=Paper=
{{Paper
|id=Vol-2874/paper24
|storemode=property
|title=Performance Analysis of Two-Way Communication Retrial Queueing Systems With Non-Reliable Server and Impatient Customers in the Orbit
|pdfUrl=https://ceur-ws.org/Vol-2874/paper24.pdf
|volume=Vol-2874
|authors=Ádám Tóth,János Sztrik
}}
==Performance Analysis of Two-Way Communication Retrial Queueing Systems With Non-Reliable Server and Impatient Customers in the Orbit==
https://ceur-ws.org/Vol-2874/paper24.pdf
Performance Analysis of Two-Way
Communication Retrial Queueing Systems
With Non-Reliable Server and Impatient
Customers in the Orbit∗
Ádám Tóth, János Sztrik
Faculty of Informatics, University of Debrecen, Debrecen, Hungary
toth.adam@inf.unideb.hu
sztrik.janos@inf.unideb.hu
Proceedings of the 1st Conference on Information Technology and Data Science
Debrecen, Hungary, November 6–8, 2020
published at http://ceur-ws.org
Abstract
Many models of two-way communication queueing systems have been
studied in recent years, they can be utilized in many fields of life like in
[7, 28, 30]. Customers have always been characterized by the phenomena of
impatience due to the long waiting time for service [4, 14, 15, 27]. In this pa-
per, we consider two-way communication systems with a non-reliable server
where primary customers may decide to leave the system after spending a
considerable amount of time in the system before getting its proper service.
The service unit can break down during its operation or in an idle state,
too. Whenever the server becomes idle it may generate requests towards the
customers’ residing in an infinite source. These requests, the so-called sec-
ondary customers, can enter the system after a random time if the service
unit is available and functional upon their arrivals. Otherwise, they return
to the source without coming into the system. Every primary customer has a
property of impatience meaning that an arbitrary request has the ability to
Copyright © 2021 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
∗ The research work of Ádám Tóth, János Sztrik was supported by the construction EFOP-
3.6.3-VEKOP-16-2017-00002. The project was supported by the European Union, co-financed by
the European Social Fund
246
quit the system after some time while its demand remains unsatisfied. Dur-
ing server failure, every individual may generate requests but these will be
forwarded immediately towards the orbit. The source, service, retrial, impa-
tience, operation, and repair times are supposed to be independent of each
other.
Keywords: Queueing, impatience, two-way communication system, abandon-
ment, finite-source, stochastic simulation, sensitivity analysis
1. Introduction
Nowadays due to the rapid technology development and the increase of traffic
growth, the study of communication systems is crucial and inevitable especially
in the topic of optimization. Not just companies or associations have some kind
of networking infrastructure but our homes as well, which results in a load of
communication processes. Therefore, the creation and modelling of new or ex-
isting telecommunication systems are needed. With the help of queuing systems
with repeated calls are perfectly useful for modelling such arising issues in main
telecommunication systems, such as telephone switching systems, call centers, or
computer systems. In the last years, a great number of articles dealt with such
scenarios like in [3, 9].
The fundamental characteristic of retrial queueing systems is that when every
service unit is busy an incoming customer remains in the system in a virtual waiting
room called the orbit. The customers who are located in the orbit try to enter the
service unit after a random time.
Impatience is a natural phenomenon and models with this property are nearer to
reality and lead us to more precise analysis. Because of the relatively great number
of situations that can occur in healthcare applications, call centers, telecommuni-
cation networks, it is no wonder that many papers were devoted to examining the
effect of impatience like [13, 18, 25]. Impatience can be interpreted in different
ways: balking customers decide not to join the queue if it is too long, jockeying
customers can move from its queue to another queue if they detect they will get
served faster, and reneging customers leave the queue if they have waited a definite
time for service. In our investigated model customers have a reneging feature.
In connection with the communication systems users (or sources) typically are
coerced to fight for the available channels or facilities. The possibility of conflict
is relatively high when several sources initiate random attempts on the channel
producing collisions and the loss of transmissions. Developing efficient methods is
important for avoiding such collisions and corresponding message delay. The effect
of collisions have been published in numerous papers for example in [19–22, 24].
The examination of the availability of the service unit is vital because a lot of
studies suppose that the service unit never fails and it is accessible on a permanent
basis. But these assumptions are quite unrealistic and in practice, on many occa-
sions, we find the service stations in failed state. This unfortunate behaviour has
a considerable influence on the system’s characteristics as well on the performance
247
measures thus it is worth investigating such systems with server breakdowns like
in the following papers [8, 12, 16, 29, 32].
Recently in a relatively high number of papers make an effort to study two-way
communication designs thanks to their usefulness to model real-life examples in the
various application field. This is especially true for call centers where the service
unit not only processes incoming calls but also operates in idle state executing
certain other works like advertising and promoting products. In such systems
utilization of the service unit is always pivotal, see for example in [1, 2, 5, 7, 11,
17, 26, 31].
