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PUBLIC SERVICE SYSTEMS: PROBABILITIES AND OPTIMIZATION
Mamatova Zilolakhon Khabibullokhonovna
Associate Professor at Fergana State University
PhD in Pedagogical Sciences
ORCID
: 0009-0009-9247-3510
:
Eshmamatova Nozimakhon Saydulla qizi
3rd-year student of Applied Mathematics,
Group 22-08, Fergana State University
:
Annotation:
Queueing theory is a field of probabilistic systems that studies random demand
flows and service processes. This theory was developed based on A. Erlang’s research on
telephone exchanges and is applied in various fields such as telecommunication, commerce,
transportation, maintenance, and others. The main elements of the system include: incoming
flow, queue, service node, and outgoing flow. The incoming flow represents the process of
random demand arrivals, while the queue reflects the order in which services are awaited.
The service node may consist of one or multiple channels, and the outgoing flow represents
the served demands. The Poisson process is the most common type of incoming flow,
characterized by stationarity, ordinariness, and lack of aftereffects. Service time often follows
an exponential distribution. The efficiency of the system is assessed using indicators such as
the utilization coefficient, waiting time in queue, time spent in the system by requests, and
both relative and absolute throughput. Queueing systems can be open or closed, with or
without waiting, and single-channel or multi-channel. The theory plays an important role in
managing queues, optimizing resources, and improving service quality in practice.
Keywords:
Queueing system, demand, arrival flow, queue, service node, departure flow,
probabilistic system, stationary flow, ordinary flow, Poisson process, service time, transition
probability, utilization coefficient, average number of requests, average waiting time in queue,
average time a request spends in the system, relative throughput, absolute throughput.
Introduction.
Operations research and optimal control are fields of study focused on
decision-making and system optimization.
1. Operations research uses mathematical models, linear programming, game theory,
and network optimization methods to efficiently allocate resources.
2. Optimal control deals with determining the best control strategies for systems. Its
main methods include Pontryagin's Maximum Principle and Bellman's dynamic
programming.
Literature Review: Queuing Theory
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A literature review on Queuing Theory is essential for understanding the fundamental
principles, historical development, and practical applications of this field. Below, key literary
sources and their contents are briefly analyzed, along with an overview of the general
directions in current research.
A.K. Erlang’s Works
Danish mathematician Agner Krarup Erlang (1878–1929) is considered the founder of
queuing theory. His work on modeling traffic in telephone exchanges, particularly the 1917
paper “Solution of Some Problems in the Theory of Probabilities of Significance in
Automatic Telephone Exchanges”, significantly contributed to the field. Erlang's studies
developed methods for determining system congestion and queue length based on models
using Poisson arrivals and exponential service distributions. These works laid the foundation
for classical queuing system models such as M/M/1 and M/M/c, and they remain relevant
today in telecommunications and network analysis. Analysis: Erlang’s contributions are
fundamental in shaping the theoretical basis of queuing systems. However, his research
primarily focused on telephone networks, and thus requires extension for application to
modern complex systems.
L. Kleinrock’s “Queueing Systems” (1975–1976)
Leonard Kleinrock’s two-volume work “Queueing Systems” plays a crucial role in the
modern development of queuing theory. The first volume (Theory) covers the mathematical
foundations of queuing systems, including Poisson processes, Markov chains, and service
time distributions. The second volume (Computer Applications) focuses on the application of
the theory in computer networks and information systems. Kleinrock's work popularized
important analytical tools such as Little’s Law and Kendall’s Notation. Analysis: Kleinrock’s
book provides a balanced approach between theory and practical application, greatly
influencing the field of computer networks. However, its mathematical rigor may be
challenging for beginners.
D. Gross and C.M. Harris – “Fundamentals of Queueing Theory” (1974, later editions up to
2013)
This book is considered one of the most important introductory texts on queuing
theory. The authors explain models such as M/M/1, M/M/c, M/G/1, and also cover the
statistical characteristics of queuing systems, simulation methods, and practical examples.
Later editions of the book expand on modern issues, including the analysis of network
systems and call centers. Analysis: The book is reader-friendly and enriched with examples
and exercises. However, it mainly focuses on classical models and does not deeply explore
complex dynamic systems.
