Computing Queueing Model Performance with Loss Model Tools

Transcription

1 Computing Queueing Model Performance with Loss Model Tools Philippe Chevalier Jean-Christophe Van den Schrieck Louvain School of Management CORE Université catholique de Louvain, Belgium Feb, 2008 Abstract Multiple class queueing models arise in situations where some flexibility is sought through pooling of demands for different services. Earlier research has shown that most of the benefits of flexibility can be obtained with only a small proportion of cross-trained operators. Predicting the performance of a system with different types of demands and operator pools with different skills is very difficult. We present an approximation method that is based on equivalent loss systems. We successively develop approximations for the waiting probability, the average waiting time and the service level. Our approximations are validated using a series of simulations. Along the way we present some interesting insights into some similarities between queueing systems and equivalent loss systems that have to our knowledge never been reported in the literature. 1

2 1 Introduction Flexibility is of increasing importance in service industries. In order to meet this competitive challenge, many companies use cross-trained staff, i.e. employees that have some expertise in more than one field. Although cross-training brings more flexibility, it is also more expensive than hiring single-skilled employees. Needless to say, such cross-trained employees are scarce too, making it impractical to have all employees capable of handling each task. Earlier research showed that a little flexibility goes a long way. In other words, it is possible to reap most of the benefits of a fully crosstrained workforce with much less cross-training. On this topic, the reader might be interested by [Wallace and Whitt, 2005] or [Chevalier et al., 2005]. The former illustrates that hiring double skilled agents permits to capture most of the variability. In the second paper, the authors find out that a good practice would be to dedicate twenty percent of a staffing budget on flexible agents. This is especially true for call center companies. Nowadays they often handle many different types of calls each requiring specific skills. Flexibility is critical as demand is stochastic and requires quick response from the service provider. This has resulted into the development of multi-class queueing models in the literature. For a review of the main models used, we refer to [Gans et al., 2003] and [Aksin et al., 2007]. On the general problem of routing and staffing in multi-skill call centers we highly advise [Koole and Pot, 2006]. Evaluating the performance of a multi-skill queueing system is a challenging issue per se. [Shumsky, 2004] proposes an interesting approximation to evaluate the performance of a queueing system with two types of demands and two types of operators, one being single-skilled and the other type being polyvalent. The author divides the state-space into different areas. This procedure permits to diminish the computation burden significantly, even for large number of operators, while keeping accurate results. In [Avramidis et al., 2008] a multi-skill queueing model is proposed that is then solved in 2

3 order to find good solutions. The final purpose is similar to this article, but we present a very different approach. Among the queueing systems, the systems with zero queue length are worth mentioning. These systems are often referred to as loss systems. Of course the no-queue assumption is very restrictive and unrealistic in most cases but these systems have the advantage of being much simpler and easier to analyse than other queueing systems. We propose to make use of this valuable advantage to approximate the performance of queueing systems. Our main objective is to build approximations for multi-class queueuing systems that would be easy to use and be quick to compute. In an earlier paper, [Chevalier and Van den Schrieck, 2006], we noticed that the relative performance of loss systems is very often a very good proxy for the relative performance of similar queueing systems. This gave us the idea of a more thorough investigation of the potential to use loss systems to build approximation techniques for the performance measurement of queueing systems. In the current paper we show that very good results can be obtained with such techniques. We propose to work with a model that assumes infinite queue length and that does not consider impatience. The main reason is that queues of infinite length are in a sense at the other extreme compared to loss systems, that have queues of zero length. We actually believe that if we can find good approximations for infinite queues, finding approximations for systems with queues of limited length which are closer to loss systems should be possible. The use of loss systems as a benchmark for the performance of queueing systems has been widely discussed. It is mentioned in [Franx et al., 2006]. [Koole et al., 2003] argue that the relative performance of a multi-skill loss system can be used to approximate the same system with queues. They also note that for a single class exponential server there is a closed form formula to compute the waiting probability based on the loss probability of an equivalent blocking system (see f.i. [Cooper, 1972]). Here we will present extensions to this formula for multiple class systems and for different 3

