Stability Of Switched Server Network And Its Applications In Traffic Signal Control

Switched server network is a class of complex networked systems, which canmodel systems with conflicting material or information flows accessing commonresources, e.g., traffic signal control system. From practical problems in traffic signalcontrol, this paper conducts research for switched server network in two aspects. First,new scheduling policies of the server are proposed for switched server system withone server. Furthermore, more general network model is proposed for multiple servernetwork system, and then signal control of urban traffic network is taken as a typicalexample, for which new signal control strategy is presented. Main research results aresummarized as follows.For switched server system, under the case that the server serves only one buffereach time, two classes of scheduling policies of the server are first presented, i.e., thescheduling policy “emptying and switching in a fixed order” and the schedulingpolicy “priority to the buffer with longest waiting time”, under which switched serversystem is periodically stable if the total load of buffers is less than one. In addition,the analytical expression of stable periodic solution to the system can be obtained.The scheduling policy “priority to the buffer with longest waiting time” can optimizeswitching sequence of the server, such that the waitting service time (or un-servedtime) of buffers is minimized. Moreover, the equivalent relation between thescheduling policy of the server for switched server system and the signal control forsignalized intersections, is established. Then, two classes of scheduling policies of theserver can be directly applied to steady-state control of signalized intersections.Furthermore, this paper generalizes the scheduling policy “emptying andswitching in a fixed order”, and considers the design problem of scheduling policiesof the server with service-time limit of buffers. The scheduling policy “maximumservice-time limit” is first proposed, under which the server assigns a maximumservice-time limit to each buffer, such that each buffer can be fairly served in the caseof large workload in the buffer. It is proved that switched server system under thescheduling policy “maximum service-time limit” is periodically stable if the total loadof buffers is less than the ratio of the minimum and the maximum of maximumservice-time limit adjustment factors of buffers. However, the minimum and themaximum green time durations of each of signal phases should be considered forsignal control of signalized intersections, which corresponds to the case that eachbuffer simultaneously satisfies the minimum and the maximum service-time limit.Thus, the scheduling policy “maximum and minimum service-time limit” is proposed,under which switched server system is periodically stable if the total load of buffers is less than the ratio of the minimum and the maximum of minimum service-time limitadjustment factors of buffers, as well as the ratio of the minimum and the maximumof maximum service-time limit adjustment factors of buffers. For most of real-worldproblems, the buffer capacity (i.e., the maximum workload the buffer canaccommodate) is finite. Based on the scheduling policy “maximum and minimumservice-time limit”, the checking condition for feasible initial states is presented in thecase of finite buffer capacity, i.e., the initial state such that the solution to switchedserver system does not violate buffer capacity constraints within first cyclic switchingof the server.Third, the scheduling policies of the server are designed for switched serversystem with the server simultaneously serving multiple buffers each time. The subsetof simultaneously served buffers is called a phase, similar to a signal phase containingmultiple controlled vehicle flows in a signalized intersection. This paper proposes thescheduling policy “maximum and minimum phase service-time limit”, which isfurther generalization of the scheduling policy “maximum and minimum service-timelimit”. It is proved that switched server system under the scheduling policy“maximum and minimum phase service-time limit” is periodically stable if the totalload of critical buffers is less than the ratio of the minimum and the maximum ofminimum service-time limit adjustment factors of phases, as well as the ratio of theminimum and the maximum of maximum service-time limit adjustment factors ofphases. Then, the proposed scheduling policy can be applied to steady-state control ofmore general type of signalized intersections.For multiple server networks, a general network model is presented, called“Dynamic Graph Hybrid System”. Signal control design of traffic network is taken asa practical example, which is a special class of multiple server networks, and thennew signal control method is proposed. The topological structure of traffic network ismodeled by directed dynamic graph, and the transition of vehicle flows between roadsections is modeled by cell transmission model. Under the assumption that thesampled period of the model is equal to common signal period of traffic network, thesignal control model of traffic network can be obtained, which is a discrete-time lineartime-invariant control system with state varibles being relative occupancy of roadsections and state matrix being unit matrix. The consensus (or balance) of states of thenetwork is an important performance index. For the signal control model of trafficnetwork, the state-feedback control law is designed such that the closed-loop systemcan reach asymptotic stable consensus.For more general case, taking asymptotic stable consensus of the system ascontrol objective, this paper presents methods for the design of the state-feedbackcontrol law for continuous-time and discrete-time linear time-invariant control systemrespectively. The row sums of state matrix of the system are assumed equal to zero for the continuous-time case, and one for the discrete-time case. In fact, the problem ofstate consensus of the system can be transformed to the problem of stability of theequilibrium set of the system. Thus, linear matrix inequality method can be applied tonumerically solve feedback gain matrix of the state-feedback control law by usingpartial stability theory.