Asymptotic optimality of the queue service probability for the radial basis function network-based queue selection rule
Introduction
Queueing systems are popular and widely used tools for analyzing the performance of communication networks, transportation system (Sahba & Balcioglu, 2011), job-shop manufacturing system (Georgiadis and Michaloudis, 2012, Shen and Buscher, 2012), and machine-repairman system (Yuan, 2012). Usually, many queueing systems are generally designed with sufficient capacity or resources to ensure that the system is stable, while preserving quality of service (QoS). However, variability and constraints, e.g. heavy traffic, timing constraints, finite queue capacity, etc., in a queueing system may have significant impacts on its performance. The study of heavy traffic began in the 1960s by Kingman (1962). Kingman’s bound implies that even small variations in service times or inter-arrival times can lead to significant delay. Apart from this, it is inevitable that a system can also encounter a periodic or temporary overload,1 either due to an unpredictable increase in load, unavailable service resources, or system breakdown. Another practical situation occurs on message scheduling in the controller area network (Mubeen, Mäki-Turja, & Sjödin, 2014). In such a context, the maximum normal utilization of the controller area network is about 30%. Under heavy disturbance, it rises to about 40% due to retransmission of corrupted messages (Johansson, Törngren, & Nielsen, 2005). The direct consequence can possibly bring about a fatal catastrophe or even a matter of life and death. The aforementioned problems become worse when the queueing system is conditioned by finite capacity and/or imposed on stringent timing constraints.
The finite-capacity queueing systems with timing constraints have been widely studied by many researchers (Osorio and Bierlaire, 2009, Sakuma and Inoie, 2014, Wu et al., 2012) and our recent publications (Chen, 2015, Chen and Yen, 2011, Chen and Yen, 2012a, Chen and Yen, 2012b, Chen and Yen, 2013, Chen and Yen, 2014). A queueing system with these restrictions occurs frequently in our real life. For instance, there may be a surge in demand for the limited hospital emergency rooms during a catastrophic event (Lakshmi & Sivakumar, 2013). In manufacturing systems, there are limited waiting rooms between workstations in assembly lines. In semiconductor wafer fabrication manufacturing, queue-time constraints occur when a set of consecutive process steps must be completed within a fixed time window (Tsai, 2008). Violations of these constraints can result in rework, scrap, and longer cycle times. To ensure product quality, the queue time between consecutive operations is often required to be shorter than a pre-specified duration. The effect is similar to imposing a limited buffer size between operations. On highways, there are rush hours during which drivers have to queue for the use of highways. Regardless of how many lanes already exist, it has finite road capacities and is vulnerable to possible congestion during weekend or holidays. In all these circumstances, the system can have excessive demands than it can handle and possibly leads to unstable levels or causes unpredictable performance degradation.
Regarding aforementioned problems, three primary considerations are (1) how the system can be stabilized, (2) how the system throughput is maximized, and (3) how the upper bounds of waiting times can be estimated. Pertaining to issues (1) and (2), several related literatures aim to avoid the degradation of throughput of the server as well as to maintain the system at its steady state. Schemes such as admission control (Cherkasova & Phaal, 2002) or specialized scheduling policies (Chen & Mohapatra, 2002), or a combination of both (Elnikety, Nahum, Tracey, & Zwaenepoel, 2004) have been studied. Furthermore, Solar (Stolyar, 2004) analyzed the parallel server queueing systems, under the assumption that the system is in heavy-traffic. That is, the system is stable but operates close to the boundary of the stability region. Shakkottai et al., 2004, Bell and Williams, 2005 also used the complete resource pooling condition to establish heavy-traffic optimality of other resource allocation models. Other researchers (Andrad’ottir and Ayhan, 2005, Andrad’ottir et al., 2001, Andrad’ottir et al., 2003, Tassiulas and Bhattacharya, 2000) concerned with the dynamic assignment of servers to maximize the system throughput in queueing networks. In addition, Venkataraman and Lin (2007) proposed scheduling algorithms that can stabilize the network at given offered loads, which also ensures that the long term average and service rate are no less than the arrival rate of each user.
Regarding issue (3), let us consider another example of the hospital queueing systems. Derlet et al., 2001, Carter and Lapierre, 1991 pointed out that the causes of hospital overcrowding include low staff availability, bed shortage, insufficient hospital space, and so forth. Actually, long waiting time is symptomatic of inefficiency and indicative of the system’s inability to satisfy patients’ demand. To ensure QoS, several criteria can be established for a satisfactory level of service. One possible gauge might be that the upper limit of waiting time should not exceed a certain value. Therefore, patients’ satisfactions are often related to the amount of time they have to wait before receiving treatments. For the mentioned criterion, the upper bound of waiting time needs to be known first for this purpose. This problem is principally related to issue (3) and is vital when the queueing system is imposed by timing constraints and with restrictions of finite capacity. In fact, a search of available literatures indicates that relatively few studies have been devoted to fully address this problem. Furthermore, due to the growth of traffic with the increasing amount of data exchanged between electronic control units in the controller area network, our motivation of this study aims at estimating the message waiting times in an overloaded circumstance.
Based on our research findings, this paper presents a machine learning approach with the radial basis function network (RBFN) (Chen & Yen, 2011) in dealing with the aforementioned issue. The RBFNs have demonstrated their usefulness in a variety of applications, including classification, prediction, and system modeling. In fact, the RBFN is a variant of the multilayer perceptron networks (Haykin, 1994) with single hidden layer and makes use of locally supported functions to calculate the Euclidian distance between the input vector and the centers of RBFs. The latest studies and applications on the RBFNs can be refereed to Dehghan and his colleagues (Dehghan and Mohammadi, 2014, Dehghan and Mohammadi, 2015, Dehghana et al., 2014, Dehghan et al., 2015, Ilati and Dehghan, 2015). With the radial basis function network as the queue selection rule (QSR), messages in queues are selected for admission into the service facility in accordance with the decision made by the RBFN. To the best of our knowledge, this study is the first attempt to combine machine learning strategy with dynamical scheduling to estimate waiting times under the overload situation. Especially, the proposed RBFN-based QSR directly leads to derive the asymptotic optimality of the queue service probability (QSP) as well as waiting time in the time domain directly.
