Pointwise Stationary Approximation Research Articles

A queuing model for ventilator capacity management during the COVID-19 pandemic.

We applied a queuing model to inform ventilator capacity planning during the first wave of the COVID-19 epidemic in the province of British Columbia (BC), Canada. The core of our framework is a multi-class Erlang loss model that represents ventilator use by both COVID-19 and non-COVID-19 patients. Input for the model includes COVID-19 case projections, and our analysis incorporates projections with different levels of transmission due to public health measures and social distancing. We incorporated data from the BC Intensive Care Unit Database to calibrate and validate the model. Using discrete event simulation, we projected ventilator access, including when capacity would be reached and how many patients would be unable to access a ventilator. Simulation results were compared with three numerical approximation methods, namely pointwise stationary approximation, modified offered load, and fixed point approximation. Using this comparison, we developed a hybrid optimization approach to efficiently identify required ventilator capacity to meet access targets. Model projections demonstrate that public health measures and social distancing potentially averted up to 50 deaths per day in BC, by ensuring that ventilator capacity was not reached during the first wave of COVID-19. Without these measures, an additional 173 ventilators would have been required to ensure that at least 95% of patients can access a ventilator immediately. Our model enables policy makers to estimate critical care utilization based on epidemic projections with different transmission levels, thereby providing a tool to quantify the interplay between public health measures, necessary critical care resources, and patient access indicators.

A Queuing Network Model of a Multi-Airport System Based on Point-Wise Stationary Approximation

A multiple-airport system (MAS) consists of more than two airports in a metropolitan area under a large block of terminal airspace that is managed by one or two air traffic control units. When the capacity of an airport or of the terminal airspace drops, flight delays occur in the MAS system. A quick estimation and predication of traffic congestion in the MAS is important yet challenging. This paper aims to develop a queuing network model of MAS using point-wise stationary queues. The model analyzes the changes of non-stationary queues under the principle of flow conservation to capture flight delay propagation in the system. Regression analyses are performed to examine the relationship between the arrival and departure efficiencies of different airports. The model is validated with the data of Guangdong–Hong Kong–Macao Greater Bay Area airports. Simulation results show that the model can effectively estimate flight delays in the MAS.

Performance evaluation of IoT networks: A product density approach

Mobile cellular-based Internet of Things (IoT) networking is set to be enhanced with the addition of two important pillars of 5G — massive Machine Type Communications and ultra-Reliable Low Latency Communications. Temporal traffic uncertainty from diverse applications in IoT networks mandates novel modeling approaches for performance evaluation. This paper introduces a novel stochastic point process approach that can be used to evaluate the time-dependent performance of IoT base stations. Special correlation functions called Product Densities are used to (a) evaluate time-dependent offered traffic, and (b) analyze delay performance. These performance measures are evaluated at IoT base station for Poisson as well as non-Homogeneous Poisson (Beta distributed) traffic arrival processes suggested by 3rd Generation Partnership Project(3GPP). The Product Density estimates of offered traffic are found to be more accurate than the point-wise stationary approximation (PSA) under non-stationary traffic arrival rates. Results from the proposed analytical model are compared with results from a simulation of two queuing models of the base station; infinite server model for offered traffic and multi-server model for delay performance. Analytical results from Product Density functions also correlate with the simulation outcomes suggesting that the proposed Product Density technique is effective when modeling the time-dependent performance of IoT networks subjected to non-stationary traffic conditions, which reflects the real-world scenario.

Stabilizing performance in a single-server queue with time-varying arrival rate

We consider a class of general $$G_t/G_t/1$$Gt/Gt/1 single-server queues, including the $$M_t/M_t/1$$Mt/Mt/1 queue, with unlimited waiting space, service in order of arrival, and a time-varying arrival rate, where the service rate at each time is subject to control. We study the rate-matching control, where the service rate is made proportional to the arrival rate. We show that the model with the rate-matching control can be regarded as a deterministic time transformation of a stationary G / G / 1 model, so that the queue length distribution is stabilized as time evolves. However, the time-varying virtual waiting time is not stabilized. We show that the time-varying expected virtual waiting time with the rate-matching service-rate control becomes inversely proportional to the arrival rate in a heavy-traffic limit. We also show that no control that stabilizes the queue length asymptotically in heavy traffic can also stabilize the virtual waiting time. Then we consider two square-root service-rate controls and show that one of these stabilizes the waiting time when the arrival rate changes slowly relative to the average service time, so that a pointwise stationary approximation is appropriate.