4 performance measures. Many measures are used to evaluate the performance of a queueing system. The most usual are the probability of having to wait, the average waiting time and the service level, i.e. the probability of being answered within a certain time interval. We try in this paper to develop a method for each of these measures. The outline is therefore as follows. In the next section we present the two models we intend to compare. In section 3 we briefly describe the set of simulations we made to illustrate and evaluate the approximations. This is followed by a section that presents three different approximation methods to compute the probability of waiting. In section 5 an approximation for obtaining the average waiting time is presented. Section 6 focuses on the service level. We end with a concluding section that also lists some of the possible extensions. 2 Multiclass queues We study queueing systems that handle multiple classes of demands. Each class of demand requires some specific competence from the server that will handle it. In order to respond to these demands, the queueing system comprises pools of operators that can handle some subset of the demand classes. Pools are homogeneous groups of agents, that have exactly the same competences. The area where such queueing systems are most widely used in practice is certainly call centers, where each competence might for example correspond to a language. There are other areas where such systems are used such as maintenance services, but to make this article more concrete, from now on we will focus on call centers as the underlying application. By grouping different types of calls, call centers actually try to benefit from the economies of scale made possible by pooling. There exists an abundant literature on pooling. [Mandelbaum and Reiman, 1996] reviews the different types of pooling in call centers. They analyse when pooling is adequate and the cases where pooling is counterproductive using an effi- 4

5 ciency index. We suppose that the arrival streams are Poisson processes and that the processing times are exponentially distributed. The service time is the same for all call types and in any operator pool. The routing of calls is supposed to follow a static priority index: that is, for each particular call type, the different pools that are liable to handle it are ordered. When a call arrives it is sent successively to each pool in the list according to this order, until an operator is found that can handle it. On the other hand when an operator finishes a call, the call that has arrived earliest among the different types this operator can handle is sent to him/her. If there are no calls waiting that the operator can handle the operator will remain idle. To compute the performance measures we draw on the analogy with a loss system where arrivals, operator pools and the routing of arriving calls are identical but, if no available operator can be found to handle a call immediately, the call is rejected rather than put on hold in a queue. Consequently when an operator finishes a call, he/she will remain idle until a new call arrives that is sent to him/her. Figure 1 shows a simple example of the type of systems we are studying. A crucial aspect of the analysis of multi-skill loss models is the analysis of overflows. These are flows of calls that are not answered at a given pool. They have specific characteristics that make them difficult to analyse. The major difficulty is to determine the performance of a pool when its input is an overflow or a combination of various overflows. To face this problem many approximation methods have been developped. Among others we can cite the hyperexponential method that is presented in [Franx et al., 2006] and, to a lesser extend, in [Koole et al., 2003] or the Equivalent Random Method first presented in [Wilkinson, 1956]. It is also described in [Jagerman et al., 1997]. We propose to use another method: the Hayward approximation. This approximation was first presented in [Fredericks, 1980] and it was extended to call centers in [Chevalier and Tabordon, 2003]. The idea is to take the volatility of the overflow into account by working with a new parameter: the 5

6 Figure 1: : a simple example of a multi-class, multi-pool system: (i) represents the system with queues and (ii) is the equivalent system without queues peakedness. The peakedness is the ratio of the variance over the mean of the number of operators that are busy if the analysed flow would be treated by a pool with an infinite number of operators. It is relatively easy to see that the peakedness of a Poisson process is equal to one. This follows directly from the properties of an M/G/ queueing system. For an overflow the peakedness is larger than one, reflecting the bursty nature of this type of 6

7 flow. The Hayward approximation consists in introducing the peakedness in the Erlang-B formula or its continuous version (see [Chevalier et al., 2005] for a description of the continuous formula) by dividing both parameters of this formula, namely the offered load and the number of operators, by the value of the peakedness. To summarize, we have the blocking probability : B(a, z, s) B E (a/z, s/z), (1) where B E (.,.) is the Erlang-B formula, B E (a, s) = a s s! s k=0 a k k!, (2) a is the offered load to the system, i.e. the arrival rate λ divided by the service rate µ. s is the number of operators and z is the peakedness of the incoming flow. The peakedness for an overflow can be computed exactly when the incoming flow is Poisson using the following formula: z = 1 ab(a, s) + a s ab(a, s) a. (3) In case the input is not Poisson, the peakedness can be approximated by the formula proposed in [Fredericks, 1980]: ( z out z in 1 a s B( zin z in, a z in ) + a s + z in + ab( s z in, a z in ) a ). (4) For an evaluation of the method the interested reader is refered to [Tabordon, 2002]. She shows that this approximation is both simple and accurate, making it very tractable in practice. 3 Notations and Description of the Simulation Dataset To describe our method we will use a small example with two types of calls and three pools of operators. Figure 2 depicts the structure of this 7

8 system. The generalization to more complex situations is straightforward, but it would entice a lot of cumbersome notations. The calls are refered to as type-x calls and type-y calls and are assumed to arrive according to two independent Poisson processes respectively of rate λ x and λ y. I = {x, y} is the set of call types. The system consists in two dedicated pools Q x and Q y and one cross-trained pool Q xy, this later pool being able to handle both demand streams. We assume that each call has the same (exponential) service time distribution. The number of operators in one pool is noted n j (j {x, y, xy}). We call S i (i I) the set of pools that can answer type-i calls. Figure 2: : structure of a 2 call types call center The priority rules are {x, xy} and {y, xy} for type-x and type-y calls respectively. This means that in both cases the calls are first sent to the pool 8

