Designing parsimonious scheduling policies for complex resource allocation systems through concurrency theory

Transcription

1 Designing parsimonious scheduling policies for complex resource allocation systems through concurrency theory Ran Li and Spyros Reveliotis School of Industrial & Systems Engineering Georgia Institute of Technology Abstract This work is part of a broader research program seeking to exploit the representational and the computational power of the formal modeling frameworks that are provided by Discrete Event Systems (DES) theory, in order to develop near-optimal scheduling policies for DES that model complex resource allocation. Hence, in a recently published work, we have shown how the modeling framework of Generalized Stochastic Petri Nets (GSPNs) enables the effective performance analysis and control of complex resource allocation systems by expressing the underlying scheduling problem(s) as the problem of the optimal pricing of the various random switches that are employed by the corresponding GSPN model. Furthermore, the imposition of some additional conditions that must be observed by this pricing leads to the definition of a set of policy spaces that enable a systematic and controllable trade-off between the tractability of the resulting optimization / scheduling problems and the operational efficiency of the obtained solutions. In this work, we pursue a refinement of the originally defined policy spaces that can result in more tractable formulations of the underlying scheduling problems, and we investigate the GSPN structure and properties that enable the definition of these policy spaces. The last part of the paper also reveals some further adaptability that is possessed by the presented results, by customizing these results to a particular GSPN sub-class that models the operations of a capacitated re-entrant line. 1 Introduction A prominent task in many contemporary control applications concerns the effective and efficient allocation of a finite set of reusable resources to a set of concurrently executing processes. These processes typically evolve in a sequential manner, going through a series of stages, and each stage requests the exclusive allocation of a certain subset of the system resources for its effective execution. Example applications of such a resource allocation paradigm are provided by the operations that take place in flexibly automated production systems [16, 36], the task assignment and the coordinated navigation of the fleets of the various robotic devices that are used for the support of the material handling operations in many production and service facilities [28, 33], the traffic management of the automated railway and monorail systems that have been deployed in various urban areas or are contemplated for these environments [11], and the operating systems that support the functionality of the contemporary computational platforms This work was partially supported by NSF grants CMMI and ECCS

2 [19, 9]. In all these environments, the concurrency and the flexibility that are aspired for the operation of the underlying system give rise to resource allocation patterns that are very complex. Furthermore, the extensive automation that is currently sought for the aforementioned environments implies that this complex resource allocation must be controlled not only for certain aspects of operational efficiency, but also for logical consistency and behavioral correctness. Among this last set of concerns, a particularly prominent one is that of establishing deadlockfree and live operation for the controlled system. The satisfaction of this requirement further necessitates the representation and the analysis of the considered operations at a fidelity level that is considerably higher than the representation and the analysis that have been typically pursued by the past scheduling paradigms. To reason effectively about potential deadlocks and the system liveness, we must model explicitly not only the allocation of the system processors, but also the allocation that concerns elements of a more passive nature, like the buffers and the auxiliary equipment that might be necessary in a manufacturing setting, the various links in the guidepath network of an automated guided vehicle (AGV) system, or the locks and the semaphores that are typically used for the coordination of the resource allocation that takes place in multi-threaded software; due to their limited availability and/or their restrictive nature in the context of the corresponding (automated) operations, each of these elements can give rise to highly disruptive effects if it is not properly recognized and managed by the adopted operational policies. 1 In order to address the aforementioned need, the relevant research community has introduced the modeling paradigm of the (sequential) resource allocation system (RAS) [30]. A concise and comprehensive account of the corresponding RAS theory can be found in [31]. The perusal of this material will reveal that in an effort to cope with the complexity of the scheduling problems addressed, the considered RAS theory decomposes them to (i) a logical-control problem that addresses the liveness-enforcing supervision of the underlying resource allocation function, and (ii) a performance-control problem that seeks to optimize certain performance indices for the underlying system while observing the operational constraints that are imposed by the liveness-enforcing supervisor (LES). Furthermore, the same material will also reveal that the existing RAS theory has attained tremendous progress with respect to (w.r.t.) the first of these two sub-problems. But, on the other hand, currently we lack a similar strong set of results for the counterpart problem of the RAS performance modeling, analysis and control. More specifically, when it comes to the sub-problem of the RAS performance modeling, analysis and control, a series of past studies [6, 7, 30] have established that, for a large part of the considered RAS, and under some typical performance objectives like those of throughput maximization and process delay / congestion minimization, an optimal scheduling policy can be obtained, in principle, through the formulation and solution of an (average reward) Markov Decision Process (AR-MDP) [25]. However, the practical application of this methodology is severely limited by the very large sizes of the involved state spaces. In fact, the typical requirement that is posed by the standard MDP modeling framework of having a distinct optimal action specified for each state of the underlying state space, implies that the sought policies have not only a very high computational cost, but they cannot even admit a sufficiently parsimonious representation that will facilitate an efficient real-time implementation. This last problem is currently addressed in the context of the broader MDP theory by the emerging field of Approximate Dynamic Programming (ADP) [3, 24]. 1 From a more methodological standpoint, the scheduling problems considered in this work are reminiscent of the scheduling problems that arise in the context of queueing networks with blocking [23], a queueing-theoretic model that is among the least studied by the current scheduling theory. 2

