Clustering of Arrivals in Queueing Systems: Autoregressive Conditional Duration Approach

03/31/2020 ∙ by Petra Tomanová, et al. ∙ University of Economics, Prague (Vysoká škola ekonomická v Praze) 0

Arrivals in queueing systems are typically assumed to be independent and exponentially distributed. Our analysis of an online bookshop, however, shows that there is autocorrelation structure present. First, we adjust the inter-arrivals times (i.e. durations) for diurnal and seasonal patterns. Second, we model adjusted durations by the autoregressive conditional duration (ACD) model based on the generalized gamma distribution with the generalized autoregressive score (GAS) dynamics. Third, in a simulation study, we investigate the effects of the dynamic arrival model on the number of customers, the busy period and the response time in queueing systems with single and multiple servers. We find that ignoring the autocorrelation structure leads to significantly underestimated performance measures and consequently suboptimal decisions. The proposed approach serves as a general methodology for treatment of arrivals clustering in practice.

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1 Introduction

In various applications of the operations research, it is undeniable that the characteristics of models evolve over time. The parameters of interest can depend on the time of day and season (see e.g. Kayacı Çodur and Yılmaz, 2020) as well as on their past values and other past indicators (see e.g. Bruzda, 2020). In the paper, we focus on the latter dependency in arrivals to queueing systems from the perspective of the autoregressive conditional duration models with the generalized autoregressive score dynamics.

Many standard queueing systems consider inter-arrival times to be independent due to analytical tractability. Some studies, however, explicitly consider autocorrelation and model arrivals using the Markovian arrival process (MAP) (see e.g. Adan and Kulkarni, 2003; Buchholz and Kriege, 2017; Manafzadeh Dizbin and Tan, 2019), the Markov renewal process (see e.g. Tin, 1985; Patuwo et al., 1993; Szekli et al., 1994), the moving average process (see e.g. Finch, 1963; Finch and Pearce, 1965; Pearce, 1967) or the discrete autoregressive process (see e.g. Hwang and Sohraby, 2003; Kamoun, 2006; Miao and Lee, 2013). Hwang and Sohraby (2003) argue that the time series models with few parameters are more suitable in practice than the MAP models which might be overparameterized. Simulation studies investigating the autocorrelation in arrivals include Livny et al. (1993), Resnick and Samorodnitsky (1997), Altiok and Melamed (2001), Nielsen (2007) and Civelek et al. (2009). Overall, these studies show that ignoring the autocorrelation structure, if present, leads to biased performance measures in queueing systems.

The arrival processes are also extensively studied in the financial high-frequency literature. In this field, the duration analysis deals with modeling of times between successive transactions, times until the price reaches a certain level and times until a certain volume is traded. Typically, the autoregressive conditional duration (ACD) model of Engle and Russell (1998) is utilized. Its dynamics is analogous to the generalized autoregressive conditional heteroskedasticity (GARCH) model of Bollerslev (1986). In its basic version, the ACD model is based on the exponential distribution but many other distributions are considered in the literature as well. Notably, Lunde (1999) introduces the generalized gamma distribution to the ACD model. Bauwens et al. (2004) and Fernandes and Grammig (2005) find that the generalized gamma distribution is more adequate than the exponential, Weibull and Burr distributions in financial applications. Hautsch (2003)

further finds that the four-parameter generalized F distribution reduces to the three-parameter generalized gamma distribution in most cases of financial durations. For a survey of financial duration analysis, see

Pacurar (2008) and Saranjeet and Ramanathan (2018).

A modern approach to time series modeling is the general autoregressive score (GAS) model of Creal et al. (2013), also known as the dynamic conditional score (DCS) model by Harvey (2013)

. The GAS model is an observation-driven model providing a general framework for modeling of time-varying parameters of any underlying probability distribution. It captures dynamics of time-varying parameters by the autoregressive term and the score of the conditional density function utilizing the shape of the density function. The theoretical properties of the GAS models together with their estimation by the maximum likelihood method are investigated e.g. by

Blasques et al. (2014) and Blasques et al. (2018). The empirical performance of the GAS models is studied e.g. by Koopman et al. (2016) and Blazsek and Licht (2020)

. The GARCH model based on the normal distribution and the ACD model based on the exponential distribution are both special cases of the GAS class. The ACD model based on the zero-inflated negative binomial distribution with the GAS dynamics is proposed by

Blasques et al. (2020). Many more models belonging to the GAS family with various applications are proposed in the literature – see Lucas, 2020 for a comprehensive list of papers.

