performance of the call centre is of key importance to the company. An essential ... Keywords: call centre, stochastic queuing model, efficiency, optimization.
Stochastic Queuing Models: A Useful Tool for a Call Centre Performance Optimization Alenka Brezavšček and Alenka Baggia University of Maribor, Faculty of Organizational Sciences Kidričeva cesta 55a, SI-4000 Kranj, Slovenia {alenka.brezavscek,alenka.baggia}@fov.uni-mb.si
Abstract: Contemporary service oriented companies often offer call centre customer support. Since the call centre is usually the first contact of a customer with a company, the quality of a service and efficient performance of the call centre is of key importance to the company. An essential factor of the call centre efficiency is the optimal number of operators answering the customer’s calls. Results of previous research show that stochastic queuing models can be used to analyse the efficiency of call centres. In the presented paper, a research was conducted on the case of Slovenian telecommunication provider’s call centre, to demonstrate the usefulness of stochastic queuing models in optimization of call centre performance. Keywords: call centre, stochastic queuing model, efficiency, optimization.
1 INTRODUCTION The queuing theory is a special application of stochastic process theory (see e.g. [6], [9], [16], [17]). Its applications can be found in diverse areas: a) telecommunications [2], [8]; b) computer networks traffic studies [15]; d) road traffic studies [11], [14]; e) and others. Many contributions in the literature prove that stochastic queuing models are also an useful and applicable tool to analyse the efficiency of call centres that are frequently used by organizations for marketing purposes and technical support for end users (see e.g. [4], [5], [7], [12], [13]). Relevance and usefulness of the results obtained with such an analysis depends on selection of an appropriate mathematical model, which is based on knowledge of the probability density functions of inter-arrival times (i.e. times between two successive incoming calls) and service times (i.e. call length), that are both random variables. These mathematical characteristics of the call centre can be obtained if accurate and complete data about the call centre operation are available. Contemporary technology enables automatic logging of all the events in the call centre, so data needed for the mathematical analysis of the call centre are usually available. However, lack of expert knowledge in practice prevents the companies from efficient usage of them for the call centre optimization. The number of operators in a shift is often based on the rule-of-thumb decision, and is frequently not an optimal solution. Continuing prior research [3] conducted on the case of Slovenian telecommunication provider’s call centre, we present a relatively simple usage of stochastic queuing models for planning an appropriate number of call centre operators. 2 METHODOLOGY A typical queuing system comprises of one or more service units (i.e. servers), arrivals of customers demanding the service, and the service process. Whenever all the customers can not be served at once, queues are formed. This leads to costs (losses) due to waiting which increase with the number of customers in the queue. To decrease the waiting costs and raise the service level ensuring better system performance different improvements can be implemented. However, any improvement often comes with a certain investment leading to higher costs of the queuing system operation. Figure 1 shows that it is always possible to determine the optimal service level which ensures the total costs of queuing system performance are minimal.
Total costs of queuing system performance
Costs
Costs of system operation
Costs (losses) due to customers waiting
Optimal service level
Service level
Figure 1: Costs of queuing system performance
To determine the optimal service level of a queuing system, different quantitative characteristics can be used. The values of these characteristics can be calculated using an appropriate mathematical model. Proper selection of the mathematical model is based on the following elements of the queuing system: Arrival process: Population of customers can be considered either limited (closed systems) or unlimited (open systems). Most mathematical models presume individual arrivals of customers and independent identically distributed inter-arrival times. Service mechanism is determined with the system capability, availability and probability density function of service times. Most of the mathematical models assume that service times are independent identically distributed random variables. Queuing discipline represents the way the queue is organised (e.g. First-In-FirstOut (FIFO), Last-In-First-Out (LIFO), random selection of customers or selection based on customer priorities). When there is only one sever, or there are a number of equivalent and parallel servers, the queuing system is called simple. Simple queuing models use the standard notation for describing the probability density function of inter-arrivals and service times: M – a Poission process of the number of events (i.e. customer arrivals or end of services); exponential density function of times between two successive events. G – a general distribution of times between two successive events (with a known mean and variance; e.g. normal density function). D – a deterministic situation; times between two successive events are constant. Notation M/M/r {infinity/infinity/FIFO} therefore describes the queuing system with r parallel servers, unlimited population, unlimited queue, FIFO queuing discipline, while both, inter-arrival and service times are distributed according to exponential density function (see e.g. [16]). For many types of simple queuing systems analytical solutions are available. 3 CALL CENTRE AS A QUEUING SYSTEM The presented research was conducted on the case of a Slovenian telecommunication provider’s call centre. The call centre is opened from 8:00AM till 12:00PM. It employs 8 full time operators while additional contractors are hired when needed. The schedule of operators is defined based on prior experiences. No analysis of the schedule has been performed. Customers are calling a single phone number. If at least one of the operators is available at the time of the call, he answers the call and serves the customer. If all of the operators are
busy the calling customer is not rejected but can wait for a free operator regardless the number of customers in the queue. The principal scheme of the call centre is presented in Figure 2. Customers calls (arrivals)
Waiting for free operator
Discussion with the operator (service)
Figure 2: Call centre as a queuing system
No. of calls
The call centre under consideration can be treated as a simple queuing system, where the number of servers is determined with the number of active operators and the queuing discipline is FIFO. The key element of the call centre efficiency analysis is the determination of the optimal number of active operators for different periods of the day. The number of calls in different time periods of the day on a typical working week was analysed. The working day of the call centre was divided into the following four periods: from 8:00 AM to 10:00 AM, from 10:00 AM to 1:00 PM, from 1:00 PM to 6:00 PM and from 6:00 PM to 12:00 PM. The results of analysis are presented in Figure 3. 1000 800 600 6:00PM-12:00PM
400
1:00PM-6:00PM
200
10:00AM-1:00PM
0
8:00AM-10:00AM Day
Figure 3: Number of calls in a typical working week by the time periods of the day.
