31 CHAPTER 3 CALL CENTER QUEUING MODEL WITH LOGNORMAL SERVICE TIME DISTRIBUTION 3.1 INTRODUCTION In this chapter, construction of queuing model with non-exponential service time distribution, performance measures of the call center, simulation experiment and results are presented. 3.2 MODEL CONSTRUCTION In the present work, queuing model with lognormal service time distribution is constructed to study the performance measures of call center. It is denoted by M/G/s+M with lognormal service time distribution. The basic assumptions behind the M/G/s+M queuing model with lognormal service time distribution of call center are as given below Inter arrival time of calls follows exponential distribution with the parameter 1 Service time of calls follows lognormal distribution with the parameters and The number of servers is s Abandonment time of calls follows exponential distribution with the parameter 2
32 3.3 PERFORMANCE MEASURES The call center queuing model involves large and complex model that incorporate many primitives for the call center management. The call center managers are very much interested in evaluating the performance measures of the call center. In order to measure the performance of the call center, the analysis of various performance measures is very much needed to the management of call centers. The commonly encountered performance measures in call center include service level, agent s utilization, abandoned calls, average waiting time, average queue length etc. Among the various performance measures of the call center, the important modeling primitives of complex call center queuing model in evaluating the performance measures appear to be abandoned calls and agent s utilization. These two primitives play an important role in the management of call center industry. 3.3.1 Abandoned Calls A call that is not served immediately hangs up. Otherwise, it joins the queue from which it will abandon if experiencing waiting time is greater than its patience time. This call is called abandoned calls. In industry, each abandoned calls represents a missed opportunity not only to provide a customer with excellent service but also to generate additional revenue. When trying to manage a large call center in heavy traffic, the call center management should consider the effect of abandoned calls in call center. The major drawback of call center queuing model is the ignorance of abandonment. Due to lack of understanding of abandonment behaviour, most modeler ignores the abandonment. This leads to many diffuiculties in the management of call center.
33 3.3.2 Agent s Utilization Agent s utilization is one of the important primitives of call center management. To improve the quality of service in call center, the agent s utilization is always maintained at maximum level. Due to ignorance of agent s utilization in call center, the call center faces over staffing or under staffing problem. To analyse the staffing problem in call center, agent s utilization plays an important role in call center management. 3.4 SIMULATION EXPERIMENT AND ANALYTICAL METHOD 3.4.1 Experimental Setup The value of the parameter 1 of exponential inter arrival time of calls of call center queuing model is assumed as one. The value of the parameter 2 of exponential abandonment time distribution of calls of call center queuing model is assumed as one. The values of parameters and of lognormal service time distribution of calls of the call center queuing model are allowed to vary in their intervals (, ) and (, ) respectively. The numbers of servers are assumed as four. The service discipline of model is First-Come and First-Served. 3.4.2 Simulation Procedure For analysising the abandoned calls and agent s utilization, the approach taken to the simulation experiment was based on the terminology employed by EXTEND simulation software. This software has graphical structure to design through which users can understand the relationship between various modules of simulation model. The modification is also done easily in each module.
34 The simulation model for queuing model M/G/s+M is constructed with the help of hierarchical blocks that are used to design the structure of simulation model. The hierarchical blocks are Generate call type, Queue call type and Answer call type. In Generate call type hierarchical block, the inbound calls are generated, based on the assumed distribution and routed to agents for service. If the agent is busy, then calls are placed in the waiting line which is called as queue. The Queue call type hierarchical block transfers the calls from queue to agent whenever agent is free. If waiting time is more than patience time of call, then this call becomes abandoned call which is allowed leave the block through alternative connects. The Answer call type hierarchical block allows the agents to process the calls. The processing time of the calls is based on the assumed distribution in this block. Based on the values of parameters, the mean of exponential distribution for inter arrival and abandonment time distribution, mean and standard deviation of lognormal service time distribution are calculated and are used as input values to the simulation experiment for the model of call center queuing model M/G/s+M. The formulae for calculating the mean and standard deviation of lognormal distribution are Mean = e 2 2 Standard deviation = 2 2 2 e e ( 1) The simulation experiment was carried out by fixing the mean of inter-arrival and abandonment time of calls as one time unit, number of servers as 4 and by varying the mean and standard deviation of lognormal service time distribution for the values of and in their intervals. The simulation experiment of the queuing model M/G/s+M was carried out for replications. Each replication has processed calls at 1 time unit
35 in four parallel servers. The results obtained from simulation experiment are the percentage of agent s utilization and abandoned calls for the model. 3.4.3 Analytical Procedure For the validation of the model, the analytical method is applied to find the agent s utilization. The formula for finding the agent s utilization of E ( s) queuing model is 1 where arrival rate of customer is 1, E(s) is the s mean of service time and s is the number of servers. 3.5 RESULTS BASED ON THE ANALYSIS OF AGENT S UTILIZATION The results are obtained from simulation experiment in terms of abandoned calls and agent s utilization for various input values of the model. The agent s utilization of the queuing model is analysed with the help of simulation method by fixing the parameters value of exponential inter arrival time and abandonment time distribution and allowing the parameters values of lognormal service time distribution in their range. In this section, the agent s utilizations for some of the values of parameters of the model are presented. 3.5.1 Variations of Percentage of Agent s Utilization with ( ) when = -7 Figure 3.1 shows the variations of percentage of agent s utilization for both analytical and simulation methods when value of is positive and the value of is taken as -7. When value of is greater than 4.5, the percentage of agent s utilization will be more than per cent. When value of
36 is from 3.5 to 4.5, the percentage of agent s utilization is increased from 1 per cent to per cent. When value of is less than 3.5, the percentage of agent s utilization will be less than 1 per cent Percentages of Agent's Utilization 11 5 3 1 A nalytic al method S imulation method 1 2 3 4 5 6 Figure 3.1 Variations of percentage of agent s utilization with sigma when = -7 The value of parameters of lognormal distribution is allowed to vary in their intervals. Due to this variation of value of parameters of service time distribution, the variance and mean of lognormal service time is changed. Since simulation experiment is done with help of variance and mean of service time distribution as input, the percentage of agent s utilization of call center is increased from per cent to per cent. Therefore, the performance measures of call center are varied from per cent to per cent in all the graphs.
37 3.5.2 Variations of Percentage of Agent s Utilization with when = - 4 It is observed from figure 3.2 that the variations of percentage of agent s utilization are changed from per cent to per cent for both analytical and simulation method when value of is positive and the value of is taken as -4. When the values of is greater than 3.5, the percentage of agent s utilization will be more than per cent. When values of is changed from 2.5 to 3.5, the percentage of agent s utilization is changed from 1 per cent to per cent. When value of is less than 2.5, the percentage of agent s utilization will be less than 1 per cent. Percentages of Agent's Utilization 11 5 3 1 A nalytical method S imulation method 1 2 3 4 5 Figure 3.2 Variations of percentage of agent s utilization with sigma when = - 4
38 3.5.3 Variations of Percentage of Agent s Utilization with when = - 3 The variations of the percentage of agent s utilization with sigma when = -3 is shown in figure 3.3 for both analytical and simulation method. It is increased from per cent to per cent when is positive values and the value of is taken as -3. When the value of is greater than 3.4, the percentage of agent s utilization will be more than per cent. When value of is changed from 2.5 to 3.4, the percentage of agent s utilization varies from 1 per cent to per cent. When value of is less than 2.5, the percentage of agent s utilization will be less than 1 per cent. Percentages of Agent's Utilization 11 5 3 1 A nalytic al method S imulation method 1 2 3 4 5 Figure 3.3 Variations of percentage of agent s utilization with sigma when = -3
39 3.5.4 Variations of Percentage of Agent s Utilization with when = -2 Percentages of Agent's Utilization 11 5 3 1 Analytic al method S imulation method 1 2 3 4 Figure 3.4 Variations of percentage of agent s utilization with sigma when = -2 Figure 3.4 presents the variations of percentage of agent s utilization which are increased from per cent to per cent for both analytical and simulation method when value of moves in positive direction and the value of is taken as -2. When value of is greater than 3, the percentage of agent s utilization will be more than per cent. When value of ranges from 1.5 to 3, the percentage of agent s utilization will change from 1 per cent to per cent. When value of is less than 1.5, the percentage of agent s utilization will be less than 1 per cent.
3.5.5 Variations of Percentage of Agent s Utilization with when = -1 It is observed from Figure 3.5 that the variations of percentage of agent s utilization increase from per cent to per cent for both analytical and simulation method when value of is in the positive direction and the value of is taken as -1. Percentages of Agent's Utilization 11 5 3 1 Analytic al method S imulation method.5 1 1.5 2 2.5 3 3.5 Figure 3.5 Variations of percentage of agent s utilization with sigma when = -1 When value of is greater than 2.5, the percentage of agent s utilization will be more than per cent. When value of moves from 1 to 2.5, the percentage of agent s utilization will be from 1 per cent to per cent. When value of is less than 1, the percentage of agent s utilization is less than 1 per cent.
41 3.5.6 Variations of Percentage of Agent s Utilization with When = The variations of percentage of agent s utilization are illustrated in figure 3.6 for both analytical and simulation method when the value of increases in positive direction and the value of is assumed as. The variation starts from 1 per cent to per cent. When value of is greater than 2, the percentage of agent s utilization will be more than per cent. When assumes the values from to 2, the percentage of agent s utilization will vary from 1 per cent to per cent. Percentages of Agent's Utilization 11 5 3 1 Analytic al method S imulation method.5 1 1.5 2 2.5 3 3.5 Figure 3.6 Variations of percentage of agent s utilization with sigma when =
42 3.5.7 Variations of Percentage of Agent s Utilization with when = 1 Percentages of Agent's Utilization 11 5 3 1 Analytic al method S imulation method.5 1 1.5 2 2.5 3 3.5 Figure 3.7 Variations of percentage of agent s utilization with sigma when = 1 From Figure 3.7, it is explored that the variations of percentage of agent s utilization has a growth from 3 per cent to per cent for both analytical and simulation method when value of is positive and the value of is taken as 1. When value of is greater than 1.5, the percentage of agent s utilization will be than per cent. When value of varies from to 1.5, the percentage of agent s utilization will lie from 3 per cent to per cent. 3.5.8 Variations of Percentage of Agent s Utilization with when = 2 It is observed from Figure 3.8 that the variations of percentage of agent s utilization range from per cent to per cent for both analytical
43 and simulation method when value of are positive and the value of is taken as 2. Figure 3.8 also shows that for the analytical method, the percentage of agent s utilization remains at per cent. 11 Percentages of Agent's Utilization Analytic al method S imulation method.5 1 1.5 2 2.5 3 3.5 Figure 3.8 Variations of percentage of agent s utilization with sigma when = 2 3.5.9 Variations of Percentage of Agent s Utilization with when = 3 It is observed from Figure 3.9 that the variations of percentage of agent s utilization are increased from 98 per cent to per cent for both analytical and simulation method when value of is positive and the value of is assumed as 3. The percentage of agent s utilization goes near to per cent for positive value of
44 11 Percentages of Agent's Utilization Analytic al method S imulation method.5 1 1.5 2 2.5 Figure 3.9 Variations of percentage of agent s utilization with sigma when = 3 Due to the variations of value of parameters in lognormal service time distribution of the queuing model M/G/s+M, the percentage of agent s utilization varies from per cent per cent 3.6 RESULTS BASED ON THE ANALYSIS OF ABANDONED CALLS The analysis of abandoned calls in call center is a very complex task because of the non availability of systematic method or procedures. Hence in this study, the simulation method is applied to analyse the call center. For this analysis, the variations of percentage of abandoned calls are obtained for various values
45 3.6.1 Variations of Percentages of Abandoned Calls with when µ =3 It is observed from Figure 3.1 that the variations of percentage of abandoned calls range from 51 per cent to per cent when (sigma) value is allowed in positive direction and the value of µ is assumed as 3. It is seen that the value of percentage of abandoned calls increases with an increase value of. Percentage of Abandoned Calls 5.5 1 1.5 2 2.5 Figure 3.1 Variations of percentages of abandoned calls with sigma when µ =3
46 3.6.2 Variations of Percentages of Abandoned Calls with when µ = 2 Percentage of Abandoned Calls 1 2 3 4 Figure 3.11 Variations of percentages of abandoned calls with sigma when µ = 2 It is evident from Figure 3.11 that the variations of percentage of abandoned calls slope from 9 per cent to per cent when (sigma) value is allowed to vary in positive direction and the value of µ is assumed as 2.