In the paper, we put together these three cornerstones – the queueing theory, the ACD models and the GAS models — and demonstrate that they fit together perfectly. We analyze inter-arrival times between orders from an online Czech bookshop. First, we adjust arrivals for diurnal and seasonal patterns using the cubic spline. Next, we find that the adjusted inter-arrival times exhibit strong clustering behavior – short inter-arrival times are usually followed by short times. To capture this autocorrelation, we utilize the dynamic model based on the generalized gamma distribution with the GAS dynamics in the spirit of the ACD models. We confirm that the proposed specification is quite suitable for the observed data. Finally, we investigate the effects of the proposed arrivals model on queueing systems with single and multiple servers and exponential services. In a simulation study, we show that various performance measures – the number of customers in the system, the busy period of servers and the response time – have higher mean and variance as well as heavier tails for the proposed dynamic arrivals model than for the standard static model. Furthermore, we illustrate how misspecification of the arrivals model can lead to suboptimal decisions.

The rest of the paper is structured as follows. In Section 2, we present the model based on the generalized gamma distribution with the GAS dynamics for diurnally adjusted inter-arrival times. In Section 3, we show that real data of a retail store exhibit the autocorrelation structure that is well captured by our model. In Section 4, we investigate the impact of the proposed arrivals model on the performance measures using simulations. We conclude the paper in Section 5.

2 Dynamic Model for Arrivals

2.1 Diurnal and Seasonal Adjustment

Despite that we focus on time dependence in terms of the autoregressive behavior, we deal with diurnal, weekly and monthly seasonality patterns as well. To capture the non-linear behavior of the diurnal and seasonal patterns and to properly adjust the durations, the cubic spline method is utilized. The cubic spline is a piecewise cubic polynomial with continuous derivatives up to order two at each -th fixed point called knot, . Bruce and Bruce (2017) point out that the cubic spline method is often a superior approach to the polynomial regression since the polynomial regression often leads to undesirable "wiggliness" in the regression equation.

To take into account the specifics of raw durations , we define the cubic spline with knots at as

(1)

where is the trend variable and are the basis functions. The basis functions are equal to (i) the variable , ; (ii) its square, ; (iii) its cube, ; and (iv) truncated power functions, . The trend variable is linear in time (not linear in observations), and for , to take into account the irregularly spaced observations. Moreover, the logarithmic transformation of

ensures the non-negativity of adjusted durations. Equidistant intervals are used for knots identification since intervals based on quantiles might lead to a too-small number of knots allocated to off-peak hours.

Regression parameters in (1) are estimated by the weighted least squares (WLS)

method with weights being the durations. WLS naturally compensates for the possibility that during a particular time interval either a small number of long durations or a higher number of shorter durations is observed, i.e. the number of observed durations within a time interval depends on the duration values themselves. Unlike the ordinary least squares, this approach properly weights the durations during hours that exhibit a small median but a huge dispersion. Once the parameters are estimated, the diurnally and seasonally adjusted and detrended durations

are set to exponentiated residuals from regression (1).

2.2 Generalized Gamma Distribution

Next, we consider the adjusted durations to follow the generalized gamma distribution. The generalized gamma distribution is a continuous probability distribution for non-negative variables proposed by Stacy (1962). It is a three-parameter generalization of the two-parameter gamma distribution and contains the exponential distribution and the Weibull distribution as well. The distribution has the scale parameter and the shape parameters and . We use the parametrization allowing for arbitrary values of

which is quite suitable for modeling of its dynamics. The probability density function is

(2)

where is the gamma function. The expected value and variance is

(3)

The score for the parameter is

(4)

The Fisher information for the parameter is

(5)

Note that the Fisher information for is not dependent on itself. Special cases of the generalized gamma distribution include the gamma distribution for , the Weibull distribution for and the exponential distribution for and . The generalized gamma distribution itself is contained in a larger family – the generalized F distribution with four parameters.