The novelty of the present paper is to achieve a sensitivity analysis using var-
ious distributions of service time of customers on the performance measures like
the mean waiting time of an arbitrary, successfully served and impatient customer,
the total utilization of the service unit, the probability of abandonment, etc. To
compare the effect of the different distributions on distinct metrics a stochastic
simulation program is developed based on SimPack. In this collection, you may
perform various algorithms in connection with discrete event simulation, continu-
ous simulation, and combined (multi-model) simulation. So we built a simulation
model to implement every feature of the system and to calculate and estimate the
desired measure using various values of input parameters. The obtained results
demonstrate the importance of utilized distribution under different parameter set-
tings represented by numerous figures and highlight some interesting specialties of
these types of systems.
2. System Model
In this paper, a two-way communication retrial queuing model is considered with
a non-reliable server (Figure 1). In this model, 𝑁 customer populates the finite-
source implying that the system will be stable at every moment. Each customer
generates a request (primary customers) towards the server with rate 𝜆/𝑁 , so the
inter-request time follows an exponential distribution with parameter 𝜆/𝑁 . Our
model does not have a queue at all, thus in the case of an idle server the service of
a primary customer starts immediately. The service time of these requests follows
gamma, hypo-exponential, hyper-exponential, Pareto, and lognormal distribution
with different parameters but with the same mean value. After departing from
the system, successful requests go back to the finite-source. When the server is
occupied, the incoming primary customers are forwarded to the orbit where they
may retry to attain the service unit after an exponentially distributed random
time with parameter 𝜎/𝑁 . It is assumed that server failures take place during its
operation or in idle state according to exponential distribution time with the rate
𝛾0 when it is busy and with rate 𝛾1 if it is idle. After server breakdown the repair
process begins right away and the repair time of the server is also an exponentially
distributed random variable with parameter 𝛾2 .
Every primary customer is characterized by an impatience property and in our
investigated model two different types are distinguished:
248
1. a primary customer after waiting for some time in the orbit leaves the system
without receiving its appropriate service,
2. a primary customer after waiting for some time in the system leaves the
system without receiving its appropriate service.
This decision is made after a random time which is exponential with rate 𝜏 .
After the server is becoming idle it has the possibility to produce outgoing calls
towards the customers (secondary) from an infinite source which is performed after
an exponentially distributed period with parameter 𝛾. The requirement of the
service of these customers is that the server will not be in a failed or busy state
upon their arrivals, otherwise, they are cancelled and turn back to the infinite
source without entering the system. The service time of this type of customer
follows a gamma distribution with parameters 𝛼2 and 𝛽2 . Whenever the server
breaks down during the service of a customer the primary ones are forwarded
immediately towards the orbit and the secondary ones depart the system without
continuing their service.
Figure 1. System model.
3. Simulation Results
In Table 1 the various values of input parameters are shown for the simulation.
A statistics package is utilized to estimate the desired performance measures in
our program which was developed by Andrea Francini in [10]. This code uses the
method of batch means to collect a certain number of independent samples (batch
means) by amassing consecutive 𝑛 observations of a steady-state simulation. It is
one of the most widespread and common methods to define a confidence interval
for the steady-state mean of a process. To have the sample averages approximately
independent the size of batches should be carefully selected. By calculating the
average of the sample averages of each batch we obtain the final mean value. About
this technique you can find detailed information in the following works [6, 23]. The
249
simulations are performed with a confidence level of 99.9%. The relative half-width
of the confidence interval required to stop the simulation run is 0.00001.
3.1. First Scenario
Achieving our objective we use hyper-exponential, gamma, lognormal and Pareto
distributions to investigate how these distributions of service time of primary cus-
tomers alter the performance measures. In the first scenario, the squared coefficient
of variation is greater than one. The parameters are selected that the mean and
variance would be identical and to accomplish an accurate comparison a fitting
process has been done. From this process, you can find detailed information in the
following paper [28].
Table 1. Numerical values of model parameters.
N 𝛾0 𝛾1 𝜎/𝑁 𝛾 𝛼2 𝛽2 𝜏
100 0.05 0.5 0.01 0.8 1 1 0.001
Table 2. Parameters of service time of primary customers.
Distribution Gamma Hyper-exponential Pareto Lognormal
Parameters 𝛼 = 0.037 𝑝 = 0.482 𝛼 = 2.018 𝑚 = −0.751
𝛽 = 0.015 𝜆1 = 0.385 𝑘 = 1.261 𝜎 = 1.826
𝜆2 = 0.416
Mean 2.5
Variance 169
Squared coefficient of variation 27.04
Figure 2 shows the steady-state distribution of the number of customers in the
orbit of the three investigated cases, using the two different impatient modes, and
when the customers are not impatient. From the shape of the curves, it is clearly
visible that the steady-state distributions of the cases are seemed to be normally
distributed. With this parameter setting big differences between the two impatient
modes do not develop but obviously, more customers are located averagely in the
system in the case of without impatience.
