Queues with resetting: a perspective

TL;DR

The paper tackles queue performance under intrinsic service-time fluctuations and proposes resetting as a universal strategy to mitigate delays in an M/G/1 queue with overhead. By modeling service reset events (Poissonian and sharp) as a renewal process, the authors derive moments for the restarted service $S_R$ via $\langle S_R\rangle$ and $\langle S_R^2\rangle$, and apply the Pollaczek–Khinchin formula with modified metrics to obtain $\langle N_r\rangle$ and $\langle T_r\rangle$. A central result is the existence of an optimal resetting rate $r^*$, accompanied by a universal expression for $CV_{r^*}$ that depends on overhead variability $CV_{on}$ and $\langle S_{on}\rangle$, leading to a potentially substantial reduction in mean queue length relative to the standard PK queue. The authors illustrate the theory using a log-normal service time with varying overhead distributions (deterministic, exponential, Weibull), showing that resetting can dramatically shorten queues, with sharper gains when overhead variability is modest. They also discuss extensions to sharp resetting and multi-server settings, underscoring the broader relevance to computing, stochastic optimization, and biological systems where intrinsic service fluctuations prevail.

Abstract

Performance modeling is a key issue in queuing theory and operation research. It is well-known that the length of a queue that awaits service or the time spent by a job in a queue depends not only on the service rate, but also crucially on the fluctuations in service time. The larger the fluctuations, the longer the delay becomes and hence, this is a major hindrance for the queue to operate efficiently. Various strategies have been adapted to prevent this drawback. In this perspective, we investigate the effects of one such novel strategy namely resetting or restart, an emerging concept in statistical physics and stochastic complex process, that was recently introduced to mitigate fluctuations-induced delays in queues. In particular, we show that a service resetting mechanism accompanied with an overhead time can remarkably shorten the average queue lengths and waiting times. We examine various resetting strategies and further shed light on the intricate role of the overhead times to the queuing performance. Our analysis opens up future avenues in operation research where resetting-based strategies can be universally promising.

Figures (5)