9 with specialized operators and then to the pool with polyvalent operators. The objective of this routing policy is to keep the more polyvalent operators available for future uncertain demand. We call L i the probability that a call of type i is rejected. L j i is the proportion of calls of type i rejected by Q j. As we assume (wrongly!) that the loss probability at one pool does not depend on the call type, we sometimes omit the subscript and use L j. We also use M j i as the probability that a call of type i finds an available operator among the operators of Q j. In the system presented in Figure 2, L i = L i i Lxy i = 1 Mi i M xy i. In this paper, we only present an illustrative part of the simulations we made. All the methods presented in the next three sections will be illustrated with this data set. This set consists in systems with two call types. There are six combinations of demands, as shown in Table 1. The different cases were built such as to vary the total load of the system as well as the imbalance between both arrival rates. Example λ x λ y Table 1: The six different combinations of arrivals. Each of the combinations of arrival streams presented in Table 1 is combined with three series of 5 different pool size combinations to obtain a series of 90 experiments. The set of experiments was created in order to try to more or less exhaustively test all combinations with utilizations varying from 0.7 to Table 2 provides all the information about the experiments. The expected service time is 1 in all experiments. For each of the 90 cases we conducted 15 different simulations of

10 Arrival rates series 1 series 2 series 3 Ex. λ x λ y n x n x N xy n x n y n xy n x n y n xy x {1,..., 5} x {1,..., 5} x {1, 3, 4, 5, 6} x {2,..., 6} x {1,..., 5} x {1,..., 5} x {3,..., 7} x {1,..., 5} x {1,..., 5} x {6,..., 10} x {1,..., 5} x {10,..., 14} x {4,..., 8} x {2, 4, 6, 8, 10} x {4,..., 8} x {3,..., 7} x {2, 4, 6, 8, 10} x {4,..., 8} 3 Table 2: The different settings used. In the first series, the number of operators in the polyvalent pool, n xy, varies. In the second and third series, this is respectively n x and n y which change. time units with a warm-up period of 1000 time units, making a total of time units available for analysis. For each set of simulation, we computed confidence intervals for the loss probabilities, waiting probabilities and average waiting time. The relative error was less than 5%, lying in general around 1 or 2%. For each setting, the simulations were made in such a way that the two systems receive exactly the same input. Practically our simulation tool first generates a set of arrivals for the entire simulation length. For each arrival, a service time is also generated. These values are then recorded so that they can be used in both systems. 4 Approximating the Waiting Probability The waiting probability is the performance measure for the queueing system that is closest to the loss probability. Basic algebra reveals a link between the Erlang-B formula, that gives the loss probability, and the Erlang-C formula. The latter computes the waiting probability in the M/M/s context. This link is mentioned in [Cooper, 1972] and in [Koole et al., 2003]. Formally we can write that: 10

11 C(s, a) = sb(s, a) s a(1 B(s, a)), (5) where s is the number of operators, a = λ µ is the incoming load and B(s, a) and C(s, a) are respectively the Erlang-B and -C probabilities. The Erlang- C formula is: C(s, a) = s 1 i=0 a s [(s 1)!(s a)] a i i! + a s (s 1)!(s a). (6) In a multi-skill call center, there are separate queues for each type of arrivals. The waiting probability can therefore be very different from one type of arrival to another although the cross-trained operators that can handle different types of calls will create some dependence between the different call types. In the next subsections, we present three different approaches to compute the waiting probability for each type of calls. All three methods exploit Formula (5). 4.1 The method with repartition of operators (ROM) By analogy with (5), we will estimate the waiting probability for type i calls as: where Ŵ P i = s i L i s i a i (1 L i ), (7) L i is the loss probability for type i calls if they were treated by the equivalent loss system as explained in section 2, a i is the load for type i arrivals (λ i /µ i ), s i is the equivalent number of operators that handle type i calls in the global system. 11

