International Journal of Science and Research (IJSR)
ISSN (Online): 2319-7064
Impact Factor (2012): 3.358
On Markovian Queuing Models
Tonui Benard C.1, Langat Reuben C.2, Gichengo Joel M.3
1, 2., 3
University of Kabianga, Mathematics and Computer Science Department, P.O Box 2030-20200 Kericho, Kenya
Abstract: The ultimate objective of the analysis of queuing systems is to understand the behaviour of their underlying process so that
informed and intelligent decisions can be made by the management. The application of queuing concepts is an attempt to minimize cost
through minimization of inefficiency and delays in a system. Various methods of solving queuing problems have been proposed. In this
study we have explored single –server Markovian queuing model with both interarrival and service times following exponential
distribution with parameters and , respectively, and unlimited queue size with FIFO queuing discipline and unlimited customer
population. We apply this model to catering data and estimate parameters for the same. A sensitivity analysis is the carried out to
evaluate stability of the system.
Keywords: Queuing models, Markovian models, Single-server, Interarrival times, Service times, Sensitivity analysis.
1. Introduction
also an additional default assumption: inter-arrival and
service times are independent.
Queuing theory is a branch of applied probability theory
used to describe the more specialized mathematical models
for waiting lines or queues. It uses Queuing models to
represent the various types of Queuing systems that arise in
practice. The models enable finding an appropriate balance
between the cost of service and the amount of waiting. The
concept of Queuing theory has been developed largely in the
context of telephone traffic engineering originated by A. K.
Erlang in 1909. Queuing models find applications in a wide
variety of situations that may be encountered in health care,
engineering, and operations research (Gross and Harris,
1998). Queuing systems are comprised of customer(s)
waiting for service and server(s) who serve the customer.
They are frequently observed in some areas of day-to-day
life, for example:
1)
2)
3)
4)
5)
6)
People waiting at the check-in counter of an airport
Aeroplanes arriving in an airport for landing
Online train ticket reservation system
People waiting to be served at a buffet
Customers waiting at a barber shop for a hair cut
Sequence of emails awaiting processing in a mail server
Queues are usually characterized by the arrival pattern
(Poisson, deterministic or a general distribution), Service
pattern (constant, exponential, hyper exponential, hypoexponential or general distribution), number of servers
(single server or multiple servers), maximum system capacity
(number of customers in a system can range from one to
infinity), population size (queue can have infinite or finite
length) and queue discipline ( order of service delivery can
be First In First Out (FIFO), random order, Last In First Out
(LIFO) or priorities), see Zukerman (2013) and Adan Resing
(2002).
To incorporate these features, Kendall (1953) introduced a
Queuing notation A/B/C/X/Y/Z in where: A is the interarrival
time distribution, B is the service time distribution, C is the
number of servers, X is the system capacity, Y is the
population size and Z is the queue discipline.
In this study, an infinite customer population and service in
the order of arrival (FIFO) are default assumptions. There is
Paper ID: OCT14788
2. Basic Markovian Queuing Models
In Markovian models, the analysis is conducted using the
memoryless property of exponential distribution
2.1 Markovian Single-Server Models
Queuing System: M stands for Markovian or
memoryless. The first M denotes arrivals following a
Poisson process, the second M denotes service time
following exponential distribution, 1 refers to a single
server and refers to infinite system capacity.
Queuing System: This system is a type of
2)
queue with at most N customers allowed in
the system.
1)
2.2 Markovian Multiserver Models
Queuing system: This is a Markovian
Queuing model with C number of servers
Loss system: This is also known as the Erlang
2)
loss system and its system state follows a truncated
Poisson distribution.
Finite –Capacity Queuing system: In this
3)
Queuing model, the system has a finite capacity of size K
and we assume that c < K.
Queuing system: This is a Markovian Queuing
4)
model without any queue. There are infinitely many
servers such that every incoming customer finds an idle
server immediately.
1)
Queuing models play an essential role for business process
re-engineering purposes in administrative tasks. “Queuing
models provide the analyst with a powerful tool for
designing and evaluating the performance of Queuing
systems.” (Banks, Carson, Nelson & Nicol, 2001).
Sometimes, inefficiencies in services also occur due to an
undue wait in service may be because of new employee.
Delays in service jobs beyond their due time may result in
losing future business opportunities.
Volume 3 Issue 11, November 2014
www.ijsr.net
Licensed Under Creative Commons Attribution CC BY
93
International Journal of Science and Research (IJSR)
ISSN (Online): 2319-7064
Impact Factor (2012): 3.358
Detailed discussion on Markovian models are found in
Castaneda et al. (2012). In this study we concentrate on the
Queuing System.
3.
The steady state probabilities are given by:
Queuing System
The
or simply M/M/1 Queuing system
describes a Queuing system with both interarrival and
service times following exponential distribution with
parameters and , respectively, one server, unlimited queue
size with FIFO Queuing discipline and unlimited customer
population. The M/M/1 is one of the earliest systems to be
analyzed. As it is neatly described by Chee-Hock and BoonHee (2008), the single-server queue is a place where
customers arrive individually to obtain service from a service
facility. The service facility contains one server that can
serve one customer at a time. If the server is idle, the
customer is served immediately. Otherwise, the arriving
customer joins a waiting queue. This customer will receive
his service later, either when he reaches the head of the
waiting queue or according to some service discipline. When
the server has completed serving a customer, the customer
departs.
Theorem 1: Let
be a random variable denoting the
Queuing system at any
number of customers in the
time t.
Define:
(1)
Note 1: Equation (1) represents the probability mass
function of a discrete random variable denoting the number
of customers in the system in the long run. Clearly this
distribution follows a geometric distribution with parameter
1− ρ.
Theorem 2: Let
be the average number of customers in
be the average number of
the M/M/l Queuing system and
customers in the queue. Then:
Proof: Using (1), the mean number of customers can be
found. Note that the number of customers in the system is the
sum of the number of customers in the queue and the number
of customers in service.
Hence:
=
Ls
∞
∞
nP
∑=
∑ n (1 − ρ ) ρ
n
=
n 1=
n 1
(2)
n
This yields;
Let
. When
are given by:
, the steady state probabilities
Pn =
lim P( X t =
n) =
(1 − ρ ) ρ n , n =
0,1,2,
t →∞
Proof: The stochastic process
in
Queuing system can be modeled by a birth-and-death process
with birth states
and death states
The steady state balance equations can be obtained are given
by:
As expected, these equations show that with increasing load,
, the mean number of customers in the system
i.e., as
grows and the probability of an idle system decreases.
Similarly, the average number of customers in the queue can
be computed as:
=
Lq
∞
∑
n =1
( n − 1 ) Pn
Solving the above equations, we get:
So that:
To obtain the value of
, we use the fact that
Note 2: The probability that the server is busy is another
performance measure of the Queuing system. The probability
that the server is busy when the system is in equilibrium is
known as the utilization factor (traffic intensity) and is
denoted by .
queue,
For the
(3)
ρ = 1 − P0
Hence, when
,
Paper ID: OCT14788
The number of customers in the system is of importance
from the management's perspective and interest. Besides, the
average queue size and average system size are also
important parameters that represent the quality of service.
Volume 3 Issue 11, November 2014
www.ijsr.net
Licensed Under Creative Commons Attribution CC BY
94
International Journal of Science and Research (IJSR)
ISSN (Online): 2319-7064
Impact Factor (2012): 3.358
Two more measures important from the customer's point of
view are the average time spent in the system
and the
.
average time spent in the queue
Little (1961) derived the following formula, which gives the
relation between the average number of customers in the
system
and the average time spent in the system
and also between the average number of customers in the
queue
and
.
(4)
It is justified that, using the average time spent in the
system, , the average number of the customers during this
, where is the average number of arrivals per
time is
unit time. It is very important to note that no assumption is
made on the interarrival distribution, the service time
distribution and the queue discipline.
From (2) we have
(5)
for the
Queuing system.
The two-tailed t-test yielded a P-value of 0.5601 indicating
that there is no significant difference in the two means.
Similarly, for the inter-arrival time, the following hypothesis
was tested:
: Inter-arrival time day 1 = Inter-arrival time day 2 V/s
: Inter-arrival time day 1
Inter-arrival time day 2
The results are shown below.
It can be deduced that:
Rust (2008) said that the Little’s theorem can be useful in
quantifying
the maximum achievable operational
improvements and also to estimate the performance change
when the system is modified.
4. Research Design
The study was conducted in the University of Kabianga
campus mess. Two days data was collected using
observation method because the objective of the study didn’t
rely on opinion of the customers. Two variables of interest
namely inter-arrival times and service times were recorded.
The population studied included all the arrivals between
Noon and 1.00pm of the study period. Data emerging from
balking and/or reneging was disregarded. A pilot study was
done to test the reliability of research instrument and it was
validated. Two-sample t-test was used to test of equality of
means of inter-arrival times and service times. Windowsbased Quantitative Software for Business (WINQSB) suite
and STATA software were used for computation of the
above parameters, testing of hypothesis and performing
sensitivity analysis.
Since the P-value for the two-tailed t-test is far much beyond
the 0.05 index, we concluded that there was no significant
difference between inter-arrival times at 95% level of
confidence. Therefore, we decided to pick on 38.312 seconds
and 12.9186 seconds as the effective service time and the
effective inter-arrival time respectively.
5.2 Estimation of the Queue Parameters
Applying the data analysis tools as mentioned in section 3
above, we obtain:
5. Results
5.1 Testing of Hypotheses.
For the service time, the hypothesis tested was:
: service time day 1 = service time day 2
: service time day 1 service time day 2
V/s
The results are shown below.
Theoretically, the service rate was 1.5660 customers per
minute while the inter-arrival time was 4.6445 customers per
minute. This therefore implied that the system utilization
was:
Since the system utilization factor is greater than one, the
system is unstable for analysis. Hence the main task is to
Paper ID: OCT14788
Volume 3 Issue 11, November 2014
www.ijsr.net
Licensed Under Creative Commons Attribution CC BY
95
International Journal of Science and Research (IJSR)
ISSN (Online): 2319-7064
Impact Factor (2012): 3.358
perform sensitivity analysis by making reasonable changes to
the queue parameters so as make it reach steady state suitable
for analysis.
The system utilization factor ideally indicates the percentage
of time the servers are busy. Thus if it exceeds 100%, the
queue grows indefinitely. This causes dissatisfaction among
customers and may lead to possible losses due to
inefficiency.
5.3 Sensitivity Analysis
Two major possible parameter changes were made in an
attempt to stabilize the system. Changes were made on the;
1) Number of servers holding all other factors constant.
2) Service rate holding all other initial factors constant
This yielded the following results:
Table 1: Sensitivity analysis for number of servers
No. of Arrival Service
servers rate
rate
µ
λ
1
2
3
4
5
4.645
4.645
4.645
4.645
4.645
Ls Lq Ts Tq P0 Pw
1.566
1.566
1.566 0.989 87.92 84.95 18.93 18.29 0.002 0.98
1.566 0.742 4.39 1.42 0.94 0.31 0.04 0.49
1.566 0.593 3.30 0.33 0.71 0.07 0.05 0.23
The above findings indicate that at the current rates of arrival
and service, the Queuing system would stabilize for analysis
when there are at least three servers (or service points), with
the corresponding system utilization factors. This means that
probability of a customer waiting in the queue (Pw) when
there are three servers is 0.98, while the probability of the
system having no customer (P0) would be 0.002. A customer
would spend about 18.93 minutes in the entire system (TS)
while they have to wait to get to the server for an average of
18.29 minutes (Tq). Still at three servers, the service points
would be busy 98.86% of the time with the queue length
being an average of 88 customers including those being
served (Ls). This means that those who would be waiting for
service would be about 85.
Table 2: Sensitivity Analysis for the Service Rate
Arrival
rate λ
4.4665
4.4665
4.4665
4.4665
4.4665
4.4665
Ls Lq Ts Tq P0 Pw
Service
rate
µ
1.5660
2.5660
3.5660
4.5660
5.5660
6.5660
0.9782
0.8025
0.6802
44.89
4.06
2.13
43.91 10.05 9.83 0.02
3.26 0.91 0.73 0.20
1.45 0.48 0.32 0.32
0.98
0.80
0.68
have to wait for service for an average of 9.83 minutes with a
probability of 0.98 and would end up spending an average of
10.05 minutes in the entire system. The probability of the
queue having no customer would be 0.02.
6. Conclusion
From the research findings, it is evident that the mess at the
University of Kabianga main campus experiences an
unstable Queuing system (not statistically analyzable)
leading to numerous inefficiencies like infinite growth of
waiting line. This therefore implies that customers may be
dissatisfied and the probability of them pulling away from
the premises is high. This could see the enterprise plunge
into consequent losses.
However, we developed a sample sensitivity analysis scheme
that could assist the management to make necessary changes
so that the queue conforms to statistical standards. The
adoption of our findings could be beneficial to both the
customers (by reducing time spent in the system) and the
enterprise (by optimizing operations and task force). The
number of servers could be increased to at least three to
enhance an effective queue. Alternatively, the current
service rate can be increased to not less than 4.5660≈5
customers per minute.
References
[1] Adan I. and Resing J., (2002). Queuing Theory
.Eindhoven University of Technology, Eindhoven.
[2] Allen O. (1990), Probability, Statistics and Queuing
Theory with Computer Science Applications, 2nd Edition,
academic press, Boston
[3] Banks J; Carson J; Nelson B. L. and Nicole D. M. (2001).
Discrete Event System Simulation. Prentice Hall
International series, 3rd Edition.
[4] Castaneda, L. B., Arunachalam, V. and Dharmaraja, D
(2012). Introduction to Probability and Stochastic
Processes with Applications. John Wiley & Sons, Inc.
[5] Chee-Hock N. and Boon-Hee, (2008). Queuing Modeling
Fundamentals with Applications. 2nd Edition, John Wiley
and sons, England
[6] Jain R (2008). Lecture notes on Introduction to Queuing
Theory. Washington University in Saint Louis Saint
Louis.
[7] Little J. D. C (1961), a proof for Queuing formula,
operations research volume 9(3).
[8] Rust K.,(2008) “Using Little’s Law to estimate cycle time
and cost”, proceedings of the 2008 winter simulation
conference. IEEE press.
[9] Zukerman M. (2013), Introduction to Queuing Theory
and Stochastic Teletraffic Models. arXiv:1307.2968v3.
Reprint.
The above results were obtained after making an arbitrary
assumption of a 1 unit shift for the service rate. The system
stabilizes for analysis at a service rate of at least 4.5660
customers per minute. This implies that the service point
would be busy 97.82% of the time with about 44 customers
awaiting service. The entire system would have about 45
people including the one being served. A customer would
Paper ID: OCT14788
View publication stats
Volume 3 Issue 11, November 2014
www.ijsr.net
Licensed Under Creative Commons Attribution CC BY
96