2.3 Generalized Autoregressive Score Dynamics

Finally, we consider the scale parameter to be time-varying. In the generalized autoregressive score (GAS) framework of Creal et al. (2013), the time-varying parameters are linearly dependent on their lagged values and the lagged values of the score of the conditional density. Typically, only first lag is utilized. In our case, parameter follows recursion

(6)

where is the constant parameter, is the autoregressive parameter, is the score parameter and is the score defined in (4). In the GAS framework, the score can be scaled by the inverse of the Fisher information or the square of the inverse of the Fisher information. In our case, however, both scaling functions based on the Fisher information and the unit scaling as well lead to the same model as the Fisher information does not depend on . The score for time-varying parameter is the gradient of the log-likelihood with respect to and indicates how sensitive the log-likelihood is to parameter . In the GAS model, the score drives the time variation in parameter based on the shape of the generalized gamma density function.

Let

denote the vector of parameters in the model. We can estimate

straightforwardly by the maximum likelihood method. The log-likelihood function is given by

(7)

where is the generalized gamma density function given by (2). We deliberately set aside the first term as the time-varying parameter needs to be initialized at . We set the value of to the long-term mean value . Subsequent values of , , than follow recursion (6). Finally, the parameter estimates are obtained by non-linear optimization problem

(8)

3 Empirical Evidence

3.1 Data Overview and Preparation

The data sample is obtained from the database of an online bookshop with one brick-and-mortar location in Prague, Czechia. The data cover the period of June 8 – December 20, 2018, resulting in full weeks and observations. The precision of the timestamp is one minute. Thus, zero durations might occur in the data due to two or more orders that arrive within one minute. Since the generalized gamma distribution has a strictly positive support, the zero durations are set to a small positive number. Bauwens (2006) replaces the zero durations with a value equal to the half of the minimal positive duration and argued that this is a more correct approach than their discarding. Hence, all 81 zero durations are set to 0.5 minutes accordingly.

3.2 Diurnal and Seasonal Patterns

The raw duration median is 24 minutes and the mean is 49 minutes – more than double due to long durations during nights (specifically hours between midnight and 9 AM, see Figure 1). Hours between 9 and 11 AM exhibit many short durations and several very long durations resulting in high dispersion (SD = 111.39). The rest of the rush hours (until 5 PM) shows a similar duration median but much lower dispersion (SD = 35.98). Moreover, strong weekly and monthly seasonal patterns are observed. The highest order counts (and consequently lower duration values) occur at the beginning of a week and decrease until Saturday. On Sundays, order counts increase again and exhibit the highest dispersion. During the summer months, the order counts are rather low (resulting in higher duration values) and linearly increase until December.

To obtain the diurnally and seasonally adjusted and detrended durations, the regression equation (1) is estimated with one knot for every 90 minutes and weekly aggregation resulting in the same daily seasonal component for Mondays, Tuesdays, etc. To ensure continuity between Sundays and Mondays, the sample is stacked three times consecutively and the adjusted durations are computed based on the second sub-sample. Parameters are estimated by the WLS.

The fitted values are shown in Figure 1. Note that they do not coincide with the smooth cubic spline function due to a linear trend which makes the corresponding fitted line "saw-toothed". The diurnally and seasonally adjusted and detrended durations are computed as the exponentiated residuals and for convenience, they are standardized to have unit mean. Their values range from 0.002 to 11.23 minutes.

Figure 1: Raw durations and their fitted diurnal/seasonal pattern.

3.3 Fit of the Dynamic Model

Even after the seasonal and diurnal adjustment, the durations still tend to cluster over time – long (short) durations are likely to be followed by long (short) durations. This dependence is not particularly strong but is statistically significant nevertheless as illustrated in Figure 2. To capture the autocorrelation, we utilize the dynamic model based on the generalized gamma distribution with the GAS dynamics presented in Section 2.3. For comparison, we also report the results for static and dynamic models based on special cases of the generalized gamma distribution (G.G.), namely for the exponential (Exp.), Weibull and gamma distributions.