3 Generally speaking, ADP theory tries to obtain near-optimal policies of manageable representational and computational complexity to otherwise intractable MDP problems either (i) by developing pertinent compact approximations of the underlying value function, or (ii) by further restricting the initial MDP formulation to policy spaces that admit a more parsimonious representation w.r.t. the underlying plant system. Hence, motivated by these more general developments, in a recent line of our work that was introduced in [18], we have set out to provide (re-)formulations of the scheduling problems arising in the context of the considered RAS, that (i) admit a more parsimonious representation of the target scheduling policies w.r.t. any concise representation of the corresponding RAS, and, at the same time, (ii) enable an effective and systematic trade-off between the representational and computational complexity of these policies and their operational efficiency. Instrumental in this endeavor has been the employment of the formal modeling framework of the Generalized Stochastic Petri Nets (GSPN) [1]. This framework enables an integrated modeling of the qualitative and the quantitative aspects of the underlying RAS dynamics, since many LES of the considered RAS classes admit a PN-based representation [27, 22, 8]. At the same time, the timed dynamics of the considered RAS and the corresponding scheduling policies are modeled by the typical differentiation of the GSPN transitions into timed and untimed ; the former model the execution of the various processing stages of the considered RAS, while the latter model the decision-making process that advances the RAS process instances through their corresponding processing stages and administers the allocation of the requested resources. More specifically, in the modeling framework of [18], the considered scheduling problem boils down to the specification of the probability distributions that will arbitrate the firing of the untimed transitions that are enabled at the various net markings; these distributions are known as random switches in the GSPN terminology [1]. In fact, in the GSPN-based modeling framework that was outlined in the previous paragraph, the particular class of scheduling policies that employ a distinct random switch for each marking that enables a number of untimed transitions, is equivalent to the class of the randomized stationary scheduling policies that are considered by the AR-MDP formulations. Therefore, the policies in this class exhibit the same representational and computational challenges with their MDP-based counterparts. But the explicit modeling of the structure of a scheduling policy to be employed in the considered modeling framework by the concept of the random switch enables the specification of additional policy spaces, that are pretty rich in terms of the supported behaviors, and define their constituent policies through the structure and/or the dynamics of the RAS-modeling GSPN. The careful structural analysis of the policies that are defined by these policy spaces, and of the GSPN dynamics that are induced by them, can reveal effective mechanisms for managing the pursued trade-off between (i) the representational and computational tractability of the underlying scheduling problem, and (ii) the operational efficiency of the derived policies. As a first step in this direction, the approach that was presented in [18] restricted the scheduling policies under consideration to those that are defined by the concept of static random switches: this concept essentially stipulates that all markings enabling the same set of untimed transitions will share the same random switch. Hence, the set of random switches that are employed by this last class of scheduling policies is determined by the distinct sets of untimed transitions that are simultaneously enabled at the various markings of the underlying GSPN, and not by the set of these markings itself. Since the first of the aforementioned sets usually is much smaller than the second one, it is expected that the proposed restriction for the considered scheduling policies will reduce substantially the set of the employed random switches, and the corresponding set of the decision variables that must be resolved by the finally formu- 3

4 lated optimization problem. At the same time, the resulting policy space contains the stage (or class ) - based priority policies that are frequently used in various practical settings of the considered applications, and in certain cases have been shown to be optimal [14, 20]. From a more theoretical standpoint, the scheduling policies that are defined by the static random switches of [18] essentially establish an aggregation of the underlying state space and address the considered scheduling problems through these state aggregates. But then, additional performance improvements can be attained through a controlled partial disaggregation of these aggregates; i.e., through a redefinition of the static random switches that correspond to some of these aggregates over more refined state subsets. Finally, the results of [18] also provide the necessary theory for the optimal pricing of the finally employed random switches through simulation-based optimization strategies. In this work, we pursue an alternative approach towards the development of more parsimonious policy spaces for the considered scheduling problems. This approach seeks to redefine the set of the random switches that arise naturally in the dynamics of the RAS-modeling GSPN, without compromising, however, the ability of the resulting policy space to represent the optimal scheduling policies. This is attained by identifying and eliminating some of the concurrency in the untimed dynamics of the underlying GSPN model in a way that does not impact the timed dynamics of the net, which shape its timed-based performance. The policy spaces that are defined by this refinement process are not only more parsimonious in terms of the random switches and the decision variables that are employed in the final formulation of the corresponding scheduling problem, but they also provide a crisper and more pertinent characterization of the essential conflicts in the underlying resource allocation function that are in an actual need of the pursued coordination and scheduling. Furthermore, as it will be revealed in the sequel, the approach that is pursued in this work can be complemented by the notion of the static random switch that was employed in [18]; when considered in that context, the presented approach enables the further refinement and the representational enhancement of the policy spaces that were defined in that earlier work. In fact, some numerical experimentation that is reported in the last part of the paper will reveal that the refinement that is attained by the proposed method ca be very drastic in the setting of both policy spaces that are respectively defined by the static and a more dynamic interpretation of the random switch concept. These experiments are performed on a particular RAS structure to be characterized as the capacitated re-entrant line (CRL) [29], and the presented developments will also reveal a potential for further adaptability and customization of the established theory, through the exploitation of additional special structure that might be present in the various RAS classes. In view of the above discussion, the rest of the paper is organized as follows: Section 2 provides the necessary background on the GSPN modeling framework and its employment in [18] for the effective modeling of the RAS scheduling problems that are considered in this work. Section 3 presents the main set of the results, i.e., the redefinition of the random switches that are employed by the considered GSPN models, in an effort to control the primary decision variables that appear in the mathematical programming (MP) formulation of the corresponding scheduling problems, without compromising, however, the possibility of the resulting policy spaces to express the optimal scheduling policies. Also, the results that are presented in the last part of this section reveal a notion of non-conflict that provides a more intuitive motivation and explanation of the pursued simplification. Section 4 deals with the customization of the developments in Section 3 to the considered CRL operational context, and it also presents the computational experiments that demonstrate the substantial gains of the proposed method w.r.t. the representational efficiency of the obtained policy spaces. Finally, Section 5 concludes 4

5 the paper and outlines possible directions for future work. 2 The GSPN modeling framework and the scheduling problem considered in [18] This section provides a more technical overview of the developments of [18] that are necessary for the detailed positioning of the problem that is addressed in this work. The formal characterization of this problem and the particular solutions that are proposed in this work are the content of subsequent sections of this paper. In the following discussion of this section we assume that the reader is familiar with the basic PN theory, and even with the main concepts and methodology that underlie the GSPN modeling framework, and we stress those aspects of these two frameworks that are most relevant to the results of [18]. When viewed from this standpoint, the following discussion also introduces important notation that will be used in the rest of this document. Finally, we notice that some comprehensive and excellent introductions to the basic PN theory and to the GSPN modeling framework are provided, respectively, in [21, 5] and [2, 1]. 2.1 The GSPN model considered in [18] Following the developments of [18], we represent a GSPN N by a septuple N = (P, T t, T u, W t, W u, m 0, R), where: P is the set of places; T t and T u are respectively the sets of timed and untimed transitions; W t and W u are the flow relations that are respectively defined in (P T t ) (T t P ) and (P T u ) (T u P ) and express the connectivity of the elements of T t and T u to the elements of P ; m 0 is the initial marking of N ; and, finally, the mapping R : T t R + gives r t R(t), t T t, the rate of the exponential distribution that determines the firing times for transition t. The considered nets are non-ordinary, and therefore, each pair in the flow relations W t and W u is associated with a weight, w(p, t) or w(t, p), that denotes the corresponding token flow. For a given marking m Z P + and a place p P, m(p) denotes the number of tokens placed by marking m in place p. Furthermore, the basic transitional dynamics of net N are expressed by the transition function tr(, ) : Z P + T Z P + that is defined as follows: m = tr(m, t) if m (p) := m(p) w(p, t) + w(t, p) and m(p) w(p, t) 0 for all p P ; otherwise, tr(m, t) is undefined. We shall use the notation tr(m, t)! to indicate that function tr(, ) is well-defined for the pair (m, t), and we shall say that t(, ) is enabled or fireable at m. Also, we shall extend the domain of tr(, ) from Z P + T to Z P + T in the standard manner: 2 m n = 0 tr(m, t 1... t n ) = tr(m, t 1 ) n = 1 tr(tr(m, t 1... t n 1 ), t n ) n 2 In line with typical convention in the PN literature, we shall use the notation x and x to denote, respectively, the input and the output nodes for a node x (place or transition) of 2 We remind the reader that T denotes the Kleene closure of the set T, i.e., all the finite-length strings consisting of elements from this set, including the empty string ɛ. (1) 5

