IPA Derivatives for Make-to-Stock Production-Inventory Systems With Lost Sales I. INTRODUCTION

Transcription

1 IEEE RANSACIONS ON AOMAIC CONROL, VOL. 52, NO. 8, AGS [2]. Zaborzsky, G. Huang, B. Zheng, an. C. Leung, On the phaseportrait of a class of large nonlinear ynamic systems such as the power system, IEEE rans. Autom. Control, vol. 33, no. 1, pp. 4 15, an [3] H. D. Chiang, C. C. Chu, an G. Cauley, Direct stability analysis of electric power systems using energy functions heory, applications, an perspective, in Proc. IEEE, Nov. 1995, vol. 83, no. 11, pp [4] S. shiki, Analytic expressions of unstable manifols, Proc. pn Aca. Ser. A.-Math. Sci., vol. 56, no. 6, pp , 198. [5] F. M. A. Salam, A. Araposthathis, an P. P. Varaiya, Analytic expressions for the unstable manifol at equilibrium points in ynamical systems of ifferential equations, in Proc. 22n IEEE Conf. Decision Control, Dec. 1983, vol. 3, pp [6] S. Saha, A. A. Foua, W. H. Kliemann, an V. Vittal, Stability bounary approximation of a power system using the real normal form of vector fiels, IEEE rans. Power Syst., vol. 12, no. 2, pp , [7] R. Qi, D. Cook, W. Kliemann, an V. Vittal, Visualization of stable manifols an multiimensional surfaces in the analysis of power system ynamics,. Nonlin. Sci., vol. 1, pp , 2. [8] Z. ing, Z. ia, an Y. Gao, Research of the stability region in a power system, IEEE rans. Circuits Syst. I, Funam. heory Appl., vol. 5, no. 2, pp , Feb. 23. [9] P. A. Cook an A. M. Eskicioglu, ransient stability analysis of electric power systems by the metho of tangent hypersurfaces, in Inst. Elect. Eng. Proc.-Gener. ransm. Distrib., ul. 1983, vol. 13, no. 4, pp [1]. Zaborzsky, G. Huang, B. Zheng, an. Leung, New results on stability monitoring on the large electric power systems, in Proc. 26th IEEE Conf. Decision Control, 1987, pp [11] M. Djukanovic, D. Sobajic, an Y.-H. Pao, Neural-net base tangent hypersurfaces for transient security assessment of electric power systems, Int.. Electr. Power Energy Syst., vol. 16, no. 6, pp , [12] V. Venkatasubramanian an W. i, Numerical approximation of (n 1)-imensional stable manifols in large systems such as the power system, Automatica, vol. 33, no. 1, pp , Oct [13] D. Cheng,. Ma, Q. Lu, an S. Mei, Quaratic form of stable submanifol for power systems, Int.. Robust Nonlin. Contr., vol. 14, pp , 24. [14] N. Kazantzis, Singular PDEs an the problem of fining invariant manifol for nonlinear ynamical system, Phys. Lett. A., vol. 272, no. 4, pp , ul. 2. IPA Derivatives for Make-to-Stock Prouction-Inventory Systems With Lost Sales Yao Zhao an Benjamin Melame Abstract his note applies the stochastic flui moel (SFM) paraigm to a class of single-stage, single-prouct make-to-stock (MS) prouction-inventory systems with stochastic eman an ranom prouction capacity, where the finishe-goos inventory is controlle by a continuous-time base-stock policy an unsatisfie eman is lost. his note erives formulas for infinitesimal perturbation analysis (IPA) erivatives of the sample-path time averages of the inventory level an lost sales with respect to the base-stock level an a parameter of the prouction rate process. hese formulas are comprehensive in that they are exhibite for any initial inventory state, an inclue right an left erivatives (when they iffer). he formulas are obtaine via sample path analysis uner Manuscript receive March 1, 26; revise November 16, 26. Recommene by Associate Eitor I. Paschaliis. he authors are with the Department of Management Science an Information Systems, Rutgers Business School, Newark an New Brunswick, N 8854 SA ( yaozhao@anromea.rutgers.eu; melame@rbs.rutgers.eu). very mil regularity assumptions, an are inherently nonparametric in the sense that no specific