Using time-varying tolls to optimize truck arrivals at ports

An analytical point-wise stationary approximation model is proposed to analyze time-dependent truck queuing processes with stochastic service time distributions at gates and yards of a port terminal. A convex nonlinear programming model is developed which minimizes the total truck turn time and discomfort due to shifted arrival times. A two-phase optimization approach is used to first compute a system-optimal truck arrival pattern, and then find a desirable pattern of time-varying tolls that leads to the optimal arrival pattern. Numerical experiments are conducted to test the computational efficiency and accuracy of the proposed optimization models.

Time-dependent analysis of virtual waiting time behaviour in discrete time queues

Discrete time queueing models have been shown previously to be of practical use for modelling the approximate time-dependent behaviour of queue length in systems of the form M( t)/ G/ c. In this paper we extend these models to include the time-dependent behaviour of virtual waiting time. Key components of this research are the derivation of exact (algorithmic) expressions for virtual waiting time behaviour. Statistical approximations for the distributional form of virtual waiting times are then developed and tested. These approximations reduce the computational effort involved in the evaluation of the exact expressions by a factor of over 1000 whilst still maintaining very high accuracy levels. The models have been implemented for use on a PC and are shown to be capable of modelling the time-dependent virtual waiting time behaviour of realistic queueing systems in just a few seconds. Results from the models are compared with results obtained by the Simple Stationary Approximation, the Pointwise Stationary Approximation and Modified Offered Load approximations. They clarify further some of the strengths and weaknesses of these approaches and demonstrate the potential of the Discrete Time Modelling approach.

Optimal Pricing and Admission Control in a Queueing System with Periodically Varying Parameters

We consider congestion control in a nonstationary queueing system. Assuming that the arrival and service rates are bounded, periodic functions of time, a Markov decision process (MDP) formulation is developed. We show under the infinite horizon discounted and average reward optimality criteria, for each fixed time, optimal pricing and admission control strategies are nondecreasing in the number of customers in the system. This extends stationary results to the nonstationary setting. Despite this result, the problem still seems intractable. We propose an easily implementable pointwise stationary approximation (PSA) to approximate the optimal policies, suggest a heuristic to improve the implementation of the PSA and verify its usefulness via a numerical study.

Uniform acceleration expansions for Markov chains with time-varying rates

We study uniform acceleration (UA) expansions of finite-state continuous-time Markov chains with time-varying transition rates. The UA expansions can be used to justify, evaluate and refine the pointwise stationary approximation, which is the steady-state distribution associated with the time-dependent generator at the time of interest. We obtain UA approximations from these UA asymptotic expansions. We derive a time-varying analog to the uniformization representation of transition probabilities for chains with constant transition rates, and apply it to establish asymptotic results related to the UA asymptotic expansion. These asymptotic results can serve as appropriate time-varying analogs to the notions of stationary distributions and limiting distributions. We illustrate the UA approximations by doing a numerical example for the time-varying Erlang loss model.

The Lagged PSA for Estimating Peak Congestion in Multiserver Markovian Queues with Periodic Arrival Rates

We propose using a modification of the simple peak hour approximation (SPHA) for estimating peak congestion in multiserver queueing systems with exponential service times and time-varying periodic Poisson arrivals. This lagged pointwise stationary approximation (lagged PSA) is obtained by first estimating the time of the actual peak congestion by the time of peak congestion in an infinite server model and then substituting the arrival rate at this time in the corresponding stationary finite server model. We show that the lagged PSA is always more accurate than the SPHA and results in dramatically smaller errors when average service times are greater than a half an hour (based on a 24 hour period). More importantly, the lagged PSA reliably identifies proper staffing levels to meet targeted performance levels to keep congestion low.

On the Accuracy of the Simple Peak Hour Approximation for Markovian Queues

We empirically explore the accuracy of the simple stationary peak hour approximation (SPHA) for estimating peak hour performance in multiserver queuing systems with exponential service times and periodic (sinusoidal) Poisson arrival processes. We show that the SPHA is very good for a range of parameter values corresponding to a reasonably broad spectrum of real systems. However, we do find and document that there are many situations in which this approximation will be very inaccurate. We postulate and then support empirically a set of hypotheses that link the accuracy of the SPHA and the related point-wise stationary approximation (PSA) to key parameter values and model characteristics. We also present results on the time-dependent behavior of these systems as a function of key parameters. Finally we present results which indicate that our findings, developed for models with sinusoidal input streams, may apply to a much broader range of Markovian models with more general cyclic inputs.

5388186 Differential process controller using artificial neural networks : Bose Chinmoy B Green Brook, NJ, UNITED STATES Assigned to AT&T Corp

An analytical point-wise stationary approximation model is proposed to analyze time-dependent truck queuing processes with stochastic service time distributions at gates and yards of a port terminal. A convex nonlinear programming model is developed which minimizes the total truck turn time and discomfort due to shifted arrival times. A two-phase optimization approach is used to first compute a system-optimal truck arrival pattern, and then find a desirable pattern of time-varying tolls that leads to the optimal arrival pattern. Numerical experiments are conducted to test the computational efficiency and accuracy of the proposed optimization models.

The Physics of the Mt/G/∞ Queue

We establish some general structural results and derive some simple formulas describing the time-dependent performance of the Mt/G/∞ queue (with a nonhomogeneous Poisson arrival process). We know that, for appropriate initial conditions, the number of busy servers at time t has a Poisson distribution for each t. Our results show how the time-dependent mean function m depends on the time-dependent arrival-rate function λ and the service-time distribution. For example, when λ is quadratic, the mean m(t) coincides with the pointwise stationary approximation λ(t)E[S], where S is a service time, except for a time lag and a space shift. It is significant that the well known insensitivity property of the stationary M/G/∞ model does not hold for the nonstationary Mt/G/∞ model; the time-dependent mean function m depends on the service-time distribution beyond its mean. The service-time stationary-excess distribution plays an important role. When λ is decreasing before time t, m(t) is increasing in the service-time variability, but when λ is increasing before time t, m(t) is decreasing in service-time variability. We suggest using these infinite-server results to approximately describe the time-dependent behavior of multiserver systems in which some arrivals are lost or delayed.

Mt/G/∞ Queues with Sinusoidal Arrival Rates

In this paper we describe the mean number of busy servers as a function of time in an Mt/G/∞ queue (having a nonhomogeneous Poisson arrival process) with a sinusoidal arrival rate function. For an Mt/G/∞ model with appropriate initial conditions, it is known that the number of busy servers at time t has a Poisson distribution for each t, so that the full distribution is characterized by its mean. Our formulas show how the peak congestion lags behind the peak arrival rate and how much less is the range of congestion than the range of offered load. The simple formulas can also be regarded as consequences of linear system theory, because the mean function can be regarded as the image of a linear operator applied to the arrival rate function. We also investigate the quality of various approximations for the mean number of busy servers such as the pointwise stationary approximation and several polynomial approximations. Finally, we apply the results for sinusoidal arrival rate functions to treat general periodic arrival rate functions using Fourier series. These results are intended to provide a better understanding of the behavior of the Mt/G/∞ model and related Mt/G/s/r models where some customers are lost or delayed.

The Pointwise Stationary Approximation for Mt/Mt/s Queues Is Asymptotically Correct As the Rates Increase

Green, Kolesar and Svoronos (in press) and Green and Kolesar (in press) use numerical methods to investigate the behavior of multiserver Markov queues with a Poisson arrival process having a sinusoidal arrival rate. For this model they propose an approximation for long-run average performance measures called the pointwise stationary approximation (PSA), which consists of an appropriate weighted average of the performance measure that would result at each point in time if the system were stationary with the arrival rate that applies at that point in time. In this paper we verify their conjecture that PSA is asymptotically correct as the service and arrival rates increase with the instantaneous traffic intensity held fixed (corresponding to long arrival rate cycles). We actually establish both pointwise and average versions of this result for general time-dependent birth-and-death processes.

The Pointwise Stationary Approximation for Mt/Mt/s Queues Is Asymptotically Correct As the Rates Increase

Green, Kolesar and Svoronos in press and Green and Kolesar in press use numerical methods to investigate the behavior of multiserver Markov queues with a Poisson arrival process having a sinusoidal...

The Pointwise Stationary Approximation for Queues with Nonstationary Arrivals

We empirically explore the accuracy of an easily computed approximation for long run, average performance measures such as expected delay and probability of delay in multiserver queueing systems with exponential service times and periodic (sinusoidal) Poisson arrival processes. The pointwise stationary approximation is computed by integrating over time (that is taking the expectation of) the formula for the stationary performance measure with the arrival rate that applies at each point in time. This approximation, which has been empirically confirmed as a tight upper bound of the true value, is shown to be very accurate for a range of parameter values corresponding to a reasonably broad spectrum of real systems.

The Pointwise Stationary Approximation for Queues with Nonstationary Arrivals

We empirically explore the accuracy of an easily computed approximation for long run, average performance measures such as expected delay and probability of delay in multiserver queueing systems wi...