Article citation information:

Madlenak, R., Dutkova, S., Hostakova, D., Sarkan, B. Reliability enhancement using optimization analysis. Scientific Journal of Silesian University of Technology. Series Transport. 2018, 100, 115-125. ISSN: 0209-3324. DOI:



Radovan MADLENAK[1], Silvia DUTKOVA[2], Dominika HOSTAKOVA[3], Branislav SARKAN[4]






Summary. This paper presents an optimization analysis of a queuing system for a particular post office as a tool to increase system reliability. The consideration of system reliability in terms of queuing system failures is very relevant. One way to increase reliability is to analyse the system and its parameters in order to identify its most critical flaws. We used the chi-squared goodness-of-fit test based on the validation of a null hypothesis over an alternative hypothesis. The purpose of the test was to verify the correspondence of the measured data with a theoretical probability distribution. Measurements of relevant data were performed on the specific post office that represented the subject of our research. This approach proved to be a powerful tool in system analysis and optimization. The results of such an analysis can serve as the basis for the modelling of queuing systems.

Keywords: probability distribution; service time; chi-squared goodness-of-fit test.



Each queuing system is characterized by its parameters and attributes. A queuing system for post offices is a stochastic system with an endless front, a certain average service time, customer arrival input and a certain number of compartments [2,15]. In the process of queuing system optimization, it is necessary to choose a particular post office and analyse as accurately as possible the system parameters. The results of this analysis cannot be generalized to each system; therefore, the measurements only refer to the post office in question [6,10]. The system also has many random variables, which are very difficult to capture in the system model [5,18]. However, there are several statistical tools that can help us to determine the attributes of these random variables. The tools of inductive statistics are used for the inductive analysis of empirical data. This means that the results from the statistical set can be generalized to the whole system. Discriminatory statistics, on the other hand, are only focused on a given statistical set and the conclusions concern only those statistical units [1].

 In the case of investigating real systems, the approach consists of a cluster of elements representing random variables, while it is often not possible to perform measurements to obtain all possible values of random variables. In this case, it is appropriate to use the tools of inductive statistics to determine the sample size and generalize the results to the entire system [17]. These results are significant for analysing the queuing system. Such an approach allows us to optimize the system and increase system reliability [16]. The queuing system for post offices is a system with failures such as system overloading, inefficient system use and overly long waiting lines. These factors not only reduce system reliability but also increase system costs.





While descriptive statistics have been used in various forms for several millennia, the basics of inductive statistics, as we know and use them today, were created in the last century. In 1908, William S. Gosset published the article “The probable error of a mean” in Biometrics under the pseudonym “Student”. Gosset needed statistical methods to be able to make rational decisions on the basis of a few samples about the entire population. As a result of his efforts, we have t-probability distribution, from which the well-known Student’s t-test is derived [5]. The renowned statistician, Sir Ronald A. Fisher, recognized the potential and relevance of the t-test and significantly helped to further the development of inductive statistics. The incorrect assumption of statistics is to consider them as only record tools. In practice, however, inductive statistics are very important. Though tools of inductive statistics, it is possible to analyse, for example, the sample of unemployed people who are not interested in seeking work, and to generalize the results of the research to the population in order to take action against unemployment. Inductive statistics are often used by managers and economists to identify information that can help anticipate the development of the economy, inflation and so on. In general, indicative statistics are applicable to the investigation of phenomena that cannot always be measured for some reason, such that research can only be done on the basis of a representative sample. The reason for the inability to measure all phenomena can be high measurement costs or a large population.

There are many methods of inductive statistics, such as hypothesis testing. The claim about one or more parameters is called the statistical hypothesis [9]. The process in which we decide to reject or not to reject the statistical hypothesis is called the testing of statistical hypotheses. The process of hypothesis testing is based on the formulation of two hypotheses. The first is a null hypothesis, which we decide to reject or not to reject. The second hypothesis is called the alternative hypothesis, which represents the opposite of the null hypothesis. Statistical hypotheses should be formulated so as to be quantifiable, verifiable and statistically significant [4].