12 Although L i is difficult to compute exactly, we can use the method outlined in Section 2, the Hayward approximation, to obtain a very good estimation of the value of this parameter. The load a i is given and poses no problem. The major difficulty is to determine an adequate value for s i. This is the goal of the next paragraph. In our view, there are very few elements to indicate a huge difference in the percentage of calls of each type among the calls answered at Q XY when queueing is possible than when it is not. Priority rules in the queues such a giving full priority to calls of one type over the other calls might affect these proportions compared to the proportions observed in the absence of queues. However, by ensuring that the oldest call is first answered, the general FCFS rule that is chosen here adopts a policy which is as close as possible to the loss system policy that any call would be answered if a skilled operator is available and thus the policy should reduce the possible differences in proportion between the two systems to their minimum. This leads us to test hypothesis that the fraction of the busy time for the crosstrained servers devoted to each call type is almost identical for the loss and the queueing system, in order to determine the value s i. To our knowledge no study about this property has been published so far. Figure 3 shows the comparison between the simulated loss system and the simulated queueing system. This seems to be a key finding for our approximation. The simulation study strongly supports our hypothesis. For most of the examples the proportion is nearly identical for both systems. The cases where (slight) divergences are observed correspond in general to the heavily loaded systems. Based on this observation we can estimate values s i for Equation (7) in the following way. We use the equivalent loss system to estimate the proportion of time the different pools of operators spend on each type of call. We then split the number of operators in each pool according to these proportions in subgroups for each type of call. Finally, we sum the operators for each type of call from the subgroups of each pool. 12

13 Figure 3: : Comparison of the proportion of time dedicated to the type-x calls at the cross trained pool in the loss and queueing systems. Figure 4 presents a comparison between the approximation we obtained and the simulation results for the waiting probability. Figure 4: : Comparison of the waiting probability observed in simulation with the approximation based on the equivalent loss systems. (i) gives W P x and (ii) gives W P y. 13

14 These results are quite good. They are very accurate when the waiting probability is lower. We notice again that when the system is heavily loaded our approximation is not as accurate. 4.2 The idleness rate method (IRM) The second approximation method uses another (equivalent) version of Formula (5), namely C(s, a) = B(s, a) 1 a/s(1 B(s, a)). (8) Formula (8) states that the waiting probability in a queueing system is obtained by dividing the loss probability in the equivalent loss system by the average idleness rate, i.e. one minus the average utilization of the operators, in the loss system. Based on this observation, we propose to use the following approximation for the waiting probability: Ŵ P i = L i IR i, (9) where IR i is the idleness rate observed on average at the operators that can answer type-i calls, in the loss system. The Hayward approximation is used to compute L i, like in Subsection 4.1. IR i is computed as the weighted average of the idleness rates at the pools that answer type-i calls: IR i = j S i d j i IRj, (10) where d j i = M j i k S M k i i is the proportion of the answered i-calls that are answered at pool j and IR j is the idleness rate observed at pool j in the loss system. It is computed as IR j = 1 (1 Lj )λ j in µ j n j, (11) 14

15 where λ j in is the rate of the total incoming flow to pool j. Note that at the polyvalent pool, IR j should not be divided between calls of different types to account for the shared utilization of the operators as it is already taken into account in the value of d j. Formula (9) was tested on the same dataset as the one presented in Section 3. Figure 5 compares the results obtained with this approximation to the results obtained by simulating the dataset. Figure 5: : Comparison of the waiting probability observed in simulation with the approximation based on the idleness rate. (i) gives W P x and (ii) gives W P y. The results are of good quality. They are excellent for the lower waiting probabilities. 4.3 The geometrical series method (GSM) The third method is based on an extension of Formula (8). An alternative (equivalent) presentation of the formula is: [ a ] k C(s, a) = s (1 B(s, a)) B(s, a). (12) k=0 (12) gives us a new intuition for a multi-skill extension of formula (5). A precise interpretation of the expression is not presented in this paper but we give the main elements to explain it. First, as, in opposition to the loss 15

16 system, all calls are answered in the queueing system, the answered load is greater and therefore W P L. It is possible to show that all calls that are rejected in the loss system are delayed in the presence of queues. This is the first and main source of waiting calls. They represent a proportion of calls equal to B(s, a). The service of these rejected calls interfere with the service of unblocked calls that will be put on hold. They represent a proportion of the total calls equal to (1 B(s, a)) ab(s,a) s, where (1 B(s, a)) is the proportion of unblocked (and thus answered) calls in the loss system and where ab(s,a) s is the total time required to answer the rejected calls (i.e. the total time during which the service of unblocked calls is disturbed) with a capacity equal to s. Because their own service is delayed, these calls affect in turn the service of calls and the effect is extended. From (12), we see that the total proportion of unblocked calls that are delayed is equal [ to (1 B(s, a)) ab(s,a) a ] k s s (1 B(s, a) B(s, a). The last factor reflects k=0 the geometric expansion of the delays as calls delayed prevent the immediate service of other calls. From now on we shall distinguish two types of delays: the direct delays are the delays of calls that would be rejected if there was no queue and the indirect delays are the ones due to the increased workload. What makes multi-skill systems different from single-skill systems is the presence of polyvalent pools. In our model, at Q xy, calls of type x are indirectly delayed not only because of the increased x-workload due to the presence of queues, but also because of the increased workload of calls of type y. We call this phenomenon the cross-delay effect. The cross-delay effects from the x-calls on the y-calls is the effect of the increase in the x-load on the y-calls. The main idea of this third approximation method is to evaluate the amplitude of the cross-delay effect by recursively estimating the proportion of delayed calls at each pool. We first present the general methodology for the 2-call type system and present a formula that can be used in our model. Secondly, we use the formula on our dataset and compare it to the simulation 16

