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.
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 [
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 [
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 [
The paper is organized as follows. After descriptions for the model of switched server systems in Section
A switched server system (see Figure
A switched server system with
In terms of above descriptions for switched server systems, the dynamics of the queues of jobs in buffers can be described by the following.
Whenever the buffer
In the subsequent parts, we assume that the total load of buffers satisfies
In this section, stability analysis of two scheduling policies, that is, emptying and service-time-limited policies, is presented, where the service-time-limited policy admits service-time constraints on buffers based on emptying policy.
The emptying policy is described as follows: The buffers are served by the server in terms of any cyclic sequence, for example, Whenever the server switches from serving the buffer When the buffer
where
From the statements in emptying policy, the server, with nonzero setup times, empties queues in buffers in terms of cyclic sequence. The following results hold.
Consider the switched server system described by ( There exists a unique periodic orbit The period where For the periodic orbit
The periodic orbit in Theorem
The emptying policy does not restrict service-time for buffers. However, the problem of constraints on service-time of buffers is of importance in practical applications, as stated in Introduction. In this subsection, the service-time-limited policy is presented based on emptying policy, which can be described by the following.
The first two terms When the buffer where
The information resulted from
Consider the following inequality:
The following results hold for switched server systems under service-time-limited policy.
Consider the switched server system described by (
The proof of Theorem
Consider the following inequality:
Consider the switched server system described by (
When applying service-time-limited policy with factors satisfying (C1), the statements in Theorem
Based on emptying policy, service-time-limited policy admits service-time constraints on buffers by introducing service-time lower-limit and upper-limit factors
Let
It is derived, from the significance of the periodic orbit
Consider the switched server system described by (
Furthermore, it is deduced from (
Consider the switched server system described by (
The proof of Theorem If If
From Theorem
If service-time-limited policy is applied with given
Consider the switched server system described by (
Inequalities (a) in (
For most of real-world problems about queuing systems, service-times and queues of buffers must be constrained. In this paper, inspired by practical problems in traffic control, the service-time-limited policy is proposed, which is the extension to emptying policy. Moreover, the execution of service-time-limited policy in the case of finite buffer capacities is considered, and the notion of feasibility of states under service-time-limited policy is presented. Furthermore, based on the checking condition for feasibility of states (i.e., the solution does not exceed buffer capacities within the first cycle of the server), a sufficient condition for determining feasibility of states is given.
The scheduling policy proposed in this paper admits taking into consideration service-time and queue constraints on buffers by the introduction of the notion of feasibility of states, and service-time upper-limit factors for the feasible state can be solved by testing the nonempty set
From views of traffic control, the server may serve multiple nonconflicting flows, which is our further research extension of results in the paper.
Assume that
We prove that the solution
From (
In conclusion, for any one of three possible cases, the service-time-limited policy converges to emptying policy. Thus, from results in Theorem
Consider switched server systems under service-time-limited policy with
In conclusion, we have that
Statement 1 immediately implies
Furthermore, Statement 1 still holds for
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the referees for their constructive comments. This work is partially supported by National Natural Science Foundation of China (51308005 and 61374191) and Scientific Research Project of Beijing Education Committee (PXM2015_014212_000023, PXM2015_014212_000018, PXM2015_014212_000019, and PXM2015_014212_000021).