Figure 2: The autocorrelation function (ACF) and the partial autocorrelation function (PACF) of adjusted durations.

Parameter estimates and the performance evaulation in terms of the Akaike information criterion (AIC) of both static and dynamic duration models are shown in Table 1. The AIC values are at least by 43.59 higher for dynamic models than for their static counterparts. However, the differences among dynamic models are not so striking – the highest difference is between exponential and generalized gamma distributions (by 5.94). The best performing model is the most general one – the dynamic GAS model utilizing the generalized gamma distribution. The dynamic models based on either the exponential or generalized gamma distributions in comparison with their static counterparts are further analyzed in the simulation study of queueing systems.

Model Estimate Model Fit
Spec. Dist. Lik. AIC
Static Exp. 0.00 0.00 0.00 1.00 1.00
Static Weibull -0.01 0.00 0.00 1.00 0.97
Static Gamma 0.04 0.00 0.00 0.96 1.00
Static G. G. -0.12 0.00 0.00 1.08 0.93
Dyn. Exp. 0.00 0.76 0.06 1.00 1.00
Dyn. Weibull 0.00 0.75 0.06 1.00 0.97
Dyn. Gamma 0.01 0.76 0.06 0.97 1.00
Dyn. G. G. -0.06 0.72 0.07 1.15 0.90
Table 1: The estimated parameters of the duration models with the log-likelihood value (Lik.) and the Akaike information criterion (AIC).

4 Impact on Queueing Systems

4.1 System with Single Server

We investigate the effects of various arrival models on performance measures in queueing systems using simulations. We consider models based on the exponential and generalized gamma distributions with the static and dynamic specifications. The coefficients of the models are taken from Table 1. In all models, the rate of arrivals is . First, we focus on the queueing system with single server only. We consider the service times to be independent and exponentially distributed with the rate ranging from to . We simulate the arrival and service processes and measure the number of customers in the system, the busy period of the server and the response time. The number of simulations we perform ranges from to as systems with lower require higher number of simulations.

The results are reported in Table 2. For all values of

, the systems based on the generalized gamma distribution have higher values of performance measures than the systems based on the exponential distribution in terms of the mean, standard deviation and 95 percent quantile. Similarly, systems with the dynamic specification have higher values of performance measures than the systems with the static specification. The left plot of Figure

3 shows how the probability mass function of the number of customers differs for the static and dynamic models. The dynamic model has higher probability of the empty system as there tend to be longer periods of low activity. It has also higher probabilities of large numbers of customers in the system as arrivals tend to cluster. The right plot of Figure 3 shows how the density function of the response time differs for the static and dynamic models. In the dynamic model, customers simply have to wait longer. The differences between the static and dynamic models are naturally weaker for larger .

These results carry a warning for practice. When the standard M/M/1 system is assumed but the arrivals actually follow the GAS model based on the generalized gamma distribution, the performance measures are significantly underestimated. For example, the mean number of customers and the mean response time are 22 percent lower than the actual value for . It is therefore crucial to correctly specify the model for arrivals.