The remaining part of the paper is organized as follows. Section 2 introduces the definition of queue cycle, the effective arrival rate, and the mean queue length for subsequent analysis. In Section 3, relationship between waiting time and mean waiting time is established. Section 4 is devoted to derive the queue service probability of the RBFN-based queue selection rule with support theorem. Section 5 reports and validates the results of simulation experiments. Finally, Section 6 concludes the paper and suggests our future research directions.
Section snippets
The queue cycle
Typically, the queueing system consists of a number of parallel queues attended by a single server. Thus, the control policy of the QSR focuses on determining the queue and events in that queue for admission into the service facility. Once a queue is selected for service, events within that queue are scheduled using first-come-first-served order. At this point the word “event” is a generic expression that represents many real-world phenomena, such as airplanes arriving to an airport, shoppers
Waiting time and mean waiting time
In the following analysis, we have assumed that the server is immediately available when the service is completed. Furthermore, preemption is not allowed. In other words, once a service operation has started, it cannot be interrupted by any other requests. As long as the queue is not full, the arriving message joins the tail of the dedicated queue immediately and the waiting time of this message starts to be counted. Then, any message can be ready for service until it reaches the head of the
Queue service probability of the RBFN-based QSR
Although the relationship between and is established in the last section, either of them still cannot be obtained analytically. The difficulty is due to different control strategies of QSRs we concerned with. In this paper, we focus on using the RBFN as our QSR. Theorem 1 is present to illustrate how the corresponding QSP is derived. Other possible QSRs and their QSPs can be referred to Yen (2011). Theorem 1 The queue service probability of using the RBFN-based QSR iswhere c is
Simulation
In this section, two experiments are conducted to verify the correctness of the proposed method, especially when the system is overloaded. The first experiment fixed the queue capacity (), while the second one is simulated with a constant load intensity (ρ). The former one examines the limiting behaviors of the QSP as well as waiting times, while the latter case further validates the proposed paradigm. Although our simulations focus on three-queue problem, the closed form expressions of QSP
Conclusion
The motivation of this study originates from supporting the industrial Controller Area Network (CAN) as the backbone of the Intelligent Vehicle System (IVS). The primary role of the CAN is to provide the data communications infrastructure for controlling various devices and subsystems within an IVS. Under heavy or overload fluctuations, the CAN is very vulnerable to network load and may lead to increase in the delay of the messages exchanged between the controller, actuators, and sensors, etc.
Acknowledgment
This research was supported by the Ministry of Science and Technology, Taiwan, R.O.C., under contract numbers MOST 103-2221-E-212-011 and MOST 104-2221-E-212-011.
References (42)
Neuro-fuzzy approach for online message scheduling
Engineering Applications of Artificial Intelligence
(2015)- et al.
Applications of machine learning approach on multi-queue message scheduling
Expert Systems With Applications
(2011) - et al.
A probabilistic approach to estimate the mean waiting times in the EDF polling
Computers & Industrial Engineering
(2013) - et al.
A meshless technique based on the local radial basis functions collocation method for solving parabolic–parabolic Patlak–Keller–Segel chemotaxis model
Engineering Analysis with Boundary Elements
(2015) - et al.
The numerical solution of Fokker–Planck equation with radial basis functions (RBFs) based on the meshless technique of Kansa’s approach and Galerkin method
Engineering Analysis with Boundary Elements
(2014) - et al.
The numerical solution of Cahn–Hilliard (CH) equation in one, two and three-dimensions via globally radial basis functions (GRBFs) and RBFs-differential quadrature (RBFs-DQ) methods
Engineering Analysis with Boundary Elements
(2015) - et al.
Real-time production planning and control system for job-shop manufacturing: A system dynamics analysis
European Journal of Operational Research
(2012) - et al.
The use of radial basis functions (RBFs) collocation and RBF-QR methods for solving the coupled nonlinear sine-Gordon equations
Engineering Analysis with Boundary Elements
(2015) - et al.
Alternating priority versus FCFS scheduling in a two-class queueing system
Operations Research Letters
(2012) - et al.
Discrete time GI/Geom/1 queueing system with priority
European Journal of Operational Research
(2008)
An analytic finite capacity queueing network model capturing the propagation of congestion and blocking
European Journal of Operational Research
The impact of transportation delays on repairshop capacity pooling and spare part inventories
European Journal of Operational Research
An approximation analysis for an assembly-like queueing system with time-constraint items
Applied Mathematical Modelling
Solving the serial batching problem in job shop manufacturing systems
European Journal of Operational Research
Dynamic production control in parallel processing systems under process queue time constraints
Computers & Industrial Engineering
Reliability analysis for a k-out-of-n: G system with redundant dependency and repairmen having multiple vacations
Applied Mathematics and Computation
Throughput maximization for tandem lines with two stations and flexible servers
Operations Research
Server assignment policies for maximizing the steady-state throughput of finite queueing systems
Management Science
Dynamic server allocation for queueing network with flexible servers
Operations Research
Dynamic scheduling of a parallel server system in heavy traffic with complete resource pooling: Asymptotic optimality of a threshold policy
Electronic Journal of Probability
Cited by (1)
Sustainable queuing-network design for airport security based on the Monte Carlo method
2018, Sustainability (Switzerland)