0

13

0 0.000047 0

10

0.3

1 0 0 0.00047 0

2 1 2 0.0023 0

3 2 6 0.0077 1

4 3 12 0.0194 3

5 7 35 0.0386 6

6 8 48 0.0641 11 1.125

7 12 84 0.0912 15 0.75

8 17 136 0.1135 19 0.235

9 21 189 0.1256 21 0

10 24 240 0.125 21 0.375

11 22 242 0.1132 19 0.409

12 18 216 0.0939 16 0.222

13 9 117 0.0719 12 1

14 8 112 0.0511 9 0.125

15 7 105 0.0339 6 0.142

16 3

6

48 0.0211 3

6 0

17 3 51 0.0123 2

18 0 0 0.0068 1

Total 165 1643 165 4.683


Table 3: TAG NUMBER FOR ARRIVAL DISTRIBUTION

S.No Time No of Patients Probability Cumulative Probability Tag numbers

1 0-6 0 0 0 0

2 6-7 0 0 0 0

3 7-8 1 0.01 0.01 0

4 8-9 2 0.01 0.02 0 – 1

5 9-10 3 0.02 0.04 2 - 3

6 10-11 7 0.04 0.08 4 – 7

7 11-12 8 0.04 0.12 8 – 11

8 12-13 12 0.07 0.19 12 – 18

9 13-14 17 0.10 0.29 19 – 28

10 14-15 21 0.13 0.42 29 – 41

11 15-16 24 0.15 0.57 42 – 56

12 16-17 22 0.13 0.70 57 – 69

13 17-18 18 0.11 0.81 70 – 80

14 18-19 9 0.06 0.87 81 – 86

15 19-20 8 0.05 0.92 87 – 91

16 20-21 7 0.04 0.96 92 – 95

17 21-22 3 0.02 0.98 96 – 97

18 22-23 3 0.02 1 98 - 100

Total - 165

Table 4: SERVICE DISTRIBUTION FOR DOCTOR I

S.No Time No of Patients Probability

1 0 - 4 5 0.08

2 4 - 8 7 0.11

3 8 - 12 15 0.23

4 12 - 16 13 0.20

5 16 - 20 12 0.19

6 20 - 24 7 0.11

7 24 - 28 3 0.05

8 28 - 32 2 0.03

Total - 64 1


Table 5: CHI-SQUARE TEST FOR DOCTOR I

X

No of Patients

f

fX

P(x)

Ei

2

1 5 5 0.2 13 4.923

2 7 14 0.154 10 0.9

3 15 45 0.2 13 0.307

4 13 52 0.10 7 5.143

5 12 60 0.10 7 3.571

6 7 42 0.10 7 0

7 3 5 21 0.05 4 7 0.57

8 2 16 0.0324 3

Total 64 255 - 64 15.414

Lvel of significance: 𝛼 = 0.01 Test statistic: Under𝐻0, the test statistic is

2 = ∑(𝑂𝑖−𝐸𝑖)2

𝐸𝑖

𝑘𝑖=1 ; �̅� = 3.9 ; λ = 26 ; P(x) = λ 𝑒−𝜆𝑥

Degrees of freedom = 6. Tabulated value of 2 for 6 degrees of freedom at 1% level

of significance is 16.812. Since 2 < 20.01, we accept 𝐻0 and conclude that the

exponential distribution is a good fit to the given data.

Table 6: TAG NUMBER FOR SERVICE DISTRIBUTION OF DOCTOR I

S.No Time No of Patients Probability Cumulative Probability Tag numbers

1 0 - 4 5 0.08 0.08 0 – 7

2 4 - 8 7 0.11 0.19 8- 18

3 8 - 12 15 0.23 0.42 19 – 41

4 12 - 16 13 0.20 0.62 42 – 61

5 16 - 20 12 0.19 0.81 62 – 80

6 20 - 24 7 0.11 0.92 81 – 91

7 24 - 28 3 0.05 0.97 92– 96

8 28 - 32 2 0.03 1 97– 100

Total - 65 1 - -


Table 7: SERVICE DISTRIBUTION FOR DOCTOR II S.No Time No of Patients Probability

1 0 - 5 11 0.13

2 5 - 10 13 0.15

3 10 - 15 16 0.19

4 15 - 20 17 0.20

5 20 - 25 14 0.16

6 25 - 30 8 0.09

7 30 - 35 5 0.06

8 35 - 40 2 0.02

Total 86

Table 8: CHI-SQUARE TEST FOR DOCTOR II

X

No of

Patients

f

fX

P(x)

Ei

2

1 11 11 0.206 18 4.4

2 13 26 0.2 18 1.9

3 16 48 0.12 11 1.5

4 17 68 0.1 9 3.7

5 14 70 0.1 9 1.7

6 8 48 0.1 9 0.125

7 5 7

35 0.1 9 12

2.1

8 2 16 0.0311 3

Total 86 322 86 15.425

Level of significance : 𝛼 = 0.01 Test statistic: Under 𝐻0, the test statistic is

2 = ∑(𝑂𝑖−𝐸𝑖)2

𝐸𝑖

𝑘𝑖=1 ; �̅� = 3.7 ; λ = o.27 ; P(x) = λ 𝑒−𝜆𝑥 Degrees of

freedom = 6. Tabulated value of 2 for 6 degrees of freedom at 1% level of

significance is 16.812 . Since 2 < 20.01, we accept 𝐻0 and conclude that the

exponential distribution is a good fit to the given data.

