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Queueing Theory: the Simulation Model Essay (Book Review)

Background of the Problem

Communication is an essential component of a successful organization. Many areas of the modern business world indicate the need for robust communication channels with the customers as well as other stakeholders. Consequently, evidence exists that the poor state of communication can be traced to the absence of means of delivering feedback. Call centers (and their more recent iteration – contact centers) are currently recognized as one of the primary means of interacting with the customer. They are most often used for delivering service information to the clients, although other functions, such as marketing purposes, become recognized as viable options. They are an established element of many industries, including telecommunication, banking, market research agencies, and a wide range of remote shopping services. For many companies, call centers are a primary means of communication with their clientele. Therefore, it is possible to assume that their efficiency determines the performance of a company at least to some degree

A call center is organized as a hub with telecommunication equipment operated by personnel responsible for accepting calls from the customers. Under most conditions, the amount of incoming calls is sufficient for creating excessive load on the system and leading to the formation of a queue. In some settings (e.g. fire departments and hospitals) a certain amount of emergency calls is expected, creating the need for prioritizing. In addition, the duration, arrival, and abandonment of withheld calls is either random or regulated by a complex interconnection of principles. Finally, the human factor should be considered, since call center creates a considerably stressful environment for employees. Therefore, the issue of call center efficiency can be approached from a stochastic perspective, and queueing theory can be applied in order to seek ways for optimizing its performance.


Several theoretical mathematical models exist that allow the desired level of call center optimization. However, the applicability of each given model must be conclusively determined before its implementation in order to avoid unnecessary expenses and obtain relevant outcomes. Therefore, the objective of the study is to analyze the arrival and service patterns of the customers of a Slovenian call center using the logged data, which would permit selecting an appropriate theoretical queueing model for its optimization. The results of the study will allow us to identify a suitable number of servers for each of the high-load periods of the working day.

Process Flow Description

The paper deals with the call center of a telecommunication provider situated in Slovenia. The customers use a single phone number to contact the center, after which they are placed in a single queue until one of the servers (the call center operator) becomes available, at which point the customer is getting served. The center uses eight operators, although the staff can be expanded with the help of independent contractors if necessary. If none of the servers is available at the time of a new arrival, the call is queued until one becomes free. The schedule of operators is not based on the analysis. The center operates from 8 AM till 12 PM, seven days a week. These characteristics are consistent with a simple queuing system with FIFO as a queueing discipline and a maximum number of servers as eight.

Input and Output Data

In order to acquire an understanding of the optimal distribution of the servers, it is necessary to analyze the data on inputs (the arrival of customers). The data used for the analysis comprised arrivals throughout a typical working week. In order to systematize the results, the arrivals were divided into four time periods of 8 AM to 10 AM, 10 AM to 1 PM, 1 PM to 6 PM, and 6 PM to 12 PM. The analysis showed a significantly higher arrival rate during the periods of 10 AM to 1 PM and 1 PM to 6 PM, with one notable exception of a surge in arrivals during the 6 PM to 12 PM in one day explained by an unusual malfunction of one of the services. It was also established that the arrivals are significantly less frequent during the weekend days, which led to the decision to exclude the weekend days data from the analysis.

The departure of the served customers (the output data) is determined largely by the duration of the call. In order to focus on the meaningful output data, the calls which are shorter than one minute are excluded from the equation since they are unlikely to contribute to the formation of the queue. Once this is done, the exponential density function can be utilized to illustrate the distribution of service times throughout the working day.

The Simulation Model

The theoretical mathematical model suitable for the call center in question is M/M/r queueing model. The selection was justified by two parameters: the distribution of times of serving the customers and the frequency distribution of the times between arrivals. The data was disaggregated by the time period in both cases in order to identify the possible differences. The times between arrivals in all four periods are consistent with the exponential density function, and the distribution of service times can be concluded to fit the same function as long as the assumption holds that the shortest calls should be excluded. The assumption is justifiable by the fact that these calls pose the smallest threat to the center’s performance and are therefore outside the scope of the study.

Conclusions and Results

In order to determine the optimal results, four performance indicators were identified, with the desired value calculated for each one based on the available data.

Expected Waiting Time

In order to calculate the minimal number of servers (rmin) necessary for a set goal, we need to first estimate the arrival rate (α) and the service rate (ϭ) during an average working day. According to the calculations, the arrival rate ranged from 1.053 to 0.532 and corresponded to the fairly stable service rate of 0.336 to 0.356. According to the formula, the minimal number of servers ranges from 2 during the least intense period to 4 during the most active one. Considering the desired Wq of 20 seconds, and a rmin value of the steady-state explained above, we can calculate the minimal number of servers required to fulfill the expected requirement and new expected service time with a modified rmin. The results show that it is possible to achieve the desired service time by increasing the number of operators to six in the most intense period and to four in the least active one.

Expected Number of Waiting Customers

The second expected goal is the number of callers (Nq) in a queue that does not exceed 1.5. Considering the steady-state condition of the minimal number of servers given above, the Nq ranges from 3.76 in the morning to 1.82 during the peak hour. By iterating the increased rmin, we can determine the new minimal number sufficient for meeting the requirement at five operators in the busiest period and three operators during the least active (and the most lengthy) period.

Number of Customers Served Immediately

The third goal is the situation where the majority of callers (one out of five) are connected to the operator immediately. In other words, the desired probability of a customer entering a queue (Pwait) does not exceed 20%. According to the calculations, the iterated rmin of six servers decreases the Pwait from 0.543 to 0.110 during the most active arrival period, and the largest Pwait of 0.714 can be reduced to 0.134 by allocating a rmin of 5.

Service Level

The fourth requirement is compliance with the informal industry rule according to which no less than 80% of the callers should wait 20 seconds or less before being served, which can be formulated as SL (0.33 min) ≥ 0.8. By iterating rmin it becomes clear that three to six servers are a sufficient number for assuming an improvement of 0.803 to 0.921.

The study allows us to conclude that six operators are a sufficient minimal amount of servers for achieving the formulated outcome under the condition of the right scheduling. The study demonstrates stochastic queueing models can be applied as a tool for optimizing a call center’s performance.


All sources used in the paper are appropriately referenced in the text and listed in accordance with the Harvard citation style.

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"Queueing Theory: the Simulation Model." IvyPanda, 15 Sept. 2020, ivypanda.com/essays/queueing-theory-the-simulation-model/.

1. IvyPanda. "Queueing Theory: the Simulation Model." September 15, 2020. https://ivypanda.com/essays/queueing-theory-the-simulation-model/.


IvyPanda. "Queueing Theory: the Simulation Model." September 15, 2020. https://ivypanda.com/essays/queueing-theory-the-simulation-model/.


IvyPanda. 2020. "Queueing Theory: the Simulation Model." September 15, 2020. https://ivypanda.com/essays/queueing-theory-the-simulation-model/.


IvyPanda. (2020) 'Queueing Theory: the Simulation Model'. 15 September.

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