6 the PN N, and we shall also employ the natural extension of this notation to entire sets of t nodes of the same type. At times, we shall also use the notation m 1...t n 1 m2 instead of m 2 = tr(m 1, t 1... t n ). Finally, for any given transition sequence σ T, we use the notation σ to denote the scoring or the Parikh vector of this sequence; i.e., σ is a vector of dimensionality equal to T, with its i-th component reporting the number of the appearances of the transition t i in σ. We also adopt the following notation from [18]: R(N ) Z P + is the set of all markings of the GSPN N that are reachable from its initial marking m 0 under the (standard) transition firing scheme that is defined by tr(, ). For a marking m R(N ): E u (m) is the set of the enabled untimed transitions at m; E t (m) is the set of the enabled timed transitions at m; and E(m) = E u (m) E t (m). If E u (m) =, then m is characterized as a tangible marking; otherwise, m is a vanishing marking. The sets of all reachable tangible and vanishing markings of the GSPN N are denoted, respectively, by R T (N ) and R V (N ), and they form a partition of R(N ). Finally, it is further assumed that the considered GSPNs N are reversible; i.e., for every marking m R(N ) there exists a transition sequence t 1... t n such that m t 1...t n m0. 3 Given a marking m R(N ), in the following we shall also use the notation R u V (m) and Ru T (m) to denote, respectively, the sets of all vanishing and tangible markings that are reachable from marking m R(N ) by firing some transition sequence in Tu. Hence, if m R V (N ), then m R u V (m). On the other hand, if m R T (N ), then R u V (m) = and Ru T (m) = {m}. In the sequel, R u V (m) will be characterized as the untimed vanishing reach from marking m, and R u T (m) will be the untimed tangible reach from m. The reader should also notice that the net reversibility assumption that was introduced in the previous paragraph further implies that, for all markings m R V (N ), R u T (m) ; i.e., the considered GSPN N cannot be trapped in a subspace of R(N ) that consists entirely of vanishing markings. In fact, since the modeling paradigm of [18] interprets the untimed transitions as (resource allocation) decisions, it is further assumed in [18] that the untimed vanishing reach R u V (m) of any marking m R V(N ) corresponds to a subspace that is an acyclic digraph. Finally, for every marking m R T (N ), we also define the non-empty set R 1 T (m) m : t E(m) s.t. m m t Ru T (m ); in plain terms, R 1 T (m) denotes the set of all the tangible markings that are reachable from the tangible marking m through the firing of some transition in E(m) and an additional (possibly empty) sequence of untimed transitions. It is well known from the basic GSPN theory that the reachability space R(N ) of any given GSPN N, when endowed with the transitional dynamics that characterize the timed and the untimed transitions of N, constitutes a semi-markov process [32], to be denoted by SM(N ). The transitional dynamics of SM(N ) at any tangible marking m R T (N ) are determined by the exponential race that is defined by E t (m), i.e., the enabled transitions at m. On the other hand, the transitional dynamics of SM(N ) at any vanishing marking m R V (N ) are defined by (i) the set E u (m), i.e., the set of the enabled untimed transitions at m, and (ii) a probability distribution Ξ(m) that is supported by E u (m) and regulates the firing of the transitions in this set; in the GSPN terminology, Ξ(m) is known as the random switch (r.s.) that is associated with vanishing marking m. For the considered GSPN, the specification of a well defined set of random switches Ξ(m), m R V (N ), induces, for every marking m R T (N ), a one-step transition probability distribution p( m) with corresponding support 3 In the case of RAS-modeling GSPN, reversibility is ensured by (i) the assumption that the corresponding RAS starts its operation empty, and (ii) the role of the applied LES, which ensures that every initiated process instance can run successfully to its completion. 6

7 the set R 1 T (m) that was defined in the previous paragraph. In this way, the timed dynamics that are expressed by the semi-markov process SM(N ) can be reduced to a continuous-time Markov chain (CTMC), M(N ), that is defined on R T (N ). Finally, if the CTMC M(N ) satisfies the standard ergodicity assumptions of the relevant Markov chain theory (e.g., c.f. [32]), it will also possess an equilibrium distribution π [π(m), m R T (N )] that defines the long-term behavior or, the steady-state dynamics of the underlying GSPN N. 2.2 The GSPN-based scheduling problem addressed in [18] An additional attribute of the RAS-modeling GSPN N considered in [18] is the finiteness of its reachability set R(N ). 4 This last property, when combined with the presumed reversibility of the considered nets, further implies that the corresponding CTMC M(N ) will be ergodic for any selection of random switches {Ξ(m) : m R V (N )} that retains the net reversibility. A practical way to establish this last requirement is by maintaining some randomization in each random switch Ξ(m) that guarantees a minimum positive selection probability δ for each element of the set E u (m). Given a net N from the considered GSPN class, in the sequel we shall use the notation Π(δ) to denote the set containing all the possible selections of random switches {Ξ(m) : m R V (N )} that satisfy this randomization requirement. Hence, Π(δ) defines a set of control policies for net N that are characterized by well-defined steady-state dynamics. However, the practical applicability of the policies in Π(δ) is (severely) limited by the fact that they require the specification of a separate random switch Ξ(m) for every marking m R(N ); in most practical applications, R(N ) will grow very fast w.r.t. any parsimonious representation of the septuple that defines the net N itself. Hence, [18] also proposes the restriction of the considered class of control policies to the particular subset of Π(δ) that satisfies the following additional constraint: m, m R V (N ) with E u (m) = E u (m ), Ξ(m) = Ξ(m ) (2) As explained in the introductory section, any selection of random switches that observes Eq. (2) is characterized as static, since it essentially defines the transitional dynamics at any marking m R V (N ) on the basis of the information that is provided in the more structural entity of the corresponding set E u (m), and not the information that is provided in the marking m itself. In the sequel, the set of control policies in Π(δ) that satisfies the additional constraint of Eq. (2) will be denoted by Π S (δ). We complete our overview of the developments in [18], by adding that these developments also presume the existence of a function f( ) that is defined on R T (N ) and specifies an (immediate) reward rate that is collected by net N when visiting the corresponding tangible marking. The availability of function f( ) enables the definition of an optimization problem that concerns the selection of a control policy from the set Π S (δ) that maximizes the collected long-term average reward. The primary decision variables for this optimization problem are the selection probabilities, ξ, appearing in the various random switches that are employed by the policies in Π S (δ). Letting ξ denote a vector that collects all these decision variables, and η( ξ) denote the resulting value for the aforementioned objective function, the considered optimization problem 4 The finiteness of R(N ), for any RAS-modeling GSPN N, results from the finiteness of the various resource types of this RAS and the fact that each processing stage requires at least one resource unit for its execution. 7