Figure 3: The probability mass functions of the number of customers in the system and density functions of the response time for the static and dynamic arrival models based on the generalized gamma distribution in the queueing system with single server and .
Queueing System No. of Customers Busy Period Response Time
Spec. Dist. M SD 95% M SD 95% M SD 95%
1.10 Static Exp. 10.00 10.49 31.00 10.00 45.83 39.81 10.00 10.00 29.96
1.10 Static G. G. 10.38 10.94 32.00 10.36 47.49 41.31 10.38 10.42 31.13
1.10 Dyn. Exp. 12.38 13.30 39.00 10.88 54.47 41.43 12.38 12.58 37.53
1.10 Dyn. G. G. 12.83 13.84 40.00 11.24 56.07 43.05 12.83 13.07 38.92
1.20 Static Exp. 5.00 5.48 16.00 5.00 16.58 22.07 5.00 5.00 14.98
1.20 Static G. G. 5.18 5.71 17.00 5.17 17.20 22.85 5.18 5.20 15.56
1.20 Dyn. Exp. 5.97 6.81 20.00 5.36 19.47 23.35 5.97 6.14 18.27
1.20 Dyn. G. G. 6.21 7.09 20.00 5.56 20.22 24.39 6.21 6.37 19.00
1.30 Static Exp. 3.33 3.80 11.00 3.33 9.23 14.81 3.33 3.33 9.99
1.30 Static G. G. 3.45 3.96 11.00 3.44 9.58 15.31 3.45 3.46 10.35
1.30 Dyn. Exp. 3.87 4.62 13.00 3.54 10.73 15.72 3.87 3.99 11.88
1.30 Dyn. G. G. 4.04 4.85 14.00 3.67 11.18 16.40 4.04 4.18 12.40
1.40 Static Exp. 2.50 2.96 8.00 2.50 6.12 10.97 2.50 2.50 7.49
1.40 Static G. G. 2.58 3.08 9.00 2.58 6.35 11.32 2.58 2.59 7.75
1.40 Dyn. Exp. 2.84 3.52 10.00 2.63 7.04 11.56 2.84 2.93 8.71
1.40 Dyn. G. G. 2.96 3.70 10.00 2.73 7.35 12.07 2.96 3.06 9.08
1.50 Static Exp. 2.00 2.45 7.00 2.00 4.47 8.61 2.00 2.00 5.99
1.50 Static G. G. 2.06 2.54 7.00 2.06 4.63 8.87 2.06 2.07 6.18
1.50 Dyn. Exp. 2.23 2.87 8.00 2.09 5.08 9.04 2.23 2.30 6.84
1.50 Dyn. G. G. 2.32 3.00 8.00 2.16 5.31 9.41 2.32 2.40 7.12
Table 2: Mean values (M), standard deviations (SD) and 95%-quantiles (95%) of the number of customers in the system, the busy period of the server and the response time in various queueing systems with single server.

4.2 System with Multiple Servers

Next, we consider the queueing systems with multiple servers. We base the simulations on the same setting as in the previous section. The only difference lies in the service structure. We consider the number of servers ranging from to with the individual service rate . Such values result in the same server utilizations as in the previous section. Again, we measure the number of customers in the system, the busy period of the servers and the response time. By the busy period, we mean the full busy period, i.e. the duration of the state in which all servers are busy.

The results are reported in Table 3. The findings are very similar to the system with single server – the generalized gamma distribution and the dynamic specification increase all performance measures. When incorrectly assuming the M/M/c system, the specification error is distinct but not as high as in the case of single server. For example, the mean number of customers and the mean response time are 14 percent lower when assuming the M/M/11 system than the actual value for arrivals based on the generalized gamma distribution with the dynamic specification.

We illustrate how misspecification of the arrival model can affect decision making in the following toy example. Let us consider that there are two types of costs associated with the operation of the system – the cost of running one server per unit of time and the cost of the queue longer than customers per unit of time . The analytic department is faced with the question of how many servers to operate. Th composition of costs for different number of servers is shown in Figure 4. The optimal number of servers according to the static model is while it is for the dynamic model. An analyst assuming the static model believes that the total optimal costs per unit of time are while they actually are for suboptimal choice of 12 servers. An analyst correctly specifying the dynamic model finds out that the lowest possible costs are 132.53 for the optimal choice of 13 servers. The decision based on the misspecified arrival model therefore results in total cost increase of 8 percent.

Figure 4: Costs related to the number of servers and long queue for the static and dynamic arrival models based on the generalized gamma distribution in the queueing systems with multiple servers and .
Queueing System No. of Customers Busy Period Response Time
Spec. Dist. M SD 95% M SD 95% M SD 95%
11 Static Exp. 16.82 10.66 38.00 10.01 46.01 39.81 16.82 13.80 43.86
11 Static G. G. 17.15 11.12 39.00 10.36 47.51 41.28 17.15 14.08 44.68
11 Dyn. Exp. 19.08 13.49 46.00 12.37 58.51 49.89 19.08 15.70 49.83
11 Dyn. G. G. 19.45 14.01 47.00 12.81 60.38 51.71 19.45 16.07 50.90
12 Static Exp. 12.25 5.84 24.00 5.01 16.63 22.12 12.25 10.84 33.57
12 Static G. G. 12.38 6.07 24.00 5.17 17.22 22.85 12.38 10.91 33.85
12 Dyn. Exp. 13.09 7.19 27.00 6.08 21.06 27.61 13.09 11.34 35.38
12 Dyn. G. G. 13.26 7.46 28.00 6.33 21.80 28.78 13.26 11.46 35.78
13 Static Exp. 10.95 4.39 19.00 3.34 9.28 14.85 10.95 10.27 31.30
13 Static G. G. 11.02 4.53 19.00 3.44 9.60 15.31 11.02 10.30 31.40
13 Dyn. Exp. 11.36 5.23 21.00 3.99 11.63 18.33 11.36 10.45 32.01
13 Dyn. G. G. 11.46 5.46 22.00 4.16 12.14 19.11 11.46 10.51 32.24
14 Static Exp. 10.44 3.76 17.00 2.51 6.14 10.99 10.44 10.10 30.53
14 Static G. G. 10.48 3.88 18.00 2.59 6.39 11.35 10.48 10.11 30.59
14 Dyn. Exp. 10.66 4.36 19.00 2.95 7.64 13.43 10.66 10.17 30.85
14 Dyn. G. G. 10.71 4.54 19.00 3.08 8.01 14.03 10.71 10.20 30.97
15 Static Exp. 10.20 3.46 16.00 2.01 4.48 8.63 10.20 10.04 30.22
15 Static G. G. 10.23 3.56 16.00 2.07 4.69 8.95 10.23 10.04 30.25
15 Dyn. Exp. 10.33 3.93 17.00 2.34 5.53 10.46 10.33 10.07 30.38
15 Dyn. G. G. 10.35 4.07 18.00 2.43 5.81 10.93 10.35 10.08 30.44
Table 3: Mean values (M), standard deviations (SD) and 95%-quantiles (95%) of the number of customers in the system, the full busy period of servers and the response time in various queueing systems with multiple servers and .

5 Conclusion

We analyze the dependence of inter-arrival times in queueing systems and demonstrate the negative impact of arrival model misspecification on decision making. To capture the autocorrelation structure of inter-arrival times, we propose to utilize the dynamic model based on the generalized gamma distribution with the GAS dynamics. We argue that this approach is superior to the standard model assuming the exponential distribution with a constant rate since it leads to a more faithful representation of the mean and extreme values of the arrival process. Our approach consists of three steps.

  1. We construct a suitable model for capturing the diurnal and seasonal dependencies which takes into account a specific time-structure of durations. We utilize a cubic spline approach and propose to estimate the parameters by the weighted ordinary least square method to properly adjust durations during hours that exhibit a small median but a huge dispersion.

  2. We argue that the GAS models based on the generalized gamma distribution and its special cases fit the data better than their static counterparts. This is due to the fact that the static models ignore the autocorrelation structure which is still present even after the proper diurnal and seasonal adjustment.

  3. We compare both static and dynamic models in the simulation study of queueing systems with single and multiple servers and exponential services. We show that ignoring the autocorrelation structure leads to biased performance measures. The number of customers in the system, the busy period of servers and the response time have higher mean and variance as well as heavier tails for the proposed dynamic arrivals model than for the standard static model. Most importantly, we show how the trust in the standard static model for inter-arrival times leads to suboptimal decisions and consequently to a profit loss.

A proper treatment of arrival dependence is of a great importance since its ignorance generates extra costs. Our approach is useful for process simulations and consequently for process optimization and process quality assessment.

Acknowledgements

We would like to thank organizers and participants of the 7th International Conference on Management (Nový Smokovec, September 26–29, 2018), 30th European Conference on Operational Research (Dublin, June 23–26, 2019), 15th International Symposium on Operations Research in Slovenia (Bled, September 25–27, 2019) and 3rd International Conference on Advances in Business and Law (Dubai, November 23–24, 2019) for fruitful discussions.

Funding

The work on this paper was supported by the Internal Grant Agency of the University of Economics, Prague under project F4/27/2020 and the Czech Science Foundation under project 19-08985S.

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