Queueing Theory and its Application
Submitted by: Vineet Gupta (11/BBS/0244) Yatin Lekhi (11/BBS/0227)
To: Ms. Gurjeet Kaur Department of Business Studies DEEN DAYAL UPADHYAYA COLLEGE UNIVERSITY OF DELHI
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Declaration th
We, Vineet Gupta and Yatin Lekhi, students of Bachelor of Business Studies (4 semester),
in Deen Dayal Upadhyaya College, University of
Delhi, hereby
declare that we have made this academic project titled “Queueing Theory” as a part of the internal assessment for the subject “Production and Operations Management” , for academic year 2012-13. The project is submitted for the first time and here only and the information submitted therein is true to the best of my knowledge. k nowledge.
I sincerely thank Ms. Gurjeet Kaur and my friends for the help extended by them for the successful completion of the project pr oject report.
DATE: ----
Countersigned
Candidate’s signature
Vineet Gupta (11/BBS/0244)
Yatin Lekhi (11/BBS/0227)
(Ms. Gurjeet Kaur)
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Acknowledgements
i.
We would like to thank Ms. Gurjeet Kaur for her direction, assistance, and guidance.
ii.
Ms. Gurjeet’s r ecommendations and suggestions have been invaluable for the project.
iii.
Special thanks should be given to our peers who have helped us in many ways.
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Table of Contents
Topic
Page Number
Declaration
2
Acknowledgements
3
Introduction
5
Single Queueing Node
6-7
Application to Telephony
8-10
Queueing Networks
11-13
Utilization
14
Role of Poisson Process, Exponential Distributions
15-16
Limitations of Queueing Theory
17
Case Study for Restaurant Queuing Model
18-27
Bibliography
28
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Introduction
Queueing theory is the mathematical study of waiting lines, or queues. In queueing theory a model is constructed so that queue lengths and waiting times can be predicted. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide service.
Queueing theory started with research by Agner Krarup Erlang when he created models to describe the Copenhagen telephone exchange. The ideas have since seen applications including telecommunications, traffic engineering, computing and the design of factories, shops, offices and hospitals.
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Single Queueing Node
Single queueing nodes are usually described using Kendall's notation in the form A/ B/C where A describes the time between arrivals to the queue, B the size of jobs and C the number of servers at the node.
Agner Krarup Erlang, a Danish engineer who worked for the Copenhagen Telephone Exchange, published the first paper on what would now be called queueing theory in 1909. He modeled the number of telephone calls arriving at an exchange by a Poisson process and solved the M/D/1 queue in 1917 and M/D/k queueing model in 1920. In Kendall's notation
M stands for Markov and means arrivals occur according to a Poisson process
D stands for deterministic and means jobs arriving at the queue require a fixed amount of service
k describes the number of servers at the queueing node (k = 1, 2,...). If there are
more jobs at the node than there are servers then jobs will queue and wait for service.
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The M/M/1 queue is a simple model where a single server serves jobs that arrive according to a Poisson process and have exponentially distributed service requirements. In an M/G/1 queue the G stands for general and indicates an arbitrary probability distribution. The M/G/1 model was solved by Felix Pollaczek in 1930, a solution later recast in ‟ probabilistic terms by Aleksandr Khinchinand now known as the Pollaczek
Khinchine formula. After World War II queueing theory became an area of research interest to mathematicians.
Work on queueing theory used in modern packet switching networks was performed in the early 1960s by Leonard Kleinrock.
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Application to Telephony
The public switched telephone network (PSTN) is designed to accommodate the offered traffic intensity with only a small loss. The performance of loss systems is quantified by their grade of service, driven by the assumption that if sufficient capacity is not available, the call is refused and lost. Alternatively, overflow systems make use of alternative routes to divert calls via different paths † even these systems have a finite traffic carrying capacity.
However, the use of queueing in PSTNs allows the systems to queue their customers' requests until free resources become available. This means that if traffic intensity levels exceed available capacity, customer's calls are not lost; customers instead wait until they can be served. This method is used in queueing customers for the next available operator.
A queueing discipline determines the manner in which the exchange handles calls from customers. It defines the way they will be served, the order in which they are served, and the way in which resources are divided among the customers. Here are details of four queueing disciplines:
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First in first out
This principle states that customers are served one at a time and that the customer that has been waiting the longest is served first.
Last in first out
This principle also serves customers one at a time, however the customer with the shortest waiting time will be served first. Also known as a stack.
Processor sharing
Service capacity is shared equally between customers.
Priority
Customers with high priority are served first.
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Queueing is handled by control processes within exchanges, which can be modelled using
state
equations. Queueing
systems
use
a
particular
form
of state
equations known as a Markov chain that models the system in each state. Incoming traffic to these systems is modelled via a Poisson distribution and is subject to Erlang’s queueing theory assumptions viz.
Pure-chance traffic ‟ Call arrivals and departures are random and
independent events.
Statistical equilibrium ‟ Probabilities within the system do not
change.
Full availability ‟ All incoming traffic can be routed to any other
customer within the network.
Congestion is cleared as soon as servers are free.
Classic queueing theory involves complex calculations to determine waiting time, service time, server utilization and other metrics that are used to measure queueing performance.
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Queueing Networks
Networks of queues are systems a number of queues are connected by customer routing. When a customer is serviced at one node it can join another node and queue for service, or leave the network. For a network of m the state of the system can be described by an m ‟ dimensional vector ( x1 ,x2,..., xm) where xi represents the number of customers at each node. The first significant results in this area were Jackson networks, for which an efficient product-form stationary distribution exists and the mean value analysis which allows average metrics such as throughput and sojourn times to be computed.
If the total number of customers in the network remains constant the network is called a closed network and has also been shown to have a product ‟ form stationary distribution in the Gordon ‟ Newell theorem. This result was extended to the BCMP network where a network with very general service time, regimes and customer routing is shown to also exhibit a product-form stationary distribution.
Networks of customers have also been investigated; Kelly networks where customers of different classes experience different priority levels at different service nodes.
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Mean field limits
Mean
field
models consider
the
limiting
behaviour
of
the empirical
measure (proportion of queues in different states) as the number of queues ( m above) goes to infinity. The impact of other queues on any given queue in the network is approximated by a differential equation. The deterministic model converges to the same stationary distribution as the original model.
Fluid limits
Fluid models are continuous deterministic analogs of queueing networks obtained by taking the limit when the process is scaled in time and space, allowing heterogenous objects. This scaled trajectory converges to a deterministic equation which allows us stability of the system to be proven. It is known that a queueing network can be stable, but have an unstable fluid limit.
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Heavy traffic/diffusion approximations
In a system with high occupancy rates (utilisation near 1) a heavy traffic approximation can be used to approximate the queueing length process by a reflected Brownian
motion, Ornstein ‟ Uhlenbeck
process or
more
general diffusion
process. The number of dimensions of the RBM is equal to the number of queueing nodes and the diffusion is restricted to the non-negative orthant.
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Utilization
Utilization is the proportion of the system's resources which is used by the traffic which arrives at it. It should be strictly less than one for the system to function well. It is usually represented by the symbol . If
then the queue will continue to
grow as time goes on. In the simplest case of an M/M/1 queue (Poisson arrivals and a single Poisson server) then it is given by the mean arrival rate over the mean service rate, that is,
where
is the mean arrival rate and
where
is the mean arrival rate,
is the mean service rate. More generally:
is the mean service rate, and c is the number of
servers, such as in an M/M/c queue.
In general, a lower utilization corresponds to less queueing for customers but means that the system is more idle, which may be considered inefficient.
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Role of Poisson Process, Exponential Distributions
A useful queueing model represents a real-life system with sufficient accuracy and is analytically tractable. A queueing model based on the Poisson process and its companion exponential probability distribution often meets these two requirements.
A Poisson process models random events (such as a customer arrival, a request for action from a web server, or the completion of the actions requested of a web server) as emanating from a memoryless process. That is, the length of the time interval from the current time to the occurrence of the next event does not depend upon the time of occurrence of the last event. In the Poisson probability distribution, the observer records the number of events that occur in a time interval of fixed length. In the (negative) exponential probability distribution, the observer records the length of the time interval between consecutive events. In both, the underlying physical process is memoryless.
Models based on the Poisson process often respond to inputs from the environment in a manner that mimics the response of the system being modeled to those same inputs. The analytically tractable models that result yield both information about the system being modeled and the form of their solution.
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Even a queueing model based on the Poisson process that does a relatively poor job of mimicking detailed system performance can be useful. The fact that such models often give "worst-case" scenario evaluations appeals to system designers who prefer to include a safety factor in their designs. Also, the form of the solution of models based on the Poisson process often provides insight into the form of the solution to a queueing problem whose detailed behavior is poorly mimicked. As a result, queueing models are
frequently
modeled
as Poisson
the exponential distribution.
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processes through
the
use
of
Limitations of Queueing Theory
The assumptions of classical queueing theory may be too restrictive to be able to model real-world situations exactly. The complexity of production lines with product-specific characteristics cannot be handled with those models. Therefore specialized tools have been developed to simulate, analyze, visualize and optimize time dynamic queueing line behavior.