• Figure 1: Schematic of a queuing system under resetting. Jobs arrive at the queue with a rate $\lambda$ and they are being served at the service station according to a first-come, first-served policy. The server has two components: a service time $S$ followed by an overhead time $S_{on}$. It can complete a task in time $S_u=S+S_{on}$ prior to resetting (upper branch). Otherwise, resetting can occur in time $R$ (lower branch), following which the service is renewed. The process repeats by itself until the task is completed which is possible only via the upper branch.
• Figure 2: Panel (a) The mean service time $\langle S_r \rangle$ from Eq. (\ref{['mean under restart']}) as a function of the resetting rate for different $\langle S_{on} \rangle$. The service time $S$ is sampled from the log-normal distribution whose density is given by Eq. (\ref{['log normal dist']}). Here, we have set $\langle S \rangle = 1$, and $\alpha = 1.5$. The circles show the optimal resetting rate $r^*$ for different $\langle S_{on} \rangle$, where $\langle S_{r} \rangle$ attains consecutive minima. Panel (b) shows the mean queue length at optimal resetting i.e., $\langle N_{r^*} \rangle$ as a function of the squared $CV_u$. Note that $\langle S \rangle$ is fixed at unity and $\alpha$ is varied to control the stochastic fluctuation in $CV_u$ via Eq. (\ref{['cv_deterministic_equation']}), Eq. (\ref{['cv_exponential_equation']}) and Eq. (\ref{['cv_weibull_equation']}) respectively. The overhead time is taken from three different distributions: deterministic ($CV_{on}=0$), exponential ($CV_{on}=1$) and Weibull ($CV_{on}>1$) while we have set $\langle S_{on} \rangle=0.5$ fixed for all the plots. The Pollaczek–Khinchin formula (shown by the dashed line) gives the familiar linear dependence of Eq. (\ref{['PK-1']}) when $r^*=0$. However, as we vary $CV_u$ further, the optimal resetting rate $r^*$ becomes finite and thus the plots for $\langle N_{r^*} \rangle$ deviate from the same with $r^*=0$. The transition points (where the deviation occurs) can be corroborated with the theory as explained in the main text. Indeed, resetting at an optimal rate can significantly shorten the mean queue length for any $S_{on}$ with $CV_{on} \{<1,=1,>1 \}$. In all the plots, we have set $\lambda=0.4$.
• Figure 3: Mean queue length at the optimal Poisson resetting as a function of squared $CV_u$ for different $\langle S_{on} \rangle$ (as mentioned in the plot) and variability drawn from deterministic ($CV_{on}=0$), exponential ($CV_{on}=1$) and Weibull ($CV_{on}=1.4624$) distributions respectively. The underlying service time is drawn from log-normal distribution, whose density is given by Eq. (\ref{['log normal dist']}). Here, we set $\langle S \rangle=1$ and vary $\alpha$ to control $CV_u$ via Eq. (\ref{['cv_deterministic_equation']}), Eq. (\ref{['cv_exponential_equation']}) and Eq. (\ref{['cv_weibull_equation']}) respectively. The dashed slanted lines indicate the Pollaczek-Khinchin formula in the absence of resetting. The left vertical dashed line with $CV_u<1$ indicates the order $\langle N_{r^*=0}^I \rangle> \langle N_{r^*=0}^{II} \rangle>\langle N_{r^*=0}^{III} \rangle$ since $\langle S_{on}^I \rangle>\langle S_{on}^{II} \rangle>\langle S_{on}^{III} \rangle$ and thus resetting is seen to have no impact on the queue length. However, as soon as the optimal resetting rate becomes finite, we see a deviation from the PK formula and the order of the curves is reversed as can be seen from the right vertical dashed line with $CV_u>1$. In this case, we observe the order $\langle N_{r^*}^I \rangle< \langle N_{r^*}^{II} \rangle < \langle N_{r^*}^{III} \rangle$ for a given $CV_u$. Thus, optimally conducted resetting is seen to have more pronounced effect on the queues with smaller fluctuations in the overhead time albeit having a larger $\langle S_{on} \rangle$ compared to the queues with larger fluctuations and smaller mean.
• Figure 4: Panel (a): The mean service time from Eq. (\ref{['mean_sharp']}) as a function of $\tau$ for different overhead time $S_{on}$. The service time $S$ is taken from the log-normal distribution whose density is given by Eq. (\ref{['log normal dist']}). In each case, the optimal resetting rate $\tau^*$ can be identified by the solid circle, where $\langle S_{\tau} \rangle$ attains the global minimum. Here, we set $\langle S \rangle=1$, $\alpha=1.5$ and $\lambda=0.4$. Panel (b) shows $\langle N_{\tau^*} \rangle$ as a function of the underlying $CV_u^2$ (which varies as we change the control parameter $\alpha$). The Pollaczek-Khinchin formula gives the usual linear variation (Eq. (\ref{['sharp-PK']})) as long as $\tau^*=0$. However, as soon as $\tau^*>0$, mean queue length gradually shortens and we see a deviation from that of the underlying process indicating the advantage gained by sharp resetting. Here, the overhead time $S_{on}$ is drawn from three distinct distributions with different $CV_{on}$ as mentioned in the plot.
• Figure 5: The difference between the mean queue length under Poisson optimal resetting and sharp optimal resetting as a function of squared $CV_u$. The service $S$ is taken from the log-normal distribution (density is given in Eq. (\ref{['log normal dist']})) where we fix $\langle S \rangle=1$, and $\langle S_{on}\rangle=0.5$ while $\alpha$ is kept as the control parameter for $CV_u$. The overhead times $S_{on}$ are taken from different distributions: deterministic with $CV_{on}=0$, exponential with $CV_{on}=1$ and Weibull with $CV_{on}>1$. In each case, the difference $\langle N_{r^*} \rangle-\langle N_{\tau^*} \rangle$ is found to be positive conferring that optimal sharp resetting performs better than the optimal Poisson resetting. In this plot, we have set $\lambda=0.4$.