8 can be concisely expressed as follows: s.t. max ξ η( ξ) (3) ξ ξ, δ ξ (4) Ξ, δ 1.0 ξ (5) ξ Ξ ξ In the above formulation, the constraints of Eq. (4) express the requirement that, in the considered class of policies, Π S (δ), every enabled untimed transition has a minimal selection probability of δ. On the other hand, the constraints of Eq. (5) are necessitated by the fact that for every employed random switch Ξ, the selection probability for one of the elements of the corresponding set E u is modeled only implicitly by the difference in the right-hand-side of these constraints; hence, the constraints of Eq. (5) express the same requirement that is expressed by the constraints of Eq. (4), but for those selection probabilities that are modeled implicitly. The last part of [18] outlines a methodology for the solution of the MP formulation of Eqs (3) (5) through some relevant theory from the sensitivity analysis of Markov reward processes [4] and stochastic approximation [15]; we refer the reader to [18] for the relevant details. 3 A refined policy space for the considered scheduling problem 3.1 Preamble It is clear from the discussion of Section 2 that, in any MP formulation for the considered scheduling problems along the lines of Eqs (3) (5), the number of the employed decision variables ξ is determined by (i) the sets of the enabled untimed transitions, E u (m), at each marking m R V (N ), and (ii) the sets of the distinct random switches that are defined w.r.t. each of these transition sets. In this section we propose a refinement of the sets E u (m) that can lead to a substantial reduction of, both, the set of the random switches and the corresponding set of the decision variables that will appear in the final MP formulation; controlling the size of these two sets is very important for enhancing the effectiveness and the efficiency of the stochastic approximation methods that are used in the solution of this formulation. The primary departing point for the subsequent developments of this section is the observation that the performance of any given policy P from the policy spaces Π(δ) and Π S (δ) that were introduced in Section 2, is essentially determined by the structure of the corresponding CTMC M(N ; P), i.e., (i) the set of the reachable tangible markings under policy P, R T (N ; P), and (ii) the one-step transition probability distributions p( m; P), m R T (N ; P), that are induced by the random switches employed by P. Hence, any two policies P 1 and P 2 from these policy spaces will present the same performance as long as they evolve the underlying GSPN N over the same set of tangible markings and with the same set of one-step transition probability distributions. This last remark motivates the following definition: Definition 1 For the considered scheduling problem: i. Two policies P 1 and P 2 from the policy space Π(δ) will be characterized as performanceequivalent if and only if (iff) they define the same CTMC M(N ) for the underlying GSPN N. 8

9 ii. For any two subspaces Π 1 (δ) and Π 2 (δ) of Π(δ), we shall say that policy space Π 1 (δ) carries the performance potential of policy space Π 2 (δ) iff there exists an optimal policy P in Π 2 (δ) that has a performance-equivalent policy P in Π 1 (δ). In view of the above discussion on the challenges that are posed to the solution of the MP formulation of Eqs (3) (5) by highly dimensional vectors ξ, we would like to identify a policy space Π(δ) that carries the performance potential of the original policy space Π(δ) and results in the smallest possible vector ξ for the corresponding MP formulation of Eqs (3) (5). However, the effective resolution of this last problem would necessitate a holistic view of the underlying reachability space R(N ), and, thus, it is challenged by complexity considerations similar to those that arise in the context of the MDP formulations that were discussed in the previous sections. Hence, in the rest of this work, we take a more pragmatic stance to this problem, seeking to develop a heuristic approach that will redefine the sets of the activated untimed transitions at the various vanishing markings of the considered GSPN models, based on some logic that will depend on some more local attributes of these markings. More specifically, in the rest of this section we provide a systematic methodology that will enable the identification of a set collection {Ẽu(m) E u (m), m R V (N )}, such that the policy space Π(δ) that is induced by these sets (i) will effect a significant reduction to the numbers of the random switches and of the decision variables ξ that are employed by the corresponding MP formulation of Eqs (3) (5), and, at the same time, (ii) will carry the performance potential of the original policy space Π(δ). Furthermore, as it will be shown in the following, the methodology that is developed in this work can be combined with the notion of the static random switch that is defined by Eq. 2, by applying this equation on the revised transition sets Ẽu(m); the resulting policy space will be denoted by Π S (δ), and the corresponding MP formulation of Eqs (3) (5) will involve a number of decision variables ξ that is substantially smaller than the number of the decision variables that are employed by the same formulation when applied on the original policy space Π S (δ). In addition, the revised policy space Π S (δ) retains the possibility of enhancing its performance potential through controlled partial disaggregation of the state aggregates that are defined by the transition sets Ẽu(m) through Eq. (2). Finally, the subsequent developments will also establish that the parsimony provided to the MP formulation of the considered scheduling problem by the revised sets Ẽu(m) essentially results by the ability of the presented methodology to identify and control the real conflicts that arise in the underlying RAS, while avoiding the arbitration of any elements of concurrency that do not correspond to conflicting behavior. 3.2 Some defining properties for the sets Ẽu(m) According to the discussion of the previous section, the sought set collection {Ẽu(m) E u (m), m R V (N )} must induce a policy space Π(δ) that carries the performance potential of the policy space Π(δ), and also controls effectively the complexity of the corresponding scheduling problem. In order to address the first of these two requirements, we stipulate the following condition for the sought sets Ẽu(m). Condition 1 The untimed tangible reach R u T (N ) for any vanishing marking m R V (N ), under the revised sets Ẽu(m), m R V (N ), must remain the same with the untimed tangible reach of m under the original sets E u (m), m R V (N ). 9

10 It is easy to see that Condition 1 retains the communication structure among all the tangible markings m R T (N ). Furthermore, the acyclic nature of the subspace that is induced by the untimed vanishing reach R u V (m) of any vanishing marking m R V(N ), implies that we can effect any desired probability distribution that will regulate the transition from marking m to its untimed tangible reach R u T (m), by pricing appropriately the random switches that appear in the untimed vanishing reach of m. This last remark, when applied to the vanishing markings m that result from some tangible marking m R T (N ) through the firing of a transition t E(m), further implies that the random switches that are induced by any set collection {Ẽu(m) E u (m), m R V (N )} that satisfies Condition 1 will materialize any onestep transition probability distribution p( m), for the considered tangible marking m, that is also realizable in the original policy space Π(δ). Hence, Condition 1 is sufficient for ensuring the realization, under the refined sets Ẽu(m), of any CTMC M(N ) that is realizable in the original policy space Π(δ). But then, the above discussion leads to the following conclusion: Proposition 1 The policy space Π(δ), induced by any collection {Ẽu(m), m R V (N )} that satisfies Condition 1, carries the performance potential of the original policy space Π(δ). On the other hand, it is also important to notice that Condition 1 cannot guarantee that the policy space Π S (δ) carries the performance potential of its counterpart policy space Π S (δ). This limitation is due to the coupling that is introduced by Eq. 2 in the pricing of the random switches that correspond to distinct vanishing markings. Nevertheless, Condition 1 still guarantees that every CTMC M(N ) that is realizable in the original policy space Π(δ), is also realizable in the policy space Π S (δ) through adequate disaggregation of the state aggregates that are defined by the employed sets Ẽu(m) E u (m), m R V (N ), and the corresponding version of Eq. 2. When it comes to the second objective that must be observed by the refined sets Ẽu(m) E u (m), m R V (N ), i.e., the control of the representational and the computational complexity of the resulting scheduling problem as manifested by the number of the decision variables ξ in the corresponding MP formulation of Eqs (3) (5), we already remarked in Section 3.1 that any complete solution procedure would require the processing of the entire reachability graph R(N ). Since this is an intractable proposition, we propose to use as a general guideline for the selection among the set collections {Ẽu(m) E u (m), m R V (N )} that satisfy Condition 1, the notion of minimality that is established in the following condition. Condition 2 For any vanishing marking m R V (N ) and any transition t Ẽu(m ), the substitution of the set Ẽu(m ) by the set Ẽu(m ) \ {t} in the set collection {Ẽu(m), m R V (N )} will result in the violation of Condition 1. The reader should notice that the notion of minimality that is introduced by Condition 2 does not determine uniquely the target sets Ẽu(m), m R V (N ). Furthermore, different collections {Ẽu(m), m R V (N )}, satisfying Conditions 1 and 2 will result in different dimensions for the variable vector ξ that is employed by the corresponding MP formulation of Eqs (3) (5). Hence, in general, there is a need for additional selection logic that will resolve this ambiguity, and possibly enable the further optimization of the final selection of these sets in various application contexts. We shall revisit this issue in the next section, where we detail a particular algorithm for the determination of the target sets Ẽu(m), m R V (N ). 10

11 3.3 Computing the refined sets Ẽu(m) In this section, first we provide a more explicit characterization of the sets Ẽu(m), m R V (N ), that satisfy Condition 1, and subsequently we propose some further logic that can facilitate the selection of the final collection {Ẽu(m), m R V (N )} among all those that satisfy both Conditions 1 and 2. These two developments will also suggest a very straightforward algorithm for the effective computation of the collection {Ẽu(m), m R V (N )} that is targeted by the aforementioned logic. We start these developments with the following proposition. Proposition 2 Consider a marking ˆm R V (N ) and a set ˆT E u ( ˆm). Then, the substitution of the set Ẽu( ˆm) by the set ˆT in any collection {Ẽu(m), m R V (N )} that satisfies Condition 1, will retain the satisfaction of this condition for the considered marking ˆm iff the set ˆT satisfies the following condition: u RT (m ) = R u T ( ˆm) (6) m : t ˆT, ˆm m t Proof: We remind the reader that Condition 1 requires that the selected transition sets will establish, for every vanishing marking m R V (N ), an untimed tangible reach equal to R u T (m ) (i.e., the corresponding untimed tangible reach which results from the original collection {E u (m), m R V (N )} that is defined by the natural dynamics of net N ). Also, for the needs of this proof, we introduce the following additional notation: (a) For any marking m R(N ), R u T (m ; Ẽ) will denote the untimed tangible reach of m defined by the collection {Ẽu(m), m R V (N )} that is considered by Proposition 2. (b) On the other hand, for any marking m R(N ), R u T (m ; ˆT ) will denote the untimed tangible reach of m that is defined by the collection {Ẽu(m), m R V (N )} under the modification described in Proposition 2 (i.e., with the transition set Ẽu( ˆm) replaced by the set ˆT defined in the proposition). Finally, we also remind the reader that, in the considered GSPN class, the markings m in R u V ( ˆm) (i.e., the untimed vanishing reach of the considered vanishing marking ˆm) induce, by assumption, an acyclic subspace of R(N ). But then, the substitution of the set Ẽu( ˆm) by the set ˆT, as suggested by Proposition 2, does not impact the untimed tangible reaches of the markings m that result from ˆm by firing any transition in ˆT ; i.e., m s.t. t ˆT : ˆm t m, R u T (m ; ˆT ) = R u T (m ; Ẽ) (7) Since the considered set collection {Ẽu(m), m R V (N )} satisfies Condition 1, it also holds that m s.t. t ˆT : ˆm t m, R u T (m ; Ẽ) = Ru T (m ) (8) Hence, the left-hand-side of Eq. (6) essentially expresses R u T ( ˆm; ˆT ) the untimed tangible reach of the considered marking ˆm for the collection of the untimed transition sets that is defined in Proposition 2 and the validity of the claimed result becomes obvious. The following corollary can be established easily from Proposition 2 and an inductive argument on the acyclic graphs that correspond to the untimed vanishing reaches of the vanishing markings m R V (N ); we leave the relevant details to the reader. Corollary 1 A collection {Ẽu(m) E u (m), m R V (N )} satisfies Condition 1 iff every set Ẽ u (m), m R V (N ), satisfies the condition of Eq. (6). 11