The number of incoming calls is significantly lower during the weekend than during the working days. Therefore, the weekend days were excluded from further analysis. The number of calls in the particular period is similar for all working days. The exception is the last period on Tuesday. This deviation was caused by the unexpected downtime of one of the main provider’s services. The third period (1:00PM to 6:00PM) is the most frequent period, while the morning and evening periods are less burdened. 3.1 Queuing model selection To select the appropriate mathematical model, the distribution of inter-arrival times and the distribution of service times have to be analysed. Figure 4 shows the frequency distribution of inter-arrival times while Figure 5 shows the frequency distribution of service times in different periods of a typical working day. We can conclude from Figure 4 that inter-arrivals times in all four periods fit the exponential density function. It can be seen from Figure 5 that probability density function of service time follows an asymmetric function (e.g. lognormal density function). When calls
40
8:00 AM - 10:00 AM
No. of calls
No. of calls
shorter than one minute are omitted, we can assume that also the distribution of service times can be described by the exponential density function in all four periods of a working day. Since these short calls do not cause queues and therefore do not threaten the efficiency of the call centre, our assumption is justified.
30
100
10:00 AM - 1:00 PM
80 60
20 40 10 20
0
0
120
Min
No. of calls
No. of calls
Min
1:00 PM - 6:00 PM
100
20
6:00 PM - 12:00 PM
80
10
60 40 20
0
0
Min
Min
20
Service time
Serice time
Figure 4: Frequency distribution of inter-arrival times in different periods of a typical working day. 8:00 AM - 10:00 AM
15
10
30
10:00 AM - 1:00 PM
25
20 15 10
5 5 0
0
1:00 PM - 6:00
Min Service times
Service time
Min 40 35
30 25
15
6:00 PM - 12:00 PM
10
20 15
5
10 5 0
0
Min
Min
Figure 5: Frequency distribution of service times in different periods of a typical working day.
From the description of the call centre organization and from the analysis of arrival and service patterns we can conclude that the call centre under consideration can be described by the M/M/r {infinity/infinity/FIFO} queuing model. 3.2 Selection of optimization criterion In the process of the call centre optimization, the expected waiting time was selected as a key performance criterion. Assuming the M/M/r {infinity/infinity/FIFO} queuing model the expected waiting time can be calculated according to the following equation (see e.g. [10]):
r 1 E (Wq ) S r !1 2 r r
(1)
The symbols in (1) denote: r – the number of servers – the arrival rate; 1 is the expected time between two successive arrivals – the service rate; 1 is the expected service time
– the traffic intensity calculated as
r
S – the sum which can be calculated as follows:
r S 1 r 2!
r r r 1! r ! r 1
2
r
1 1
(2)
Equation (1) makes sense when S < ∞. This condition is met when < 1. The condition < 1 ensures that the steady state distribution exists. In such a case the infinite queues are not formed and the queuing system still operates after a long run. The minimum number of servers rmin needed to satisfy the steady state condition is the lowest integer that fulfil the equation
r
(3)
3.3 Call centre optimisation To optimize the call centre performance we set up the following requirement: the expected waiting time in a particular time period should not be longer than 20 seconds. The mathematical formulation of our requirement is: E (Wq ) 20sec 0,33min . Results of our analysis for a particular period of a working day are given in Table 1. From the field data the parameters and were estimated (columns 1 and 2). The value rmin needed to satisfy the steady state condition was determined according to (3) and corresponding E (Wq ) was calculated using (1) and (2) (columns 4 and 5). The minimal number of servers r needed to fulfil our performance requirement was determined by iteration using (1) and (2) (columns 6 and 7). Period 8:00AM - 10:00AM 10:00AM - 1:00PM 1:00PM - 6:00PM 6:00PM - 12:00PM
Table 1: Results of call centre optimization. E (Wq ) rmin [min-1] [min-1] [min]
minimal
E (Wq )
r
[min]
(1)
(2)
(3)
(4)
(5)
(6)
(7)
0,847 1,053 0,877 0,532
0,336 0,342 0,356 0,356
2,25 3,08 2,46 1,49
3 4 3 2
4,34 1,72 3,57 3,55
5 6 5 4
0,16 0,11 0,14 0,08
4 CONCLUSION The usefulness of stochastic queuing models is demonstrated in the case of Slovenian telecommunication provider’s call centre. We analysed the arrival and service patterns, and established that the call centre under consideration can be described by the M/M/r {infinity/infinity/FIFO} queuing model. The expected waiting time was selected as a key performance criterion in the process of the call centre optimization. The aim was to
determine the minimal number of servers in a particular period of a working day to ensure the expected waiting time should not exceed 20 seconds. This requirement will be fulfilled when in the period 10:00AM - 1:00PM at least six operators are available. In the periods 8:00AM 10:00AM and 1:00PM - 6:00PM we need at least five operators, while in the last period (6:00PM - 12:00PM) four operators are enough. Results obtained prove that stochastic queuing models represent an applicable and useful tool for a call centre performance optimization. Such models enable rather easily determination of an appropriate number of active operators regarding a specific key performance criterion. This is a preliminary condition to ensure the optimal service level and therefore minimal cost of queuing system performance. Discrete event simulation is also a viable option for accurate performance modelling and subsequent decision support. Some authors (e.g. [1]) argue that, the analytical approach is not accurate enough, as it does not mimic randomness. In future research we will simulate the presented case with a discrete event simulation tool, where for describing the probability density function of service times an asymmetric function will be used. References [1] [2] [3]
[4]
[5] [6] [7] [8] [9] [10] [11] [12]
[13] [14] [15] [16] [17]
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