For example; the mathematical models often assume infinite numbers of customers, infinite queue capacity, or no bounds on inter-arrival or service times, when it is quite apparent that these bounds must exist in reality. Often, although the bounds do exist, they can be safely ignored because the differences between the real-world and theory is not statistically significant, as the probability that such boundary situations might occur is remote compared to the expected normal situation. Furthermore, several studies show the robustness of queueing models outside their assumptions. In other cases the theoretical solution may either prove intractable or insufficiently informative to be useful.
Alternative means of analysis have thus been devised in order to provide some insight into problems that do not fall under the scope of queueing theory, although they
are
often
scenario-specific
because
they
computer simulations or analysis of experimental data.
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generally
consist
of
Case Study for Restaurant Queuing Model
Abstract
—
Restaurants would avoid losing their customers due to a long wait on the line. Some restaurants initially provide more waiting chairs than they actually need to put them in the safe side, and reducing the chairs as the time goes on safe space. However, waiting chairs alone would not solve a problem when customers withdraw and go to the competitor’s door; the service time may need to be improved. This shows a need
of a numerical model for the restaurant management to understand the situation better. This paper aims to show that queuing theory satisfies the model when tested with a real-case scenario. We obtained the data from a restaurant in Jakarta. We then derive the arrival rate, service rate, utilization rate, waiting time in queue and the probability of potential customers to balk based on the data using Little’s Theorem
and M/M/1 queuing model. The arrival rate at Sushi Tei during its busiest period of the day is 2.22 customers per minute (cpm) while the service rate is 2.24 cpm. The average number of customers in the restaurant is 122 and the utilization period is 0.991. We conclude the paper by discussing the benefits of performing queuing analysis to a busy restaurant.
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I.
INTRODUCTION
There are several determining factors for a restaurant to be considered a good or a bad one. Taste, cleanliness, the restaurant layout and settings are some of the most important factors. These factors, when managed carefully, will be able to attract plenty of customers. However, there is also another factor that needs to be considered especially when the restaurant has already succeeded in attracting customers. This factor is the customers queuing time. Queuing theory is the study of queue or waiting lines. Some of the analysis that can be derived using queuing theory include the expected waiting time in the queue, the average time in the system, the expected queue length, the expected number of customers served at one time, the probability of balking customers, as well as the probability of the system to be in certain states, such as empty or full. Waiting lines are a common sight in restaurants especially during lunch and dinner time. Hence, queuing theory is suitable to be applied in a restaurant setting since it has an associated queue or waiting line where customers who cannot be served immediately have to queue (wait) for service. Researchers have previously used queuing theory to model the restaurant operation, reduce cycle time in a busy fast food restaurant, as well as to increase throughput and efficiency.
This paper uses queuing theory to study the waiting lines in Sushi Tei Restaurant at Senayan City, Jakarta. The restaurant provides 20 tables of 6 people. There are 8 to 9 waiters or waitresses working at any one time. On a daily basis, it serves over 400 customers during weekdays, and over 1000 customers during weekends. This paper seeks to illustrate the usefulness of applying queuing theory in a real-case situation.
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II.
SUSHI TEI QUEUING MODEL
The data were obtained from Sushi Tei through interview with the restaurant manager as well as data collections through observations at the restaurant. The daily number of visitors was obtained from the restaurant itself. The restaurant has been recording the data as part of its end of day routine. We also interviewed the restaurant manager to find out about the capacity of the restaurant, the number of waiters and waitresses, as well as the number of chefs in the restaurant. Based on the interview with the restaurant manager, we concluded that the queuing model that best illustrate the operation of Sushi Tei is M/M/1. This means that the arrival and service time are exponentially distributed (Poisson process). The restaurant system consists of only one server. In our observation the restaurant has several waitresses but in the actual waiting queue, they only have one chef to serve all of the customers. Figure 1 illustrates the M/M/1 queuing model.
Figure 1. M/M/1 Queuing Model For the analysis of the Sushi Tei M/M/1 queuing model, the following variables will be investigated:
λ : The mean customers arrival rate µ : The mean service rate
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„
ρ : λ /µ : utilization factor
„
Probability of zero customers in the restaurant: P 0 = 1 ρ
–
„
Pn : The probability of having n customers in the restaurant. n
P n = P 0 ρ = (1 ρ ) ρ
n
–
L: average number of customers dining in the restaurant.
Lq: average number in the queue.
„
W: average time spent in Sushi Tei, including the waiting time.