Research Methodology
The research methodology in queuing systems focuses on analyzing random processes,
queues, and service efficiency using mathematical modeling
,
statistical methods
,
and
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simulation
.
Below is a concise and meaningful analysis of research methodology suitable for
the topic “Queuing Systems: Probability and Optimization.”
Findings and Analysis
The emergence and development of queuing theory stem from the need to study
probabilistic systems arising in various fields of human activity. The formation of this theory
was significantly influenced by the work of Danish scientist A. Erlang, whose research on
improving telephone exchanges is of particular importance. Queuing theory uses requests
arriving at random time intervals to provide quantitative characteristics of a system and
develops methods for analyzing it. A queuing system involves both incoming requests and
the service-providing entity. A key feature of such systems is the random nature of demand
,
meaning that the timing and volume of incoming requests are unpredictable. Additionally, the
time required to service each request is also a random variable. As a result, a mismatch often
arises between the rate at which requests arrive and the speed at which they are serviced. This
leads to either the formation of queues or idle service facilities.
Examples of real-world queuing systems include:
Long-distance telephone exchange services;
Checkout counters in supermarkets;
Vehicle traffic at signal-controlled intersections;
Waiting lines in reception areas;
Aircraft awaiting take-off clearance at major airports;
Technicians servicing faulty equipment or machinery;
Manuscripts awaiting publication in a publishing house.
Basic Elements of Queuing Systems
The key structural components of queuing systems include:
1.
Incoming request stream
;
2.
Queue for service
;
3.
Service node
;
4.
Outgoing request stream
.
Based on these, a queuing system model can be constructed, illustrated in
Figure 1
.
Incoming Stream
The incoming stream represents the process of service-requesting entities entering the
system. Requests may arrive individually or in groups. Examples of individual arrivals
include ships entering a port, customers entering a store, or clients visiting a repair workshop.
Examples of group arrivals include freight train wagons arriving at a cargo station or diners
entering a restaurant for meals.
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In some cases, requests may leave the system without being serviced. This may occur
due to long waiting times, lack of space in the waiting area, or other reasons. If the system
has no space for waiting, it is referred to as a loss system (or queue-less system). If all
requests in the system are eventually serviced, it is called a queuing system
.
The source of requests may be finite or infinite. If the system receives requests from an
infinite source
,
it is known as an open queuing system
.
Queue Discipline
The order in which requests are serviced plays a major role in queuing systems. In
practice, the “First Come – First Served” (FCFS) discipline is most commonly used.
Sometimes, “Last Come – First Served” (LCFS) or “Random Order of Service” (ROS) may
also be applied. Queues may also be organized according to a priority rule, where some
requests are serviced before others based on predefined criteria.
Service Node
Once it is their turn, requests proceed to the service node
.
The service node may
consist of one or several service channels (servers). A system with only one server is called a
single-channel system
,
while systems with multiple parallel servers are referred to as multi-
channel systems. A typical example of a multi-channel system would be a large supermarket
with several checkout counters
.
If servicing occurs through several stages in sequence, using
different servers, the system is known as a multi-phase system
.
A typical example would be a
manufacturing process where a product undergoes processing in a sequence of workshops.
Output Stream
Requests that have been serviced form the output stream. This output stream may
serve as the input stream for another system, as seen in multi-phase systems. If serviced
requests return to the system, we have a closed queuing system. An example of this would be
………
input
strea
output
strea
queue
Service
node
Figure 1
………
………
………
………
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the maintenance of machine tools in a manufacturing plant, where repaired machines are
returned to the production line and may require servicing again later.
Poisson Process and Service Time
The study of queuing systems often begins with the analysis of the structure of the
input stream. The simplest type is a regular stream, where requests arrive at fixed time
intervals. However, regular input streams are rare in practice. In most real-life cases, the
interarrival times of requests are random. The probability distribution of these random
variables is a key characteristic of the input stream. Much of the research in queuing theory
has been conducted on input streams that follow a Poisson process, also known as a simple
arrival process, due to its frequent appearance in practical systems and the analytical
convenience it offers.
The Poisson process has three key properties:
1.
Stationarity
: The probability of a certain number of arrivals in a time interval
depends only on the length of the interval, not on its position on the time axis.
2.
Ordinariness
: In a sufficiently small time interval
t
D
, the probability of more than
one arrival is negligibly small, i.e., it becomes a higher-order infinitesimal.
3.
Lack of Aftereffects (Memorylessness)
: The probability of a certain number of
arrivals in a time interval does not depend on the number or timing of arrivals in
previous intervals. In other words, arrivals in disjoint time intervals are statistically
independent.
From these definitions, it is clear that:
Stationarity
implies the arrival process has time-invariant probabilistic
characteristics.
Ordinariness
ensures that multiple arrivals at exactly the same moment are
practically impossible.
Lack of Aftereffects
means that the arrival process in one time segment does not
influence the process in another.
Let us denote the probability that
k
requests arrive during a time interval of length
t
in a
Poisson process as
)
(
t
p
k
. For a Poisson arrival process:
t
k
k
e
k
t
t
p
l
l
-
=
!
)
(
)
(
(1)
The distribution is valid, where
0
>
l
is the flow parameter. This formula is known as
the Poisson formula.
To understand the meaning of the
l
parameter, let us find the expected value, i.e., the
average number of arrivals within the time interval
)
,
0
(
t
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=
-
=
=
-
=
=
=
=
0
1
1
!
)
(
!
)
(
)
(
)
(
s
s
t
k
k
t
k
k
t
s
t
t
e
e
k
t
k
t
kp
t
M
l
l
l
l
l
l
Naturally, if
1
=
t
, then
l
=
)
1
(
M
. This means that
l
represents the average number of
arrivals per unit of time.
Using the Poisson formula, we get:
)
(
)
(
1
t
o
t
t
p
D
+
D
=
D
l
(2)
Here,
)
(
t
o
D
is a quantity that becomes infinitesimally small compared to
t
D
in other words,
0
)
(
®
D
D
t
t
o
,
0
®
D
t
From (2), when
t
D
is sufficiently small, we obtain the relationship:
t
t
p
D
»
D
l
)
(
1
That is, in a sufficiently short time interval, the probability of an arrival is approximately
t
D
l
.
Let T denote the interarrival time between successive requests in the system. Then,
)
(
t
T
p
<
represents the probability that the arrival time of a new request in the system exceeds
t
D
. We
calculate this probability as:
=
=
-
-
-
=
=
=
<
1
1
1
!
)
(
)
(
)
(
k
k
t
t
k
k
e
e
k
t
t
P
t
T
p
l
l
l
.
(3)
Thus, equation (3) shows that if the arrival times of requests follow a Poisson process,
then the interarrival times follow the exponential distribution.
To analyze a queueing system using mathematical methods, the service time
distribution function of each service device in the node must be known.
Let
.
.
k
x
t
be the service time for a request, and
)
(
.
.
t
t
p
k
x
<
be the probability that the service
time does not exceed
t
D
. Then,
)
(
)
(
.
.
t
t
p
t
F
k
x
<
=
is the distribution function of the
service time.
The service time distribution can vary. In practice, the exponential distribution is often used:
,
0
,
1
)
(
>
-
=
-
m
m
t
e
t
F
(4)
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Here,
m
is called the service rate (intensity). To understand the meaning of this parameter,
we calculate the expected value of the service time:
-
=
=
=
=
0
0
.
.
.
.
1
)
(
)
(
m
m
m
dt
e
t
t
tdF
t
t
M
t
k
x
k
x
Thus, the
m
parameter represents the average number of requests the service device can
handle per unit of time.
Using the exponential distribution in (4), we can calculate the probability that a request will
be served within time
t
D
as:
t
к
х
e
t
t
p
t
F
D
-
-
=
D
<
=
D
m
1
)
(
)
(
.
.
.
)
(
)
(
.
.
t
t
p
t
F
k
x
<
=
using the Taylor series expansion of the exponential function, we get:
t
t
o
t
t
t
p
к
х
D
»
D
+
D
+
-
=
D
<
m
m
)
(
1
1
)
(
.
.
(5)
The probability that the service of a request will not be completed in time
t
D
is:
t
t
t
p
t
t
p
к
х
к
х
D
-
»
D
<
-
=
D
m
1
)
(
1
)
(
.