In order to know the procedure for testing statistical hypotheses, we need to know its basic attributes (according to [11,14]):

·         feasibility (the particular test is used for a specific type of distribution)

·         hypotheses H0, H+, level of significance α

·         test statistic

·         critical value


Some errors may occur while we are testing the hypothesis. We could reject a null hypothesis, which should not be rejected. This error is called a Type I error and the probability that this error occurs is called the level of significance α. It may also transpire that we do not reject a hypothesis, which should in fact be rejected. Such an error is called a Type II error and the probability of occurrence of such an error is called β. The probability of occurrence of these errors can be eliminated by appropriate testing and a sufficient number of statistical samples [3]. There are also three types of tests.



Fig. 1. Right-tailed test of error


In the right-tailed test (Figure 1), we are also interested in the comparison of the test statistic and the critical value. In the case where the test statistic is greater than the critical value, we reject the null hypothesis.


Fig. 2. Two-tailed test of error


In the two-tailed test (Figure 2), if the test statistic is not equal to the critical value, we reject the null hypothesis.



Fig. 3. Left-tailed test of error


In the left-tailed test (Figure 3), we examine whether the calculated test statistic is greater or less than the critical value [13]. In the case where the value of the test statistic is less than the critical value, we reject the null hypothesis.





The queuing system for post offices is a system with several parameters and attributes. One of the parameters is the average service time. Customer service times are continuously random variables with a certain probability distribution. Our objective was to find out which probability distribution belongs to the measured data in order to create a model of a queuing system for a particular post office, which approximates to a real system. Once the model is created, it will be possible to analyse the model, simulate it, identify its critical points, take actions to eliminate them and thus increase the reliability of the system.

In the first step of the research, we identified the problem of optimizing the system and set a few subgoals, which led us to the main goal, that is, optimizing the system. 

In the second step of the research, we used an empirical method specifically for measurement purposes. The process can be divided into three phases:

·           a preparatory phase, which included the preparation of a paper form on which we recorded the measured values

·           a realization phase, which included the measurement of customer service time at a post office in Bytca with a stopwatch

·           processing phase, which included data processing with a form suitable for further use, setting time intervals according to a pilot measurement also performed also at the post office in Bytca


Exact methodology includes statistical methods, such as hypothesis testing [8]. In order to determine the appropriate probability distribution, we used the chi-square goodness-of-fit test to compare the observed sample distribution with the expected probability distribution [12].  The procedure is described in Figure 4.

The probability of a particular interval is given by Formulas 1-2:


     pi = F(bi) . F(ai)                                                       (1)


      pi = 1 - e - λ*bi - (1 - e - λ*ai)                                                 (2)



λ is average service time

ai is the lower interval limit

bi is the upper interval limit


The test statistic is given by Formula 3:


   T = ∑ (Xi - npi)2 / n*pi)                                            (3)



Xi is the central interval value

n is the sample size

pi is the interval probability


In the last step of the research, we reached certain conclusions about the research subject, which were based on the results of the statistical test.


Untitled Diagram (1).jpg


Fig. 4. Methodology of hypothesis testing





As our subgoal was to identify and characterize average service time as a parameter of the queuing system of a particular post office, we performed seven measurements directly at a post office in the Slovakian town of Bytca in January 2017. The measurements were always made at a different time during the post office’s opening hours, so that we could capture as many as possible situations. The result of the measurement was 700 samples, which equates to 700 customers with a certain time of service (Table 1).

Bytca is small town with 11,000 citizens. Those measurements were made behind a service line in cooperation with postal workers. As you can see in Figure 5, the post office in Bytca has seven service compartments. Three of them are universal and are therefore capable of providing most postal services. Two of them are financial, while the last compartment is mainly for receiving and sending parcels. All compartments were service lines in a single queuing system and service time measurements were made arbitrarily without distinction.

Table 1

Measurement characteristics


Serial number

Place of measurement

Date of measurement

Time of measurement




Post office, Bytca



Stopwatch, paper form



Post office, Bytca



Stopwatch, paper form



Post office, Bytca



Stopwatch, paper form



Post office, Bytca



Stopwatch, paper form



Post office, Bytca



Stopwatch, paper form



Post office, Bytca



Stopwatch, paper form



Post office, Bytca



Stopwatch, paper form




Fig. 5. Layout of post office in Bytca


The subject of measurements concerned customer service times. The customers of the post office came with various requirements, which affect the length of service time. It is very important to understand that service time is not the time that a customer spends in a transaction. We stop the stopwatch after the post worker finish the last activity associated with the service. The service time depends on the type of service that a customer requests, the number of requests and failures of information system or technology equipment. The length of the service time is also affected by the worker‘s service speed. These factors can be reflected in average service time. In the next step, measured data were divided to time intervals.