Wq : average waiting time in the queue.
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III.
RESULT AND DISCUSSION
The one month daily customer data were shared by the restaurant manager as shown in Table I.
TABLE I.
MONTHLY CUSTOMER COUNTS Mon st
1 Week 470
Tue
Wed
Thu
Fri
Sat
Sun
427
429
492
663
973
1092
2
nd
Week 456
445
536
489
597
1115 1066
3
rd
Week 421
541
577
679
918
1319 1212
4 Week 494
559
581
613
697
1188 1113
th
Figure 2. One month daily customer counts
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As can be seen in Figure 2, the number of customers on Saturdays and Sundays are double the number of customers during weekdays. The busiest period for the restaurant is on weekend during dinner time. Hence, we will focus our analysis in this time window.
A. Calculation
Our teams conducted the research at dinner time. There are on average 400 people are coming to the restaurant in 3 hours’ time window of dinner time. From this we
can derive the arrival rate as:
We also found out from observation and discussion with manager that each customer spends 55 minutes on average in the restaurant (W ), the queue length is around 36 people ( Lq) on average and the waiting time is around 15 minutes. It can be shown using (7) that the observed actual waiting time does not differ by much when compared to the theoretical waiting time as shown below.
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Next, we will calculate the average number of people in the restaurant.
Having calculated the average number of customers in the restaurant, we can also derive the utilization rate and the service rate.
With the very high utilization rate of 0.991 during dinner time, the probability of zero customers in the restaurant is very small as can be derived.
The generic formula that can be used to calculate the probability of having n customers in the restaurant is as follows:
We assume that potential customers will start to balk when they see more than 10 people are already queuing for the restaurant. We also assume that the maximum queue length that a potential customer can tolerate is 40 people. As the capacity of the restaurant when fully occupied is 120 people, we can calculate the probability of 10 people in the queue as the probability when there are 130 people in the system (i.e. 120 in the restaurant and 10 or more queuing) as follows:
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Probability of customers going away = P (more than 15 people in the queue) = P (more than 130) people in the restaurant)
B. Evaluation
The utilization is directly proportional with the mean number of customers. It means that the mean number of customers will increase as the utilization increases. The utilization rate at the restaurant is very high at 0.991. This, however, is only the utilization rate during lunch and dinner time on Saturdays and Sundays. On weekday, the utilization rate is almost half of it. This is because the number of visitors on weekdays is only half of the number of visitors on weekends. In addition, the number of waiters or waitresses remains the same regardless whether it is peak hours or off-peak hours. In case the customers waiting time is lower or in other words we waited for less than 15 minutes, the number of customers that are able to be served per minute will increase. When the service rate is higher the utilization will be lower, which makes the probability of the customers going away decreases.
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C. Benefits
This research can help Sushi Tei to increase their QoS (Quality of Service), by anticipating if there are many customers in the queue. The result of this paper work may become the reference to analyze the current system and improve the next system. Because the restaurant can now estimate of how many customers will wait in the queue and the number of customers that will go away each day. By anticipating the huge number of customers coming and going in a day, the restaurant can set a target profit that should be achieved daily. The formulas that were used during the completion of the research is applicable for future research and also could be used to develop more complex theories. The formulas provide mechanism to model the restaurant queue that is simpler than the creation of simulation model in.
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IV.
CONCLUSION
This research paper has discussed the application of queueing theory of Sushi Tei Restaurant. Here we have focused on two particularly common decision variables (as a vehicle for introducing and illustrating all the concepts. From the result we have obtained that the rate at which customers arrive in the queuing system is 2.22 customers per minute and the service rate is 2.24 customers per minute. The probability of buffer flow if there are 10 or more customers in the queue is 15 out of 100 potential customers. The probability of buffer overflow is the probability that customers will run away, because may be they are impatient to wait in the queue. This theory is also applicable for the restaurant if they want to calculate all the data daily. It can be concluded that the arrival rate will be lesser and the service rate will be greater if it is on weekdays since the average number of customers is less as compared to those on weekends. The constraints that were faced for the completion of this research were the inaccuracy of result since some of the data that we use was just based on assumption or approximation. We hope that this research can contribute to the betterment of SushiTei restaurant in terms of its way of dealing with customers.
As our future works, we will develop a simulation model for the restaurant. By developing a simulation model we will be able to confirm the results of the analytical model that we develop in this paper. In addition, a simulation model allows us to add more complexity so that the model can mirror the actual operation of the restaurant more closely.
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Bibliography 1) http://en.wikipedia.org/wiki/Queueing_theory 2) http://www.ipedr.com
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