17 results. After that we introduce the extension for more complicated systems. Let us first put aside the y-calls and only consider the x-calls. We consider however that the loss probability observed at the polyvalent pool is the same as if the y-calls were kept. By doing so we first compute the proportion of calls indirectly delayed. We shall then include the cross-delay effects and add the calls affected by these effects to the other delayed calls. With the y-calls put aside, our system is reduced to a system with one call and two successive pools, with respectively n x and n xy operators. This is equivalent to a single skill system with s = n x + n xy so that it is easy to compute W P x (\y), the waiting probability in the absence of y-calls: W P x (\y) = L x 1 a n x +n xy (1 L x ). (13) Note that L x is not the loss probability we would observe in the corresponding single skill loss system as we assume that L xy, the loss probability at Q xy, is unchanged compared to the 2-call type system. However, for reasons that will become clear later, we consider the two successive pools separately and compute W Px x (\y) and W Px xy (\y), the proportions of calls that are indirectly delayed and finally answered at the first and second pools, respectively. We have W P x (\y) = L x + W P x x (\y) + W P xy x (\y). (14) The procedure to obtain W Px x (\y) and W Px xy (\y) is not fully detailled here. This is an iterative procedure and we only present a few details that give a flavour of the whole methodology and that reflect how tricky the recursions are. A proportion of the directly delayed calls equal to p x xλ x L x is serviced at the first pool, where p j i is the probability that a waiting call of type i is finally serviced by an operator of pool j. The increase in workload is equal to p x xa x L x and causes the delay of a portion of the calls equal to Mx x p x x ax n L x x. The same occurs at the second pool and a proportion of calls equal to Mx xy p xy x ax n L x x. All these calls join the queue and are in turn answered either at the first or at the second pool and the procedure is repeated. We end up with: 17

18 W P x x (\y) = = p x xl x a x 1 p k x n k M x k k S x p x xl x 1 ρ k xmx k k S x (15) W P xy x (\y) = = p xy x L x a x 1 p k x n k M x k k S x p xy x L x 1 k S x ρ k xm k x. (16) We use ρ j i = pj i a i. Recall that M x n j x = 1 L x x and Mx xy = L x x(1 L xy x ) are the proportions of calls answered in the loss system respectively at Q x and Q xy. Note that when there is a call in the queue, all operators are busy servicing. Therefore, in the absence of y-calls, in order for the next waiting x-call to be answered at the j th pool, an operator of Q j should be the first to finish a service. In other words p j x is equal to P (j-op), the probability that an operator in Q j is first to complete a service. As all service times are exponentially distributed with rate µ it is easy to show that: p j x = nj n k k S x. (17) By plugging (17) in (16) and (16) and then the resulting equations in (14), we find back formula (13). Repeating the analysis for the y-calls yields: W P y y (\x) = p y yl y 1 k S y ρ k ym k y (18) W P xy Y (\x) = p xy y L y 1, (19) ρ k ymy k k S y 18

19 At this point, we introduce a new concept which we call the amplification. Amplification refers to the total effect induced because of the increased workload. For example the amplification of L y is L y /(1 k S y ρ k ym k y ) and the amplification factor is k S y ρ k ym k y. Let now consider the system with both types of calls. So far we have identified the effects as if the other type of call was not present in the system. To take into account the cross-delay effects from y on x, we should take into account the proportion of time when Q xy is saturated due to the y-queue. This proportion of time is equal to ay n W P xy xy y (\x). Therefore, at Q xy, the initial proportion of time when the pool is saturated is equal to ρ x xl x + ay n W P xy xy y (\x). Once again we do not give much details about the intermediate analysis which is long and cumbersome and only present the critical elements. The key fact here is that we should add a to the above mentioned amplification of delay, y n W P xy xy y (\x), the amplification that results from the increased proportion of delayed y-calls due to the cross-delay effects from the x-calls on the y-calls. Figure 6 illustrates such a double cross-delay effect. Figure 6 represents Q xy. The doted line separates the x-part from the y-part. At some point in the recursion, the load of delayed calls is equal to H. H affects other calls of the same type at the same pool, represented by (Hx), but also calls of type y, (Hy). Similarly (Hy) is the cause of additional delays for calls of both types. What we call the double cross-delay effect is represented by the bold lines and its amplitude is (Hxy). By extension, there are multiple cross-delay effects. In the end, the total amplification factor for the x-calls is equal to ρ x xm x x + ρ xy x L x xm xy x Mx xy ρ xy x M xy y + ρxy y 1 ρ k ym k. The later term accounts for all the cross-delay y k S y effects, both direct and indirect. Therefore, the total waiting probability for 19