Figure 3 demonstrates the observed differences between the applied distribu-
tions in the case of the steady-state distributions of the number of customers in
the orbit with impatience mode 2. Regardless of the distribution, every obtained
curve is similar to the normal distribution. In the case of Pareto distribution, the
mean number of customers is the highest and the effect of different distributions is
clearly observable.
The mean waiting time is presented in function of arrival intensity of primary
customers on Figure 4, 5 and 6. First, I compared the effect of impatience. Inter-
estingly, the results are almost identical between the impatient modes whether we
talk about the mean waiting time of an arbitrary, successfully served, or impatient
250
customer. However, pronounced differences appear when impatience is not present
vs. when there it is.
After that we are curious to see how the effect of different used distributions de-
velop, so Figure 7, 8 and 9 demonstrates that. Even though the mean and variance
are the same, results clearly illustrate the effect of various distributions. The high-
est values are experienced in the case of Pareto distribution and the lowest in the
case of the gamma distribution. With suitable parameter settings, we experience
the maximum property characteristic of a finite-source retrial queueing system.
Figure 10 demonstrate how the probability of abandonment of a customer
changes with the increment of the arrival intensity. By a probability of abandon-
ment, we mean the probability that a customer leaves the system without getting
its full-service requirement (through the orbit). After a slow increase of the value
of this performance measure, it stagnates which is true for every used distribution
of impatience of calls but they differ significantly from each other. At gamma dis-
tribution, the tendency of leaving the system earlier is much higher than the others
especially compared to the other distributions.
Figure 2. Comparison of steady-state distributions, 𝜆/𝑁 =0.01.
Figure 3. Comparison of steady-state distributions, 𝜆/𝑁 =0.01.
251
Figure 4. Mean waiting time of an arbitrary customer.
Figure 5. Mean waiting time of a successfully served customer.
Figure 6. Mean waiting time of an impatient customer.
252
Figure 7. Mean waiting time of an arbitrary customer.
Figure 8. Mean waiting time of a successfully served customer.
Figure 9. Mean waiting time of an impatient customer.
253
Figure 10. Comparison of probability of abandonment.
3.2. Second Scenario
The question arises whether it is true for using other parameter settings for example
when the squared coefficient of variation of the service time of primary customers
is less than one. We use the same parameters as in the previous section (Table 1),
and Table 3 contains the changed parameters of service time of primary customers.
Instead of hyper-exponential, we use this time hypo-exponential distribution be-
cause the squared coefficient of variation is always less or equal to one. We will
show some figures, Figure 11 is connected to the steady-state distribution of the
number of customers in the orbit. Analyzing the curves in more detail they com-
pletely overlap each other. As regards the shape of the curves they correspond to
normal distribution. The mean number of customers is higher in the case of every
distribution compared to the previous section.
The Figure 12 and 13 are related to the mean waiting of an arbitrary and
impatient customer. As you can see very slight differences occur in the case of
Pareto distribution the values are a little bit higher. With this parameter setting
the interesting maximum value of the mean waiting time appears as in the previous
section.
Table 3. Parameters of service time of primary customers.
Distribution Gamma Hypo-exponential Pareto Lognormal
Parameters 𝛼 = 1.8 𝜇1 = 0.6 𝛼 = 2.6733 𝑚 = 0.695374
𝛽 = 0.72 𝜇2 = 1.2 𝑘 = 1.5648 𝜎 = 0.6647
Mean 2.5
Variance 3.472
Squared coefficient of variation 0.555
254
Figure 11. Comparison of steady-state distributions, 𝜆/𝑁 =0.01.
Figure 12. Mean waiting time of an arbitrary customer.
Figure 13. Mean waiting time of an impatient customer.
255
Figure 14 demonstrates the probability of abandonment of a customer versus
arrival intensity. Not surprisingly after seeing the previous figures the difference of
achieved values are relatively close to each other. It can be stated that with these
parameters customers are more tend to leave the system from orbit.
Figure 14. Comparison of probability of abandonment.
4. Conclusion
We investigated a queueing system of type 𝑀/𝐺/1//𝑁 with impatient customers
in the orbit and in the system and an unreliable server capable of calling in re-
quests from an infinite source in this paper. Simulation has been implemented,
it is shown that stationary probability distribution of the number of customers in
the orbit and in the system correlates to the Gaussian distribution regardless of
the used distribution of service time of the primary customers. Under different
scenarios, it was displayed when the squared coefficient of variation is greater than
one the applied distributions of service time have a significant influence on the per-
formance measures like the mean waiting time of an arbitrary, successfully served
and impatient customers or the probability of abandonment even though the mean
and variance are equal in the case of every distribution. Results also indicate that
there is almost no gap between the obtained values of measures when the squared
coefficient of variation is less than one. The authors plan to continue their research
work, examining the obtained phenomenon in more detail and expand their model
with other features like collisions, outgoing calls toward the customers from the
orbit, or carrying out other sensitivity analysis on other random variables.
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