Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 347931, 10 pages

http://dx.doi.org/10.1155/2015/347931

## Stability of Switched Server Systems with Constraints on Service-Time and Capacity of Buffers

Beijing Key Lab of Urban Intelligent Traffic Control Technology, North China University of Technology, Beijing 100144, China

Received 6 February 2015; Revised 18 May 2015; Accepted 20 May 2015

Academic Editor: Sebastian Anita

Copyright © 2015 Li Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The execution of emptying policy ensures the convergence of any solution to the system to a unique periodic orbit, which does not impose constraints on service-time and capacity of buffers. Motivated by these problems, in this paper, the service-time-limited policy is first proposed based on the information resulted from the periodic orbit under emptying policy, which imposes lower and upper bounds on emptying time for the queue in each buffer, by introducing lower-limit and upper-limit service-time factors. Furthermore, the execution of service-time-limited policy in the case of finite buffer capacity is considered. Moreover, the notion of feasibility of states under service-time-limited policy is introduced and then the checking condition for feasibility of states is given; that is, the solution does not exceed the buffer capacity within the first cycle of the server. At last, a sufficient condition for determining upper-limit service-time factors ensuring that the given state is feasible is given.

#### 1. Introduction

Switched server system is a class of mathematical models for queuing systems with finite number of conflicting queues alternately served by a single server. Moreover, there exists a nonzero setup time of the server whenever the server switches from serving one queue to another one, and assume that the jobs arrive at and leave the buffer at constant rates in this paper. The evolution of the system involves continuous changes of queues in buffers and discrete switching of the server, and thus switched server system is a special class of hybrid systems [1, 2], with extensive applications in practical problems, such as manufacturing systems [3, 4] and traffic signal control systems [5–7], and more applications of this field can be referred to [8].

Fundamental synthesis problem for switched server systems is to design the scheduling policy of the server. The emptying policy (i.e., the server alternately empties queues in buffers with any fixed cyclic sequence) was proposed in [9], under which any solution to the system asymptotically converges to a unique periodic orbit analytically determined by system parameters [6]. However, the emptying policy does not impose constraints on queue-emptying time in converging process of the solution. In practical applications, the server with emptying policy must take longer time to empty buffers with larger queues, and thus other buffers have to wait longer time for service. Thus, in order to ensure fairness for all buffers, the upper bound for emptying time of each buffer based on emptying policy was considered in [10], and a conjecture about stability of the policy was given, which was further proved in [11]. Also, [12] considered distributed execution of emptying policy with upper bounds for queue-emptying time of buffers in the network with multiple servers. In most of literatures, a scheduling policy is first proposed, and then dynamic behaviors of the system are analyzed, as in [9]. In [13–16], a different idea for controlling the network was presented; that is, the steady state (a periodic orbit) of the system is first given, and then corresponding scheduling policy is derived ensuring the convergence of any solution to the steady state. However, the policies in [13–16] resulting from the given periodic orbit do not impose constraints on service time of buffers.

The problems about designs of the scheduling policies with constraints on queue serving process mainly result from practical applications. For example, in traffic intersection, the signal control for signalized intersections was modeled as switched server systems in [5, 6], and emptying policy was applied, where signal light in a signalized intersection is seen as the server; incoming links to the signalized intersection are seen as buffers, which can accommodate queues of vehicles; the lost time between phase switching is seen as the nonzero setup time of the server; and signal control law is seen as the scheduling policy of the server. However, in traffic control [17], the shortest and longest green-time constraints on each of traffic phases are necessarily imposed for feasible signal control plans, with the purpose of ensuring traffic safety for drivers and pedestrians, and controlling total delay of signalized intersections, respectively. Thus, inspired by traffic control, the emptying policy is further extended in this paper, based on which the service-time-limited policy is proposed, with lower and upper bounds on queue-emptying time of each buffer by introducing lower-limit and upper-limit service-time factors, respectively. Furthermore, the buffer capacity is finite for most of real-world problems. For example, in a signalized intersection, incoming links with finite length only accommodates finite number of vehicles. Thus, the execution of service-time-limited policy in the case of finite buffer capacities is considered, and moreover the notion of feasibility of states under service-time-limited policy is introduced, that is, the state originating in which the solution asymptotically converges to the steady state (the periodic orbit) and does not exceed buffer capacities in the converging process. Moreover, the checking condition for feasibility of states is given; that is, the solution does not exceed buffer capacities within the first cycle of the server, and a sufficient condition for determining upper-limit service-time factors ensuring that the given state is feasible is given.

The paper is organized as follows. After descriptions for the model of switched server systems in Section 2, we introduce emptying and service-time-limited policies in Section 3. Feasibility of states and checking conditions under service-time-limited policy are considered in Section 4. Conclusions and future research topics are given in Section 5.

#### 2. Descriptions of Switched Server Systems

A switched server system (see Figure 1 for illustration) consists of buffers and a single server, where the server alternately serves buffers in terms of the scheduling policy and only one buffer each time. Let denote the queue of jobs in the buffer at the moment . Because of nonnegative constraints on the queue of jobs in each buffer, the state space of the system is defined as . Assume that the jobs arrive at the buffer at a constant rate [lots/s]. Whenever the buffer , in which there are accumulative queues, that is, , is served by the server, the jobs leave the buffer at a constant rate [lots/s]; and whenever the buffer , in which there are no accumulative queues, that is, , is served by the server, the jobs leave the buffer at the constant rate . Both and are called arriving rate and service rate of jobs in the buffer , respectively, and is called the load of the buffer . Whenever the server switches from serving the buffer to the buffer , there exists a nonzero setup time , , , during which the server is in idle.