20 Figure 6: : Illustration of a cross-delay effect. delayed calls is a cross-delay effect of the initial delay (H). The proportion (Hxy) of the x-calls is: W P x = L x + ρ xy y Mx xy L y 1 ρ k ymy k k S y / 1 k S x ρ k xm k x ρxy y Mx xy ρ xy x My xy 1 k S y ρ k ym k y (20) = L x + ρ xy y Mx xy L y 1 σy k k S y / 1 k S x σ k x ρxy y Mx xy ρ xy x My xy 1 k S y σ k y, (21) where σ j i = ρj i M j i. 20

21 All parameters in Equation (21) are known from the loss system except for p j i. Equation (17) is not valid here because of the shared capacity. The likelyhood of being answered by Q xy as an available polyvalent agent may answer an older call from another type. Therefore, p x x = P (x-op)[p (xy-op)p (y-oldest)] k, (22) k=0 where P(i-oldest) is the probability that a call of type i has waited for the longest time. Computing p j i in those circumstances is tricky and we propose to approximate it and to use p x x = nx s x, (23) p xy x = 1 p x x. (24) The approximation relies on the hypothesis tested in Figure 3. We actually assume that the likelyhood that the oldest waiting call is of type i is equal to the proportion of time a polyvalent operator answers a type-i call. Note that with this is approximation, (21) and (7) are equal when a x = a y and N x = N y. Formula (21) has been tested with the usual dataset of 90 examples. The results are presented in a similar way as in the two previous subsections. They are displayed in Figure 7. We see that the results obtained with the approximation are very accurate. The quality is not affected in heavily loaded situations. This is however at the cost of simplicity. Compared to (7) and (9), Formula (21) is more complicated. Increasing the number of calls and the number of different pools complicates the formula even more. As an example, if there are three types of calls (x, y and z), three dedicated pools and one polyvalent pool (i.e. we add one dimension), (21) becomes: 21

22 Figure 7: : Comparison of the waiting probability observed in simulation with the approximation based on the geometric series approach. (i) gives W P x and (ii) gives W P y. W P x = L x + ρ xyz y +ρ xyz z Mx xyz M xyz x L y + (ρ xyz z My xyz L z )/(1 σz k ) k S z 1 σy k (σz xyz σ xyz k S y y )/(1 σz k ) k S z L z + (ρ xyz y Mz xyz L y )/(1 σy) k k S y 1 σz k (σy xyz σ xyz k S z z )/(1 σy) k k S y / 1 σx xyz σy xyz σx xyz σz xyz k S x σ k x 1 + σ xyz z /(1 k S z σ k z ) 1 σy k (σz xyz σ xyz k S y y )/(1 σz k ) k S z 1 + σ xyz y /(1 k S y σ k y) 1 σz k (σy xyz σ xyz k S z z )/(1 σy) k. (25) k S y We could not find a generalization of Formulas (21) and (25) so far. 22

23 However it is possible to describe the different parts of both formulas. The numerator represents the total increase in workload compared to the situation with only x-calls and no queue. The first term accounts for the additional x-calls that are answered. The second term represents the total y-calls that are delayed independently of the x-calls. Note that this term is obtained by analyzing the system with all x-calls excluded. The third term represents the total amount of z-calls that are delayed independently of the x-calls. The structure is the same as in the second term. Both later terms only affect the x-calls at the polyvalent pool and therefore they are multiplied by M xyz x. The terms that form the amplification factor in the denominator represent the amplifications caused at each pool by the delayed calls. The first and second terms account for the effects of the increased x-workload at pools from S x. The third term is the amplification due to the double cross-delay effects on x from y-calls. The last term is the amplification due to the double cross-delay effects on x from z-calls. In short, to compute the waiting probability for a type of call i in a multi-skill setting, one should compute the waiting probability of the i-calls in the absence of the other calls and compute the waiting probability of the other calls in the absence of all i-calls. The total waiting probability is a combination of all those probabilities divided by an amplification factor with a similar structure as the different elements of the numerator. 4.4 Comparison of the ROM, IRM and GSM methods Figures 4, 5 and 7 are not handy to compare accurately the quality of the three approximations proposed. In Tables 3 and 4, we present the average of the difference in absolute value (Abs. Mean) between the values found with the approximations and the ones obtained by simulations. We also give the standard deviation (Abs. StD). GSM performs the best on average on the whole dataset and the weak standard deviation suggests that the quality of the approximation remains 23