Table 2

Statistical characteristics of measurement in the post office in Bytca


Class i

Class interval

Central interval

Absolute frequency

Relative frequency in%

Cumulative absolute frequency

Cumulative relative frequency in%












































In the order to determine which probability distribution is going to be tested, we plotted a histogram, which showed that it could be an exponential distribution. In fact, the service times in queuing systems generally fit to exponential distribution. So, we decided to use the chi-squared goodness-of-fit test to test data for exponential distribution. As this is a single-tailed test, we did not have to choose the type of test.


4.1. Chi-squared goodness-of-fit test


Null hypothesis H0 means that the service time is modelled according to exponential distribution; while alternative hypothesis H1 means that the service time is not modelled by exponential distribution. The level of significance reflects the probability that we reject the true hypothesis. In general, this probability must be low and therefore we have chosen

The objective of the chi-squared goodness-of-fit test is to compare the calculated test statistic and the critical value, which can be found in the chi-square distribution table. The calculation of the test statistic is given by the mathematical relationship according to Formula 3, where pi represents the probabilities of individual class intervals. These probabilities can be calculated using Formulas 1-2, where α and β are class interval boundaries, and parameter  is   average service time. In Table 3, we can observe the probability classes with test criteria values for each class interval.

After calculating the test statistic, we took the critical value χ2 - the distribution corresponding to the chosen significance level and the degree of freedom f:


                                                           χ2 0.05(7-1-1)=χ2 0.05(5)=11.0705                                  (3)


If the test statistic is less than the critical value, we do not reject the null hypothesis:


             T<χ20.05                                                      (4)




As the test statistic is not greater than the critical value, we do not reject the null hypothesis. This means that the service times at the post office in Bytca fit to exponential distribution (Figure 6).


Table 3

Probability classes with test criteria values for each class interval


Class i































































Fig. 6. Chi-squared goodness-of-fit test: exponential distribution of time service at the post office




By using the inductive statistics tool, we have found that random variable service time at a particular post office fits with exponential distribution. Service times in systems such as queuing systems in post offices fit to exponential distribution in most cases. This means that the probability of service time t+x is less than the probability of service time t, and decreases exponentially. This discovery has helped us in the process of building a model of a queuing system for a particular post office. Our samples, which are generated by a random function in the post office queuing system model, are from a uniform distribution (0,1). However, the random variables in real systems including service times do not fit to uniform distribution, such that it is necessary to determine the appropriate probability distribution. In the next step, we transformed uniform distribution samples into samples that fit to probability distribution by a given algorithm. The process of building a queuing model is one of the most important steps in system optimization.

Analysing system attributes and applying them correctly in a model are important in order to achieve the most accurate results. This model is applicable to all analogical queuing systems, but it is necessary to measure the input parameters of service time and customer input for a particular post office, as well as take into consideration different attributes of the post office, such as the number of compartments and the range of services. The reliability of a system reflects its performance and the satisfaction of customers, especially in queuing systems where customers want to be served. One of our optimization goals was to increase system reliability by optimizing the number of service compartments at individual time intervals, which ultimately led to a reduction in customer queues and customer waiting times. Such optimization can also result in maintenance cost reduction and an overall increase in system efficiency. It can also serve as a starting point in the compilation of post office staffing schedules.





This paper was supported by the scientific grant VEGA 1/0721/18 for research on the economic impacts of visual smog on transport using methods of neuroscience.





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Received 02.03.2018; accepted in revised form 16.08.2018



Scientific Journal of Silesian University of Technology. Series Transport is licensed under a Creative Commons Attribution 4.0 International License

[1] Faculty of operation and economics of transport and communications, University of Zilina, Zilina, Slovakia, Email:

[2] Faculty of operation and economics of transport and communications, University of Zilina, Zilina, Slovakia, Email:

[3] Faculty of operation and economics of transport and communications, University of Zilina, Zilina, Slovakia, Email:

[4] Faculty of operation and economics of transport and communications, University of Zilina, Zilina, Slovakia, Email: