Stochastic grey-box modeling of queueing systems: fitting birth-and-death processes to data

Abstract

This paper explores grey-box modeling of queueing systems. A stationary birth-and-death (BD) process model is fitted to a segment of the sample path of the number in the system in the usual way. The birth (death) rates in each state are estimated by the observed number of arrivals (departures) in that state divided by the total time spent in that state. Under minor regularity conditions, if the queue length (number in the system) has a proper limiting steady-state distribution, then the fitted BD process has that same steady-state distribution asymptotically as the sample size increases, even if the actual queue-length process is not nearly a BD process. However, the transient behavior may be very different. We investigate what we can learn about the actual queueing system from the fitted BD process. Here we consider the standard $GI/GI/s$ queueing model with $s$ servers, unlimited waiting room and general independent, non-exponential, interarrival-time and service-time distributions. For heavily loaded $s$-server models, we find that the long-term transient behavior of the original process, as partially characterized by mean first passage times, can be approximated by a deterministic time transformation of the fitted BD process, exploiting the heavy-traffic characterization of the variability.

Appendix: Additional simulation results for the \(GI/GI/1\) queue

We now supplement Fig. 1 and Table 1 in Sect. 2.1, which display estimated birth rates $\bar{\lambda }_k$ and estimated death rates $\bar{\mu }_k$ for several $GI/GI/1$ queues with traffic intensity $\rho = 0.8$. Here Fig. 6 and Table 6 show the corresponding estimates of birth and death rates for $\rho = 0.9$.

Fig. 6

Fitted birth rates $\bar{\lambda }_k$ (left) and death rates $\bar{\mu }_k$ (right) for nine $GI/GI/1$ models with $\rho = \lambda = 0.9$ and $\mu = 1$

Table 6 Estimates of the asymptotic fitted birth rate $\bar{\lambda }$, death rate $\bar{\mu }$, traffic intensity $\bar{\rho }$ and speed ratio $\bar{\omega }$ via (2.2) for the nine $GI/GI/1$ models with $\rho =0.9$ in Fig. 6

As in Sect. 2.1, we estimated the rates from $30$ independent replications of $1$ million customers. This large sample size is sufficient for $95\,\%$ confidence intervals of the state-dependent rates to be within $1\,\%$ of the rates for states $k$ with steady-state probability $\alpha _k \ge 0.01$.

Paralleling Fig. 4 in Sect. 3.3, we also estimated the steady-state queue-length probabilities $\alpha _k$ and their logarithms. The statistical precision is less with the higher traffic intensity $\rho = 0.9$ instead of $\rho = 0.8$, as expected from [48]. To illustrate, the estimate of $\alpha _{10}$ in the $H_2/M/1$ model with $\rho = 0.9$ was $\bar{\alpha }_{10} = 0.03251$. The sample standard deviation from the $30$ replications was $0.000351$. Using the Student $t$ distribution with $29$ degrees of freedom, the $t$ value for a two-sided $95\,\%$ confidence interval is $2.045$. Thus, the halfwidth of the $95\,\%$ confidence interval is $(2.045 \times 0.000351)/\sqrt{30} = 0.37340 \times 0.000351 = 0.000131$, which is less than $0.5\,\%$ of the estimated value (Fig. 7).

Fig. 7

Estimated steady-state probabilities $\bar{\alpha }_k$ (left) and their logarithms $\log _\mathrm{e}{\bar{\alpha }_k}$ (right) for nine $GI/GI/1$ models with $\rho = \lambda = 0.9$ and $\mu = 1$

This is contrasted with the case with traffic intensity $\rho = 0.8$. The estimate of $\alpha _{10}$ in the $H_2(2)/M/1$ model with $\rho = 0.8$ was $\bar{\alpha }_{10} = 0.02839$. The sample standard deviation from the $30$ replications was $0.000282$. Using the procedure as above, the halfwidth of the $95\,\%$ confidence interval was found to be $0.000105$, which is less than $0.4\,\%$ of the estimated value, about $0.37\,\%$.