24 Measure ROM IRM GSM Abs. Mean Abs. StD Table 3: Comparison of te mean and standard deviation of the approximation error measured in absolute value for the x-calls Measure ROM IRM GSM Abs. Mean Abs. StD Table 4: Comparison of te mean and standard deviation of the approximation error measured in absolute value for the y-calls constant. The results presented in Tables 3 and 4 confirm what we could see from Figures 4 and 5: the performance of ROM and the performance IRM are very close. The former is slightly better both in terms of average and standard deviation but the differences are so small that we can not conclude anything strong. Tests on a larger dataset may help. We present in Tables 5 and 6 the same statistics (Mean and StD) but computed on the difference between approximation and simulation values, both postive and negative. Measure ROM IRM GSM Mean StD Table 5: Comparison of te mean and standard deviation of the approximation error for the x-calls The average difference between both types of value is weak for all three approximation, strongly suggesting that none of the method causes a bias. According to the dataset GSM is the best method. However, it is not 24

25 Measure ROM IRM GSM Mean StD Table 6: Comparison of te mean and standard deviation of the approximation error for the y-calls easy to use in practice as there is no ready-to-use method for larger settings, either in terms of number of skills and in terms of number of pools. We therefore advise either ROM or IRM. For the remaining of the paper we shall use ROM as it performs slightly better. 5 The Average Waiting Time Another measure of importance is the average waiting time before being served. For an M/M/s system, it is possible to compute it using the following formula: k=0 a s W T = 1 µ (s 1)!(s a) 2 p 0 (26) ( s 1 ) 1 a k p 0 = k! + as s (27) s! (s a) We can express it from the Erlang-C : W T = 1 C(s, a) µ (s a) (28) This last formula can be interpreted as the expected service time multiplied by the waiting probability and divided by the average idleness rate of all servers. We use this observation to build our approximation. 5.1 Bounding the average waiting time Intuitively the waiting calls all benefit to some extend from the idle capacity of all pools thanks to the first come first served rule. Indeed, for a call of a 25

26 particular type that is waiting, the fact that calls of other types are handled quickly increases the probability of this call being the one that has waited longest when an adequate operator becomes available. From this we can derive bounds on the estimations of the waiting time. Indeed, a lower bound on the waiting time is obtained if we suppose all call types fully benefit from the total idleness rate of all operators of all pools. This would give the following estimate for the average waiting time: Ŵ T i,low = 1 Ŵ P i ( ). (29) µ n j a k j S i On the other hand we can derive an upper bound on the waiting time if we suppose that the system behaves as if there was no interaction between the different call types. To compute this we use the equivalent number of operators dedicated to type-i calls as derived in section 4. k I Ŵ T i,up = 1 Ŵ P i µ (s i a i ) (30) We use the same simulation data as in the preceeding section to illustrate these bounds. In Figure 8 we compare these bounds (vertical axis) with the observed average waiting times (horizontal axis). We observe that the values obtained by computation are very good bounds on the waiting time: the upper bounds lie above the 45 degrees line while the lower bounds are below. 5.2 An approximation for the waiting time The previous results confirm that our interpretation of Equation (28) seems to give good results. In order to improve our approximation of the average waiting time we try to estimate the idleness capacity, IC j, of each pool of operators, i.e., the exceeding capacity when taking into account all calls that are answered on average at the pool. For a given call type we then sum the idleness capacities of all the pools that are liable to handle that call type. 26

27 Figure 8: : the bounds on the average waiting time, as functions of the simulated waiting time. (i) gives the bounds on W T x and (ii) on W T y (the axes have a logarithmic scale) Consequently we need to find the proportion of calls that are answered by each pool of operators. We have to find a way to approximate that proportion and for the same reasons than the ones used to justify our approximation for s i in Subsection 4.1, we propose to benchmark the situation in the loss system to the situation in the equivalent queueing system. More precisely, we propose to approximate AP j i,q, the proportion of calls of type i that are answered at pool j in the queueing situation by AP j i,l the proportion of the answered calls i that are answered by an operator of pool j in the queueing system. In fact AP j i,l = M j i k S i M k i. This proposal is further justified by the results presented in Figure 9 which clearly show that this proportion is roughly the same in a loss system and in the corresponding queueing system. We observe that in general the proportion is a bit higher in the loss system than in the equivalent queueing one. The difference is however suf- 27