47 3.6.3 Variations of Percentage of Abandoned Calls with when µ = 1 Percentage of Abandoned Calls 5 3 1-1 1 2 3 4 Figure 3.12 Variations of percentage of abandoned calls with sigma when µ = 1 It is depicted from Figure 3.12 that the variations of percentage of abandoned calls shows an upward trend from per cent to per cent when (sigma) value is allowed in positive direction and the value of µ is assumed as 1. When the value of is less than 1.5, the percentage of abandoned calls is less than 1 per cent. 3.6.4 Variations of Percentage of Abandoned Calls with when µ = Figure 3.13 exhibits that the variations of percentage of abandoned calls witness a positive trend ranging from per cent to per cent when (sigma) value is allowed in positive direction and the value of µ is assumed as
48. Moreover, when value of is less than 2, the percentage of abandoned calls will be less than 1 per cent. Percentage of Abandoned Calls 5 3 1-1 1 2 3 4 Figure 3.13 Variations of percentage of abandoned calls with sigma when µ = 3.6.5 Variations of Percentage of Abandoned Calls with when µ = -1 Figure 3.14 reveals that the variations of percentage of abandoned calls experience a positive trend varying from per cent to per cent when (sigma) value is allowed in positive direction and the value of µ is assumed as -1. Further, when value of is less than 2.5, the percentage of abandoned calls would be less than 1 per cent. The percentage of abandoned calls gradually increase when is greater than 2.5.
49 Percentage of Abandoned Calls 5 3 1-1 1 2 3 4 Figure 3.14 Variations of percentage of abandoned calls with sigma when µ = -1 3.6.6 Variations of Percentage of Abandoned Calls with when µ = -2 It is understood from Figure 3.15 that the curve on the variations of percentage of abandoned calls slope up from per cent to per cent when (sigma) value is allowed in positive direction and the value of µ is assumed as -2. It is added that when the value of is less than 2.7, the percentage of abandoned calls will be less than 1 per cent. The curve rises smoothly when is greater than 2.7
5 Percentage of Abandoned Calls 5 3 1-1 1 2 3 4 Figure 3.15 Variations of percentage of abandoned calls with sigma when µ = -2 3.6.7 Variations of Percentage of Abandoned Calls with when µ = -3 Percentage of Abandoned Calls 5 3 1-1 1 2 3 4 5 Figure 3.16 Variations of percentage of abandoned calls with sigma when µ = -3
51 Figure 3.16 shows that the variations of percentage of abandoned calls vary from per cent to per cent when (sigma) value is met the positive values and the value of µ is assumed as -3. It is seen that the percentage of abandoned calls is less than 1 per cent when value of ranges from to 3. 3.6.8 Variations of Percentages of Abandoned Calls with when µ = -4 Percentage of Abandoned Calls 5 3 1-1 1 2 3 4 5 Figure 3.17 Variations of percentage of abandoned calls with sigma when µ = -4 Figure 3.17 finds that the variations of percentage of abandoned calls vary between per cent and per cent when (sigma) value is witnessed at an upward trend and the value of µ is assumed as -4. Further, when value of is less than 3.5, the percentage of abandoned calls will be below 1 per cent.
52 3.6.9 Variations of Percentage of Abandoned Calls with when µ = -7 Figure 3.18 presents that the variations of percentage of abandoned calls slope up from per cent to per cent when (sigma) value is allowed in positive direction and the value of µ is assumed as -7. While the value of is below to 4, the percentage of abandoned calls will be less than 1 per cent. Percentage of Abandoned Calls 5 3 1-1 2 4 6 Figure 3.18 Variations of percentages of abandoned calls with sigma when µ = -7 Due to the variations of value of parameters in lognormal service time distribution of the queuing model M/G/s+M, the percentage of abandoned calls varies from per cent per cent