probability law nee be postulate. It is further shown that all IPA erivatives uner stuy are unbiase an fast to compute, thereby proviing the theoretical basis for online aaptive control of MS prouction-inventory systems. Inex erms Infinitesimal perturbation analysis (IPA), lost sales, make-to-stock (MS), prouction-inventory systems, stochastic flui moels (SFMs). I. INRODCION his note erives infinitesimal perturbation analysis (IPA) erivatives of selecte ranom variables for a class of make-to-stock (MS) systems in stochastic flui moel (SFM) setting, where the traitional iscrete arrival, service, an eparture stochastic processes are replace by corresponing stochastic flui-flow rate processes. We henceforth refer to this approach as IPA-over-SFM. he IPA erivatives provie sensitivity information on system metrics with respect to control parameters of interest, an as such can serve as the theoretical unerpinnings for online control algorithms. Comprehensive iscussions of IPA erivatives an their applications can be foun in Fu an Hu [3] an Cassanras an Lafortune [1]. he IPA-over-SFM approach has been successfully applie to theoretical stuies of various queuing an prouction-inventory systems; see, e.g., [2], [6], [4], an [7]. hese stuies constrain the system to start from a prescribe initial inventory state, an only consier cases where the left an right IPA erivatives coincie. In contrast, Zhao an Melame [8] consiere any initial inventory state for MS systems with backorers an erive sie IPA erivative formulas as neee. he goal of this note is to erive IPA erivatives for MS systems with lost sales, an to show them to be unbiase. First, we erive IPA erivatives for the time averages of inventory level an lost sales with respect to the base-stock level for all initial inventory states, incluing sie erivatives when they iffer. We are only aware of one note [6] aressing IPA-over-SFM queues with finite buffers, which can be use to moel MS systems with lost sales, though it constrains the initial conition to an empty buffer. Secon, we erive IPA erivatives for the aforementione metrics with respect to a prouction rate parameter, incluing sie erivatives when they iffer. We point out that the assumptions in [6] preclue iffering left an right IPA erivatives. As will become evient in the sequel, MS systems with lost sales are also analytically more challenging than MS systems with backorers, because the inventory state of the former has an extra bounary, a fact that results in more elaborate formulas. he computation of IPA erivatives for all initial conitions is motivate by two reasons. he first reason is to enable potential applications of IPA erivatives to online control of MS systems riven by nonstationary processes among others. he intent is to ajust the system parameters over time accoring to the changing statistics of the unerlying processes, but not necessarily to optimize it. Clearly, IPA-base online control applications manate the computation of IPA erivatives for all initial states, as well as all sie erivatives when they iffer, since a control action can change the system parameters at any state (which is then consiere as the new initial state). It makes little sense to wait for the system to return to certain selecte inventory states as this coul suspen control actions over extene perios of time. he secon reason is that the transient IPA erivatives compute here are exact an unbiase, whereas their asymptotic counterparts may not provie aequate approximations. Furthermore, in orer to compute asymptotic IPA erivatives, we still nee to obtain their transient counterparts before sening time to infinity. Digital Object Ientifier 1.119/AC /$ IEEE

2 1492 IEEE RANSACIONS ON AOMAIC CONROL, VOL. 52, NO. 8, AGS 27 he rest of this note is organize as follows Section II presents the MS moel. Sections III an IV erive IPA erivative formulas an prove their unbiaseness. Section V conclues the note. We will use the following notational conventions. Let the inicator function of set A be 1 A an x =maxfx; g. A function f (x) is sai to be locally ifferentiable at x if it is ifferentiable in a neighborhoo of x; it is sai to be locally inepenent of x if it is constant in a neighborhoo of x. II. MS MODEL WIH LOS SALES Consier a single-stage, single-prouct MS system, consisting of a prouction facility an an inventory facility. he two facilities are couple he latter sens orers to the former, while the former prouces stock to replenish the latter. he prouction facility is comprise of a queue that houses a prouction server, precee by an infinite buffer that hols incoming prouction orers. We assume that the prouction facility has an unlimite supply of raw material, so it never starves. he inventory facility satisfies incoming emans on a first-come firstserve (FCFS) basis, an is controlle by a continuous-time base-stock policy with a base-stock level S > (the case S = correspons to the just-in-time policy as a simple special case). Demans arrive at the inventory facility an are satisfie from inventory on han (if available). Otherwise, when an inventory shortage is encountere, the incoming eman is satisfie by the amount of inventory on han, an any shortage of inventory becomes a lost sale. he MS system can operate in one of the following two moes. In the normal operational moe, the inventory level oes not excee S. he prouction facility strives to replenish the inventory facility to its base-stock level, but no higher. In the overage operational moe, the inventory level excees S (for example, this coul result from a control action that lowere S below the current inventory level). Prouction is then temporarily suspene until the inventory level reaches or crosses S from above, at which point normal operational moe resumes. he MS system with lost sales can be moele as an SFM, where I(t) is the (flui) volume of inventory on-han at time t, X(t) is the (flui) volume of outstaning orers at time t, (t) represents the rate of incoming emans at time t, an (t) represents the prouction rate at time t. Finally, (t) is the (flui) loss rate of sales at time t. Let [;] be a finite time interval; for example, may esignate the time perio separating applications of control actions. he notion of sample path events pertains to a property of a time point along a sample path (not to be confuse with the orinary notion of events as aggregates of sample paths); the istinction can be iscerne by context. Similarly to [6], we efine two types of sample path events An exogenous event occurs either whenever a jump takes place in the sample path of f(t)g or f(t)g, or when the time horizon is reache. An enogenous event occurs whenever a time interval is inaugurate, in which X(t) =or X(t) =S. hroughout this note, we make the following mil regularity assumptions (cf. [6]). Assumption 1 1) he eman rate process f(t)g an the prouction rate process f(t)g have right-continuous sample paths that are piecewiseconstant with probability 1 (w.p.1.). 2) Each of the processes f(t)g an f(t)g has a finite number of iscontinuities in any finite time interval w.p.1, an the time points at which the iscontinuities occur are inepenent of the parameters of interest. 3) No multiple sample path events occur simultaneously w.p.1. In overage operational moe, the system satisfies the relations (=t )I(t) =(t);(t) =, an X(t) =. In normal operational moe, the system satisfies the conservation relation X(t)I(t) =S (2.1) he lost-sales rate process is given by (t) = [(t) (t)] 1 fi(t)=;(t)>(t)g ; t. he outstaning orers process is governe by the sie stochastic ifferential equation (=t )X(t) =if X(t) =an (t) (t) or X(t) =S an (t) (t); otherwise, (=t )X(t) =(t) (t). We consier the following performance ranom variables or simply metrics the inventory time average L I( ) = (1= ) I(t) t an the lost-sales time average L ( ) = (1= ) (t) t. he control parameters of interest are the base-stock level at the inventory facility an a scaling parameter of the prouction rate at the prouction facility. Let 2 2 enote a generic parameter of interest with a close an boune omain 2. We write S;(t; );L I(;);L (;), an so on to inicate the epenence of a performance ranom variable on its parameter of interest. Our objective is to erive formulas for the IPA erivatives (=)L I(;) an (=)L (;) in SFM setting, using sample path analysis, an to show them to be unbiase. Finally, we make the following assumption regaring the initial state. Assumption 2 he initial inventory level oes not epen on, i.e., I(;)=I; III. IPA DERIVAIVES WIH RESPEC O HE BASE-SOCK LEVEL his section erives IPA erivatives (incluing sie ones) for the inventory time average, L I(;) an the lost-sales time average L (;), both with respect to the base-stock level S. It will exhibit the requisite formulas for any initial inventory state. Assumption 3 1) S =, where ) he processes f(t)g an f(t)g are inepenent of the parameter. 3) For each 2 2, the sie erivatives of L I (;) an L (;) exist w.p.1. Let (Q j;r j);j = 1;...;, be the orere extremal subintervals of [; 1), such that I(t; ) <Sfor all t 2 (Q j ;R j ). hat is, the enpoints Q j an R j are obtaine via inf an sup functions, respectively. By convention, if any of these enpoints oes not exist, then it is set to 1. Furthermore, let Z j 2 (Q j ;R j ) be the first time point in this interval at which I(t; ) =, provie such a point exists; otherwise, let Z j =R j. By [8, Observation 3], Q 1 <R 1 <Q 2 <R 2 <... <Q <R. We consier three initial states I < S;I > S, an I = S. he last state cannot be exclue because it may happen in applications where inventory levels are iscrete. On the event fi <Sg fi >Sg, we will make use of the hitting time S = minft 2 [; 1] I(t; ) =Sg, if it exists; otherwise, efine S =1. On the event fi = Sg, we make use of the hitting times an. On the event fq 1 > g,efine = minft 2 [;Q 1 ) (t) >(t)g, if it exists; on the event fq 1 = g [fq 1 > g f(t) = (t); t 2 [;Q 1)g] efine =R 1,ifR 1 exists; otherwise, efine =1. Finally, efine = minft 2 [;] (t) > g, if it exists; otherwise, efine = 1. In wors, is a hitting time of fi(t; )g, which correspons to the first time that the inventory level changes in any perturbe process fi(t; 1)g, while plays an analogous role, but for a perturbe process fi(t; 1)g. Note that is inepenent of. We will also make use of horizon-epenent ranom inices, given by S (;) = maxfj 1R j g, if it exists, an S(;)=, otherwise. hese constitute restrictions of to finite time horizons [;]. heorem 1 W.p.1, the IPA erivatives of the inventory time average with respect to the base-stock level are given for all > an 2 2 as follows 1) On the event fi <Sg

3 IEEE RANSACIONS ON AOMAIC CONROL, VOL. 52, NO. 8, AGS L I(;)= 1 (;) [minfz j1 ;gr j ] 2) On the event fi >Sg f S <Q 1 g L I(;)= 1 f1 f < g[minfz 1 ;g S ] (;) [minfz j1 ;gr j ]g 3) On the event fi >Sg f S =Q 1 g LI(;)= 1 f1 ( S) f < g ( [minfz1;gs] S) (;) [minfz j1 ;gr j ]g 4) On the event fi = Sg f =R 1 g L I(;)= 1 (;) [minfz j1 ;gr j ] 5) On the event fi = Sg f <R 1 g L I(;)= 1 f1 f < g[minfz 1 ;g ] (;) [minfz j1 ;gr j ]g 6) On the event fi = Sg f <Q 1 g L I(;)= 1 f1 f < g[minfz 1 ;g ] (;) [minfz j1 ;gr j ]g 7) On the event fi = Sg f = Q 1 g L I(;)= 1 f1 ( ) f < g ( ) [minfz 1;g ] (;) [minfz j1;gr j]g