Table 9: TAG NUMBER FOR SERVICE DISTRIBUTION OF DOCTOR II

S.No Time No of

Patients

Probability Cumulative

Probability

Tag

numbers

1 0 - 5 11 0.13 0.13 0 – 12

2 5 - 10 13 0.15 0.28 13 – 27

3 10 - 15 16 0.19 0.47 28 – 46

4 15 - 20 17 0.20 0.67 47 – 66

5 20 - 25 14 0.16 0.83 67 – 82

6 25 - 30 8 0.09 0.92 83 – 91

7 30 - 35 5 0.06 0.98 92 – 97

8 35 - 40 2 0.02 1 98 - 100

Total 86


Table 10 : DISTRIBUTION FOR DOCTOR CHOOSEN

S.No Server Probability Cumulative

Probability

1 1 0.5 0.5

2 2 0.5 1

Table 11: TAG NUMBER FOR DOCTOR CHOOSEN

S.No Server Probability Cumulative

Probability

Tag numbers

1 1 0.5 0.5 0 - 49

2 2 0.5 1 50 - 99

Table 12: SIMULATION FOR MULTI - SERVER MODEL

S.No

Ra

nd

o

m

nu

mb

er

Inte

r

arri

val

tim

e

Actu

al

tim

e R

an

do

m

nu

mb

er

Servic

e

Tim

e R

an

do

m

nu

mb

er

Server

ch

oo

sen

DOCTOR 1

DOCTOR 2

Wa

itin

g

tim

e o

f

cu

sto

me

r i

n

qu

eu

e

Wa

itin

g

tim

e o

f

cu

sto

me

r i

n

ho

spit

al

Qu

eu

e

len

gth

Service

begins

Service

ends

Idle

time

Service

begins

Service

ends

Idle

time

1 58 16 6.16 65 20 22 1 6.16 6.36 16 - - - - 20 -

2 68 16 6.32 91 30 98 2 - - - 6.32 7.02 32 - 30 -

3 53 15 6.47 30 12 01 1 6.47 6.59 11 - - - - 12 -

4 07 10 6.57 66 20 71 2 - - - 7.02 7.22 - 5 25 1

5 99 22 7.19 32 12 58 2 - - - 7.22 7.37 - 3 15 1

6 33 14 7.33 29 12 14 1 7.33 7.45 34 - - - - 12 -

7 42 15 7.48 11 8 28 1 7.48 7.56 3 - - - - 8 -

8 45 15 8.03 43 16 68 2 - - - 8.03 8.18 26 - 16 -

9 54 15 8.18 40 12 69 2 - - - 8.18 8.33 - - 12 -

10 15 12 8.30 65 20 46 1 8.30 8.50 34 - - - - 20 -

11 20 13 8.43 82 25 53 2 - - - 8.43 9.08 10 - 25 -

12 46 15 8.58 73 20 33 1 8.58 9.18 8 - - - - 20 -

13 57 16 9.14 15 10 78 2 - - - 9.14 9.24 6 - 10 -

14 81 18 9.32 70 20 97 2 - - - 9.32 9.57 8 - 20 -

15 78 17 9.49 65 20 86 2 - - - 9.57 10.17 - 8 28 1

16 12 12 10.01 33 12 59 2 - - - 10.17 10.32 - 16 28 1

17 50 15 10.16 54 16 75 2 - - - 10.32 10.52 - 16 32 1

18 95 20 10.36 87 24 70 2 - - - 10.52 11.22 - 16 40 1

19 82 18 10.54 27 10 02 1 10.54 11.06 96 - - - - 10 -

20 35 14 11.08 37 12 57 2 - - - 11.22 11.37 - 14 26 1

21 82 18 11.26 99 32 46 1 11.26 11.58 20 - - - - 32 -

22 63 16 11.42 94 35 66 2 - - - 11.42 12.17 5 - 35 -

23 03 9 11.51 12 8 09 1 11.58 12.06 - - - - 7 15 1

24 90 19 12.10 56 16 21 1 12.10 12.26 4 - - - - 16 -

25 39 14 12.24 51 20 48 1 12.26 12.42 - - - - 2 22 1

26 77 17 12.41 80 20 52 2 - - - 12.41 1.06 24 - 20 -


27 88 19 1.00 66 20 91 2 - - - 1.06 1.26 - 6 26 1

28 63 16 1.16 03 4 02 1 1.16 1.20 34 - - - - 4 -

29 47 15 1.31 69 20 80 2 - - - 1.31 1.36 5 - 20 -

30 92 20 1.51 72 20 97 2 - - - 1.51 2.16 15 - 20 -

31 57 16 2.07 24 12 67 2 - - - 2.16 2.26 - 9 21 1

32 23 13 2.20 43 16 37 1 2.20 2.36 60 - - - - 16 --

33 17 12 2.32 0 5 92 2 - - - 2.32 2.37 6 - 5 -

34 56 15 2.47 07 4 55 2 - - - 2.47 2.52 10 - 4 -

35 38 14 3.01 83 24 28 1 3.01 3.25 25 - - - - 24 -

36 55 15 3.16 63 20 80 2 - - - 3.16 3.36 24 - 20 -

37 01 8 3.24 86 24 28 1 3.25 3.29 - - - - 1 25 1

38 41 14 3.38 39 15 80 2 - - - 3.38 3.53 2 - 15 -

39 03 9 3.47 46 16 52 2 - - - 3.53 4.08 - 6 22 1

40 14 12 3.59 30 15 06 1 3.59 4.11 30 - - - - 15 -

41 69 16 4.15 10 8 61 2 - - - 4.15 4.20 7 - 8 -

42 28 13 4.28 02 4 29 1 4.28 4.32 17 - - - - 4 -

43 98 22 4.50 19 12 52 2 - - - 4.50 5.00 30 - 12 -

44 98 22 5.12 07 4 78 2 - - - 5.12 5.17 12 - 4 -

45 88 19 5.31 45 16 70 2 - - - 5.31 5.46 14 - 16 -

46 89 19 5.50 13 8 88 2 - - - 5.50 6.00 4 - 8 -

47 90 19 6.09 31 12 10 1 6.09 6.21 97 - - - - 12 -

48 91 19 6.28 03 4 84 2 - - - 6.28 6.33 28 - 4 -

49 75 17 6.45 12 8 24 1 6.45 6.53 24 - - - - 8 -

50 57 16 7.01 12 8 15 1 7.01 7.09 8 - - - - 8 -

Total 781 - - 753 - - - - 521 - - 268 109 862 13

Simulation Calculation

Average Arrival Time = 15.62 hr

Average Service Time = 15.06 min

Average Waiting Time of a Patient in Queue = 2.18 𝑚𝑖𝑛

Average Waiting Time of a Patient in a hospital = 17.24

Average Number of Patients in the Queue = 0.26

Average Idle Time of doctor 1 = 7.08

Average Idle Time of doctor 2 = 10.02

Analytical Calculation

Average Arrival time = 15.52 hr

Average Service time of doctor 1 = 13.96 min

Average Service Time of doctor 2 = 16.1 min

Average Waiting Time of a Patient in Queue = 1.66

Average Waiting Time of a Patient in first doctor’s clinic= 15.6𝑚𝑖𝑛

Average Waiting Time of a Patient in second doctor’s clinic = 17.76 𝑚𝑖𝑛

Number of Patients in the Queue = 0.1

Number of Patients in first doctor’s clinic = 1

Number of Patients in second doctor’s clinic = 1


4. Numerical Study

Comparison of Arrival and Service in Simulation and Analytical method

0

2

4

6

8

10

12

14

16

18

λ Ls

Simulationmethod

Analyticalmethod

14.5

15

15.5

16

16.5

17

17.5

λ Ws

Simulationmethod

Analyticalmethod

0

5

10

15

20

μ1 Ws

Simulationmethod

Analyticalmethod


CONCLUSION In this paper, we have presented a simulation table for queueing system with a

multiple service station. Here, we have developed a model for a multi-speciality

hospital and to decide whether two doctors are enough or to increase the number of

doctors in future. Numerical examples illustrate that analytical and simulation

methods are almost same. This has given the feasibility of the system. The main

purpose of this study is to develop an efficient procedure in a hospital for the future.

REFERENCES

[1] Syed Shujauddin Sameer (2014), “Simulation: Analysis of Single Server

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13.5

14

14.5

15

15.5

16

16.5

17

17.5

μ2 Ws

Simulationmethod

Analyticalmethod

0

2

4

6

8

10

12

14

16

18

20

Lq Ls Ws Wq

Simulationmethod

Analyticalmethod


[2] Erlang,A.K (1909), “The Theory of Probabilities of Telephone

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[3] Banks,J., Carson,J.S., Nelson,B.L., and Nicol,D.M(2001), “Discrete Event

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[6] Shanmugasundaram,S. and Punitha,S (2014), “A Study on Multi Server

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[7] Wijewickrama,A.K.A (2006), “Simulation Analysis for Reducing Queues in

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a study on m/m/c queueing model under monte carlo …€¦ · · 2016-11-05a study on m/m/c...

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