28 Figure 9: : Comparison of the proportion of calls treated by the cross-trained operators in the loss and the queueing systems. ficiently small for it to be overlooked. Our approximation will thus be computed as follows: 1. we compute the overflows at each level of the loss system, from this we deduce for each call type the proportion that is handled by each pool. 2. We extrapolate these proportions for the queueing system (where all calls are treated, contrary to the loss system). 3. We compute the rates of calls that will be handled at each pool. 4. We deduce IC j for each pool j. 5. We use Equation (28) where we replace the Erlang-C result, C(s, a), by the waiting probability value found using one of the approximations presented in Section 4 and the denominator with the sum of the idleness capacities for the pools that handle the corresponding call type. 28

29 This gives us the following formula: Ŵ T i = 1 Ŵ P i ( ) (31) µ IC j j S i We tested Formula (31) on the dataset described in Section 3. Ŵ P i has been computed using ROM. The final results are presented in Figure 10. Figure 10: : The approximation of the average waiting time for the x-calls (i) and the y-calls (ii) based on computations. The axes have a logarithmic scale. The results of Figure 10 show again that the quality of the approximation is quite good, with some deterioration for the heavily loaded systems. Notice that we switched to a logarithmic scale in order to have more evenly spread values. The relative accuracy of our approximation is not as good as for the waiting probability though. 6 The Service Level A third measure of performance is the service level. It gives the proportion of calls that are being answered within a given time. In other words this is the proportion of calls that do not wait more than a given limit. This measure is 29

30 important as there exist regulations in some countries that impose minimum performances in terms of service level and as many contracts in call center industry use service level as the performance measure. See [Avramidis et al., 2008] or [Hasija et al., 2008] for some applications involving service level measurements. In a single-skill M/M/s setting, it is easy to compute a service level because the distribution of the waiting time is known. Conditionally on the fact that an arrival has to wait the waiting time is exponentially distributed (see f.i. [Khintchine, 1960] or [Gross and Harris, 1998]). So we have: P (W T < t W T > 0) = 1 e (sµ λ)t. (32) With this formula, it is easy to compute the service level. As the probability of waiting is given by (6), the total proportion of calls that are answered within a time t, is the product of (6) and (32) plus the proportion of calls that are answered immediately. In short: P (W T < t) = W P (1 e (sµ λ)t ) + 1 W P (33) = 1 W P e (sµ λ)t. (34) If we analyse these formulas, we see that the parameter of the exponential distribution is sµ λ, which is the idleness rate of the servers. Using the idleness rates we computed in the previous section we can thus build an approximation for the service level. P (W T i < t W T i > 0) = 1 e ( j S i IC j )t (35) P (W T i < t) = 1 Ŵ P ie ( j S i IC j )t. (36) The approximation has been tested on the same data set as in the preceeding sections. In order to test the validity of the conditional waiting probability approximation, we first present simulation results for the conditional service level. This is equivalent to testing Equation (36). Figure (11 30

31 Figure 11: : Approximation of the conditional probability for five different maximum waiting times 31

32 i. to v.) present the results for maximum waiting time of 5, 10, 25, 50 and 100 percent of the average service time. As it may be observed, the results are particularily good for smaller maximum waiting times. There are some deviations at the higher ones (Cases iv and v) for the smallest probabilities. Oncemore this corresponds to the heavily loaded systems: the waiting time is usually very high in these cases, resulting in a small proportion of calls answered within the proposed bounds. In Figures (12 i. to iii.), we present the service level as it is approximated. Figure 12: : Approximation of the service level for three different maximum waiting times 32

33 We observe that although the approximation is accurate in most cases, there are an several cases for which the approximation is of lesser quality. This is once again the more heavily loaded cases. A comparison of Figures (11) and (12) reveals that most of the difference comes from the earlier approximation on the waiting probability. We should note however that the approximation tend to underestimate the service level compared to what the results observed by simulation. 7 Conclusion In this paper a method was presented to approximate the most important performance measures of multi-class queueing systems based on equivalent loss systems. We successively developed approximations for the waiting probability, the average waiting time and the service level. Our approximations were validated using a series of simulations. Along the way we presented some interesting insights into some similarities between queueing systems and equivalent loss systems that have to our knowledge never been reported in the literature. The accuracy of our approximations is generally quite good, one should nevertheless be aware that the quality of the approximations degrades for heavily loaded systems and for longer waiting times. Although many call centers work close to saturation, which are cases where we observed some deviation, the methods provide fairly good approximations even in these cases. More importantly, in terms of relative performance the approximations presented here perform particularly well. Another important point is that all methods are quite easy to compute. These observations make our method quite appropriate to be used in practice. The relative error of our method is well within the precision of the estimates that can often be obtained for the arrival rate or service rate. There are many possible extensions to the work presented here. We really believe that the approximation methodology developed in this article could be applicable in many situations. Indeed, loss models seem in general to be 33

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