12 Corollary 1 enables a localized computation, at any marking m R V (N ), of all the sets Ẽu(m) that can be part of any collection that will satisfy Condition 1. This decomposing effect can be further exploited towards the effective identification of such collections {Ẽu(m) E u (m), m R V (N )} that will also satisfy Condition 2. In particular, the above developments further imply that any collection {Ẽu(m) E u (m), m R V (N )} consisting of minimal-cardinality subsets of the corresponding parent sets E u (m) that satisfy the condition of Eq. (6), will also satisfy Condition 2 (since, if this condition was violated by some particular set Ẽu(m ), then this set would not be among the minimal-cardinality subsets of E u (m ) that satisfies the condition of Eq. (6)). Hence, we have the following proposition: Proposition 3 A collection {Ẽu(m) E u (m), m R V (N )} consisting of minimal-cardinality subsets of the corresponding parent sets E u (m) that satisfy the condition of Eq. (6), will satisfy both Conditions 1 and 2. At this point, the following remarks are in order: (a) By minimizing the cardinality of the selected sets Ẽu(m), m R V (N ), the selection logic that is implied by Proposition 3 is generally conducive to the primary objective of minimizing the number of the decision variables ξ. (b) On the other hand, the resulting selection scheme still needs further arbitration / disambiguation in the case of vanishing markings m R V (N ) with more than one minimal-cardinality subsets of the corresponding set E u (m) that satisfy the condition of Eq. (6). Ideally, this conflict should be arbitrated in a way that leads to the minimal-dimensionality vector ξ for the corresponding MP formulation of Eqs (3) (5). Since, however, such an arbitration scheme is not practically feasible in the context of a real-time resolution of the considered ambiguity, in this work we propose to resolve this ambiguity by taking, at every vanishing marking m R V (N ), the minimal-cardinality set Ẽu(m) E u (m) that satisfies the condition of Eq. (6) and it is also lexicographically minimal w.r.t. the ordering that is defined by the natural (or, perhaps, some other) indexing of the transition set T u. Then, at every visited vanishing marking m R V (N ), the corresponding set Ẽu(m) can be effectively obtained through the execution of the computation that is depicted in Algorithm 1. Algorithm 1 first computes the set E u (m), and, subsequently, for each transition t in this set, it computes (a) the marking m (t) that results from the firing of transition t in m, and (b) the untimed tangible reach of m (t) under the original collection {E u (m ), m R V (N )} of [18]. Also, the algorithm computes the untimed tangible reach of marking m as the union of the untimed tangible reaches of the markings m (t), t E u (m). In the last phase of its computation, Algorithm 1 enumerates the subsets of the set E u (m), sequencing them first in increasing cardinality, and second (i.e., the subsets with equal cardinality) lexicographically according to some ordering that is imposed on the elements of E u (m); 5 for every transition set T that is generated by this enumeration, Algorithm 1 checks whether this set satisfies the condition of Eq. (6) in Proposition 2, in which case it terminates returning the set T as the sought set Ẽu(m). The computational complexity of Algorithm 1 is determined primarily by (a) the computation of the untimed tangible reach for every marking m (t), and (b) the enumeration of the subsets of E u (m). Both of these elements imply a super-polynomial worst-case complexity w.r.t. the 5 In lack of any bias that might result from special structure or further input, this can be the ordering that is defined by the natural indexing of the net transitions in T. 12

13 Algorithm 1 Computing the target set Ẽu(m) at a visited vanishing marking m Input: GSP NN = (P, T t, T u, W ); current vanishing marking m Output: the transition set Ẽu(m) ˆT E u (m) for each t ˆT do m (t) tr(m, t); UT R(t) R u T (m (t)) end for UT R0 t ˆT UT R(t) for i := 1,... ˆT do for j := 1,..., C i ˆT do T the j-th subset of ˆT of cardinality i, where these subsets are enumerated lexicographically based on some ordering of the elements of ˆT if t T UT R(t) = UT R0 then return T end if end for end for size of the underlying GSPN N. 6 Since, however, (i) the computation that provides the aforementioned untimed tangible reaches is performed on acyclic subgraphs of a more local nature w.r.t. to the entire reachability space R(N ), and (ii) the sets E u (m) are usually small subsets of the transition set T, it is expected that the empirical computational complexity of Algorithm 1 will be quite benign. The next example demonstrates the execution of Algorithm 1 and the last claim about its empirical complexity. Furthermore, the example reveals some additional attributes of the collections {Ẽu(m) E u (m), m R V (N )} that are generated by the considered algorithm. Example Consider the marked GSPN N depicted in Figure 1a. In the depicted net, the black transitions are untimed, while the white ones are timed. Since the depicted marking enables the untimed transitions t 2 and t 6, it is vanishing; for further reference, we shall label this marking as m 1. The digraph of Figure 1b represents the untimed vanishing and tangible reaches of marking m 1. More specifically, in Figure 1b, the nodes depicted as single-bordered correspond to vanishing markings in the untimed reach of marking m 1, and those depicted as double-bordered correspond to tangible markings. Finally, the nodes and the edges that are depicted with thicker lines in the digraph of Figure 1b, represent the untimed vanishing and tangible reaches of the considered vanishing marking m 1 under the random switches that are returned by Algorithm 1. It is interesting to notice in the digraph of Figure 1b the extent of the reduction that is effected by Algorithm 1 in terms of (a) the reachable vanishing markings and (b) the dimensionality of the random switches that are constructed for the remaining vanishing markings. Both of these reductions simplify extensively the arbitration that must be effected on this part of the underlying state space by the corresponding scheduling problem. Furthermore, the small size of the random switches that are eventually employed at the reachable vanishing markings visited 6 We remind the reader that this last quantity is defined as the size of any parsimonious encoding / representation of the septuple that defines GSPN N. 13

14 m 1 t 0 t 2 t 6 p 0 p 1 t 1 p 7 p 8 t 0 m 2 t 3 t 6 t 2 t 5 (enabled) p 2 t 2 t 3 t 3 t 0 m 3 t 5 t6 t 6 t 3 t 3 t 5 p 3 p 11 m T1 m 4 t 4 p 4 t 6 t 5 t 3 p 5 t 5 p 9 p 10 m T2 (enabled) p 6 t 6 t 7 (a) Example: the considered GSPN marking m 1. (b) The digraph characterizing the untimed vanishing and tangible reaches of marking m 1, R u V (m 1) and R u T (m 1), under (i) the original random switches and (ii) the refinement of these random switches by Algorithm 1. Figure 1: The GSPN vanishing marking, m 1, and the digraph that characterizes its untimed tangible reach, for the example of Section 3.3. by the algorithm, also corroborates our earlier claims that, in general, the enumeration of the subsets of the sets E u (m) that is performed by Algorithm 1 at each visited vanishing marking m R V (N ), will be terminated in its early stages. Finally, it is also interesting to take a closer look in the way that Algorithm 1 treats the existing conflicts among the transitions of the sets E u (m) in its selection of the corresponding sets Ẽu(m). Hence, consider first the execution of Algorithm 1 at the original vanishing marking m 1. At this marking, the algorithm picked transition t 2 over t 6. The reader can check that these two transitions are not in conflict, since t 2 = {p 1, p 10 }, and t 6 = {p 5, p 7 }. However, selecting among non-conflicting transitions is still quite subtle. More specifically, the firing of the transition t 2 at marking m 1 will release tokens to its output places p 2 and p 8. And adding one token in p 8 will enable the untimed transition t 0, which is in conflict with t 6, since t 0 and t 6 have the common input place p 7. Hence, giving arbitrary precedence to transition t 6 over transition t 2 at marking m 1 would have a restrictive effect on the potential behavior that can be generated by the eventual firing of t 2. On the other hand, firing t 6 at m 1 will only release one token to the place p 6, which is not an input place of any untimed transitions. Thus, firing t 6 cannot add any new elements to the set E u (m), and there are no hidden conflicts by dropping t 6 from the refined random switch of m 1. Similarly, the algorithm selection at marking m 2 can be interpreted by the fact that, for the effected partition ({t 3 }, {t 0, t 5, t 6 }) of the original set E u (m 2 ), it holds that {t 3 } = {p 2, p 9 }, and {t 0, t 5, t 6 } = {p 4, p 5, p 7, p 8, p 11 }; hence, there is no conflict between these two sets. Furthermore, from the information that is provided in Figure 1b, it can be easily checked that no new untimed transitions will be enabled before reaching some tangible marking from m 2. Thus, in this case, we can eliminate either {t 3 } or {t 0, t 5, t 6 }, according to the logic that interpreted the output of Algorithm 1 on m 1. Eventually, 14

15 Algorithm 1 gives precedence to the set {t 3 } since it is the set with the smallest cardinality. Finally, at marking m 3, Algorithm 1 returns the transition set {t 0, t 5 }, and transition t 6 is eliminated from the original set of enabled untimed transitions. When considering the original set E u (m 3 ) = {t 0, t 5, t 6 }, we can see that t 0 is in conflict with t 5 and t 6, but t 5 and t 6 are not in conflict. Since in this case there are no further hidden conflicts that could have been revealed only after the firing of the non-conflicting transitions t 5 and t 6, Algorithm 1 includes only one of the non-conflicting transitions in the returned random switch, and this is transition t 5 due to the applied lexicographic ordering. The next section provides a formal characterization to the apparent connection between the selection logic of Algorithm 1 and the nature of the conflicting relations between the selected transition sets Ẽu(m) and their complements E u (m) \ Ẽu(m), that was observed in this example. 3.4 Some sufficient conditions for the condition of Eq. (6) We conclude the developments of Section 3 by presenting two results that provide sufficient conditions for the condition of Eq. (6). The first of these results highlights further the connection between Condition 1 and the notion of (non-)conflict, that was indicated in the discussion of the example of Section 3.3. Proposition 4 Consider a vanishing marking m of a GSPN N, and a partition (T 1, T 2 ) of E u (m). Furthermore, let E u ((T u \ T 1 ) ) T u \ E u (m) denote the set containing those untimed transitions of N that are not enabled in m but get enabled after firing some transition sequences in (T u \ T 1 ). Also, set ˆT 2 T 2 E u ((T u \ T 1 ) ). Finally, suppose that σ ˆT 2, t 1 T 1 such that the sequences σt 1 and t 1 σ are feasible at m. Then, the set T 1 satisfies the condition of Eq. (6) in Proposition 2. Proof: We need to show that R u T (m) = m M Ru T (m ) where M = {m R(N ) : t T 1 s.t. m = tr(m, t)}. The inclusion R u T (m) m M Ru T (m ) is obvious since T 1 E u (m). So, next, we focus on the inclusion R u T (m) m M Ru T (m ). Consider any tangible marking m T R u T (m). Then, there exists a feasible transition sequence σ n = t 1 t 2... t n leading from marking m to m T. To show that m T m M Ru T (m ), we distinguish the following two cases: Case 1: t 1 T 1. Then, m T R u T (tr(m, t 1)) m M Ru T (m ). Case 2: t 1 T 2. Then, by the last assumption of Proposition 4, there exists t i T 1 in the sequence σ n such that the transition sequence σ i 1 = t 1... t i 1 is composed by transitions in ˆT 2 only. According to the same assumption, we can exchange the firing order of t i and σ i 1 and get the transition sequence t i t 1... t i 1 which is feasible at m. But then, the sequence t i t 1... t i 1 t i+1... t n is a feasible sequence leading from marking m to m T, and m T R u T (tr(m, t i)) m M Ru T (m ). The next proposition introduces another set of sufficient conditions for the condition of Eq. (6). As it will be revealed in the proof that establishes this proposition, this new condition set is stronger than the condition set of Proposition 4; but its primary value lies in the fact that it is much easier to verify than the condition of Eq. (6) and the conditions that are stipulated by Proposition 4. More specifically, while the conditions of Eq. (6) and Proposition 4 have an exponential verification cost w.r.t. the size of the underlying GSPN N, this new condition set 15

16 is polynomially verifiable. Furthermore, in Section 4 we demonstrate how the simplicity of this condition set can lead to the specification of a very efficient and pertinent policy space Π(δ) for a particular GSPN class that models the resource allocation taking place in the CRL operations that are considered in that section. Proposition 5 Consider a vanishing marking m of a GSPN N, a partition (T 1, T 2 ) of E u (m), and further suppose that: i. ˆt T 1 such that t T 2, ˆt t = ; ii. t T 2 and p P, p t = p T u =. Then, the set T 1 satisfies the condition of Eq. (6) in Proposition 2. Proof: We shall prove the result of Proposition 5 by showing that the two conditions stated in this proposition are sufficient to satisfy the requirements of Proposition 4. We start by noticing that condition (ii) guarantees that the firing of any transition sequence σ T2 does not add any tokens to the input places of any untimed transition. Therefore, under this condition, ˆT 2 = T 2, where the set ˆT 2 is defined in the statement of Proposition 4, and we only need to show that σ T2, t 1 T 1 such that both sequences σt 1 and t 1 σ are feasible at m. Consider any transition sequence σ = t 1 t 2... t n T2 feasible at m. From condition (i), ˆt t i =, i {1, 2,..., n}. Therefore, firing σ does not decrease the number of tokens in any input place of ˆt, and ˆt is enabled at the marking m that results from the firing of σ. In other words, σˆt is feasible at m. The feasibility of ˆtσ can be proved in a similar manner. Therefore, the condition of Proposition 4 is met by setting t 1 = ˆt. 4 Customizing the random-switch refinement process for a GSPN sub-class modeling capacitated re-entrant lines The basic model of a re-entrant line is a queueing-theoretic model that has been used extensively in the past to model and analyze the workflow dynamics of single-item production lines that involve revisits to some of the line workstations. The model has been motivated by, and extensively used in the context of, the operations that take place in modern semiconductor manufacturing [14]. As it happens with most classical queueing-theoretic models that have been investigated by scheduling theory, this model assumes an infinite buffering capacity for the line workstations, and therefore, it focuses only upon the allocation of the servers that are available at these workstations. On the other hand, capacitated re-entrant lines were originally introduced in [29], in an effort to investigate the additional dynamics and complications that arise when the available buffering capacity at each of the line workstations is finite. In this section, we focus on a particular GSPN class that models the operations of a capacitated re-entrant line (CRL) and the corresponding scheduling problem. The main intention of the subsequent developments is to show how the results of Section 3 can provide a formal base for the development of a customized refinement process for the random switches that are employed in this particular setting, which is computationally more efficient than Algorithm 1. More 16

17 specifically, this new refinement process exploits special structure in the considered GSPN class in order to ensure the satisfaction of the condition in Eq. (6) while foregoing the computation of the untimed tangible reach and the enumeration of the subsets of the set E u (m) at every visited vanishing marking m. At the same time, a series of computational experiments that are reported at the end of this section indicate that this customized refinement process is very competitive with Algorithm 1 w.r.t. the number of the random switches generated and the number of the decision variables that are eventually employed in the MP formulation of Eqs (3) (5). 4.1 The considered capacitated re-entrant line and the modeling GSPN The considered CRL model The CRL model that is considered in this work is a generalization of the example CRL introduced in [18]. These models essentially constitute an abstraction for the operation of flexibly automated manufacturing cells consisting of n workstations and supporting a single job type with l sequential processing stages. 7 In the sequel, we shall employ the mapping W : {1,..., l} {1,..., n} to indicate the required workstation for each processing stage. Hence, the jobs that are processed by this cell are initially routed to workstation W(1) to execute their first processing stage, and subsequently, they proceed sequentially: after being processed for stage j at workstation W(j), the jobs will advance to workstation W(j +1). Finally, after being processed for stage l at workstation W(l), the jobs are unloaded from the system. We suppose that j {1,..., l 1}, W(j) W(j +1), and that there exists at least one pair of stages {i, j}, with W(i) = W(j); this last assumption is a formal characterization of the notion of re-entrance in the considered operations. The job transfer among the workstations and the I/O port of the cell are facilitated by a robotic manipulator. However, in the context of the considered CRL class, we assume that transport times are negligible. Furthermore, the allocation of the robotic manipulator to these job transfers does not generate any interesting behavioral dynamics as long as the allocation of the buffering capacity of the n workstations is properly handled. Therefore, the material handling operations are not modeled explicitly in the considered CRL model. 8 When it comes to the internal structure and operation of each workstation W S i, i = 1,..., n, the considered model assumes that each workstation possesses a finite number B i of buffer slots for accommodating the jobs routed to it, and a single server which processes the buffered jobs in situ; i.e., the server moves to the buffer slot of each job and processes it while the job is sitting in its buffer. Hence, a job holds its allocated buffer slot during its entire sojourn at workstation W S i, and it releases this slot only after it has completed its local processing and it has secured a buffer slot to the workstation that supports its next processing stage, or it is ready to exit the cell (in the case that i = W(l)). The processing times for the l processing stages are assumed to be exponentially distributed with means τ j = 1/µ j > 0, j = 1,..., l. 9 Finally, the CRL model presented in this work presumes a non-idling operational scheme for the station servers; i.e., a server will not idle as long as there are jobs waiting for processing In particular, CRL models like the one considered in this work can model effectively the operation of the miniaturized robotic cells that are used in the semiconductor manufacturing industry and are known as cluster tools [34]. 8 The assumption regarding the negligible job transfer times and the implicit modeling of the material handling function, that are employed by the presented CRL model, simplify the exposition of the presented developments, but they are not critical for the presented methodology. 9 More general timing distributions can be effectively handled by the presented model through their approximation by phase-type distributions [5]. 10 It is known, however, that due to the blocking that takes place in the considered operations, a non-idling 17

18 (enabled) p 0 t 0 t 1 p 8 p 1 p 7 WS 1 WS 2 p 2 t 2 t 3 p 3 p 11 t 4 p 4 p 5 t 5 p 9 p 10 I/O Port Process route: WS1 - > WS2 - > WS1 (a) An example capacitated re-entrant line. p 6 t 6 t 7 (b) The GSPN model of the example CRL of Figure 2a, at its initial marking m 0, and including the control logic of the maximally permissive LES. Figure 2: A capacitated re-entrant line and the corresponding GSPN model. Figure 2a depicts an example CRL, with n = 2, l = 3, and the process route being defined by the function W(1) = W(3) = 1, W(2) = 2 (or, more schematically, by W S 1 W S 2 W S 1 ). We also assume in this example that B 1 = B 2 = 2. We want to control the workflow dynamics of the aforementioned CRL in order to maximize its throughput. To address this objective, first we notice that the smooth operation of the considered CRL can be challenged by the finiteness of the buffering capacity of its n workstations in combination with the re-entrant nature of its material flow. For instance, in the considered example, if the buffer of workstation W S 1 is allocated to two jobs in their first processing stage and the buffering capacity of workstation W S 2 is also fully allocated, then it is impossible for the cell robot to advance any of these jobs to their next stage, and the operation of the system will be permanently stalled, or deadlocked. This deadlocking problem can be prevented by the application of an appropriate liveness-enforcing supervisor (LES). The reader is referred to [30, 31] for a comprehensive exposition of the current theory on the synthesis of effective and efficient LES for sequential RAS, including the RAS corresponding to the considered CRL. Many of these LES take the convenient form of a set of linear inequalities that control the number of process instances situated at certain subsets of the line workstations, and as discussed further below, they admit a very efficient and elegant PN-based representation. For the simple example depicted in Figure 2a, it can be easily checked that deadlocks can be avoided as long as the scheme might be sub-optimal for certain performance objectives, including the objective of the maximization of the long-term throughput of the line, which is of primary interest to the presented research program [29]. We have adopted this non-idleness assumption in this work, in an effort to simplify the exposition of the presented developments. But the presented modeling framework and the accompanying analysis can be extended to accommodate operational schemes that involve intentional idling. 18