.
.
.
Example
Customers with a high likelihood of making a purchase arrive at a retail store. Suppose
we want to test the hypothesis that the arrival process of customers follows a Poisson
distribution. To do this, we perform a statistical analysis in order to determine and construct
the parameters of the empirical distribution function.
In building the mathematical model, we consider only those customers who are
certain to make a purchase. Assume that the arrival time of each customer within the time
interval T is fixed. We form a sample based on the number of customers who actually make
purchases.
According to the statistical method, we divide the observation time interval T into n
equal subintervals: ∆t = T/n
We compute the parameter using:
α ∙ ∆t = (
i=1
r
i ∙ m
i
)/(
i=1
r
m
i
)
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Here, r is the last interval for which customer data exists (i.e., there are no intervals beyond r
with customers). We then find the relative frequency Wi of observing i customers in
t
D
:
W
i
=
m
i
i=1
r
m
j
, i = 0,1,2, . . . , r.
Next, we calculate the theoretical probability
p
i
of having i arrivals in
t
D
based on the
Poisson process:
p
i
=
α∆t
i
i! e
−α∆t
, i = 0,1,2, …, r
Let’s suppose the observation time is 300 minutes. We divide this time interval into n=100
equal subintervals, so
t
D
=3 minutes.
The hypothesis that customer arrivals at the store follow a Poisson distribution is not
contradicted by the experimental data, and therefore, it can be accepted.
The obtained results are expressed in the following table format:
The number of
customers in the
t= 3 min interval.
The
number
of
intervals
with
i
customers, denoted as
mi.
The probability
value
of
the
Poisson
distribution,
denoted as pi.
Relative
frequency Wi
1
0
0
0.010
0
2
1
5
0.043
0.05
3
2
11
0.106
0.11
4
3
13
0.163
0.13
5
4
22
0.187
0.22
6
5
18
0.172
0.18
7
6
14
0.132
0.14
8
7
9
0.087
0.09
9
8
4
0.050
0.04
10
9
2
0.026
0.02
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11
10
1
0.012
0.01
12
11
1
0.005
0.01
13
12
0
0.002
0
Explanation
: In the example considered, the arrival rate of the process is given as α·∆t = 4, 6.
The unit time is taken as t= 3 minutes. Therefore, the arrival rate per minute is α = 4, 6/3 ≈
1,53 1/min per minute, assuming that the time unit is typically taken as 1 minute.
Conclusion
:
Mass service systems are an important theoretical field that analyzes random demand
flows and service processes, and they have been developed based on A. Erlang's research.
These systems consist of the incoming flow, queue, service node, and outgoing flow, and
they are characterized by probabilistic models such as Poisson flow and exponential service
times. Open, closed, waiting, or non-waiting systems are used in various fields, such as
communication, trade, transportation, and maintenance. The system's performance is
evaluated using metrics like the utilization factor, waiting time, and throughput capacity. The
theory plays a crucial role in efficiently managing queues, optimizing resource usage, and
improving service quality.
References
:
1. Erlang, A. K. (1917). Solution of Some Problems in the Theory of Probabilities of
Significance in Automatic Telephone Exchanges. Post Office Electrical Engineers’
Journal.
2. Kleinrock, L. (1975–1976). Queueing Systems, Volume 1: Theory & Volume 2:
Computer Applications. Wiley-Interscience.
3. Gross, D., & Harris, C. M. (2013). Fundamentals of Queueing Theory (5th ed.). Wiley.
4. Medhi, J. (2002). Stochastic Models in Queueing Theory (2nd ed.). Academic Press.
5. Cooper, R. B. (1981). Introduction to Queueing Theory (2nd ed.). North-Holland.
6. Bhat, U. N. (2015). An Introduction to Queueing Theory: Modeling and Analysis in
Applications (2nd ed.). Birkhäuser.
7. Taha, H. A. (2017). Operations Research: An Introduction (10th ed.). Pearson.
8. European Journal of Operational Research (Articles on queueing systems).
9. Queueing Systems (Articles on network queueing systems).
10. Scientific publications of Tashkent University of Information Technologies and
Samarkand State University (articles on network systems and logistics).