Proof First note that in normal operational moe, an outstaning orer (a lost sale, respectively) of the MS system is equivalent to a parcel of workloa (a lost volume, respectively) in the prouction facility s queue (cf. [6] which consiers the initial state X(;) =, i.e., I(;)=S). o prove part 1), we write L I (;) = minf ; g minfq ; g (1= )[ I(t; )t I(t; )t minf ; g I(t; )t]. By stanar arguments (see [2], [6], an [8]), minfq ; g Q 2 is locally inepenent of, an in the time interval [Q 2 ;], the system always starts with full inventory in a neighborhoo of. aking the erivative of the previous equation an noting that I(t; ) is inepenent of on ft < S g, I(t; ) = S on f S < t Q 2g, an that the terms associate with (=) S vanish, part 2) follows from (2.1) an [6, Proposition 3.2]. Part 2) follows from the proof of part 1) if we replace Q 2 with Q 1. o prove part 3), we assume, without loss of generality, > R 1 Z 1 > S = Q 1. hus, L I (;) = (1= )[ I(t; )t Z R I(t; )t I(t; )t Z I(t; )t]. Clearly, I(t; ) oes not epen on on f t< R Sg. By stanar arguments (see [2], [6], an [8]), I(t; ) oes not epen on on fz 1 <t<r 1g. Onf S <t<z 1g, I(t; ) =S t [( ) ( )]. Differentiation with respect to yiels (=)I(t; ) =1[( S )( S )] (=) S. Moreover, ( ) = I S implies (by ifferentiation) S = 1 ( S ) (3.2) Hence, (=)I(t; ) = ( S )=( S ), an, after some algebra, part 3) is obtaine. o prove part 4), we write L I(;) = minf ; g 1= [ I(t; )t I(t; )t]. minf ; g By efinition of ;I(t; ) =I(t; 1) on f t g for sufficiently small 1 >. Moreover, Q 2 is locally inepenent of, an I(t; ) =S on f <tq 2 g. Part 4) now hols by the proof of part 1). Part 5) hols by a similar proof if we replace Q 2 by Q 1. Finally, for parts 6) an 7), consier 1. he system starts in overage operational moe because I = S >S(1).By efinition, is the limiting time point at which fi(t; 1)g first hits S( 1) from above as 1! ; so is equivalent to the S of parts 2) an 3). Replacing S by, an using arguments similar to those in the proof of parts 2) an 3) proves parts 6) an 7). We next erive the IPA erivatives for L (;). For any time interval [a; b], let N [a;b] be the number of intervals of the form [Q j;r j], such that Z j < R j (i.e., lost sales actually occur) an Z j 2 [a; b]. heorem 2 W.p.1, the IPA erivatives of the lost-sales time average with respect to the base-stock level are given for all > an 2 2 as follows. 1) On the event A =fi < Sg an B =fi > Sg f S <Q 1 g L (;)= N ( ; ] 2) On the event C =fi >Sg f S =Q 1g L (;)= 1 1 ( S ) fz <R ;Z < g ( S ) 3) On the event D =fi = Sg L (;)= N ( ; ] N (R ; ] 4) On the event E =fi = Sg f <Q 1 g L (;)= N ( ; ] 5) On the event F =fi = Sg f = Q 1 g L (;)= 1 1 ( ) fz <R ;Z < g ( ) N (R ; ] Proof For part 1) an the event A, we write L (;) = minf ; g minfq ; g (1= )[ (t; )t (t; )t minf ; g (t; )t]. By the proof of part 1) in heorem 1, Q2 minfq ; g is locally inepenent of, an in the time interval [Q 2;], the system starts with full inventory in a neighborhoo of. Differentiating the previous equation an noting that (t; ) is inepenent of on f t< Sg an (t; ) =on f S t Q 2g, part 1) follows from [6, Proposition 3.1]. he same proof applies to the event B of part 1), provie Q 2 is replace by Q 1.

4 1494 IEEE RANSACIONS ON AOMAIC CONROL, VOL. 52, NO. 8, AGS 27 For part 2), we assume without loss of generality R 1 <, R an write L (;) = (1= )[ (t; )t Q R (t; )t Q (t; )t]. IfZ1 = R1, then there are no lost sales in (Q 1 ;R 1 ), whereas if Z 1 < R 1, then some lost sales occurre in (Q 1;R 1). Let [ k ;V k ];k = 1;...;K, be extremal subintervals of (Q 1 ;R 1 ) over which I(t; ) =. By stanar arguments (see [6]), k is locally ifferentiable in, V k is locally inepenent of, an V R (t; )t = K k=1 [(t) (t)]t = [( 1) ( 1)] 1 (3.3) Because S =Q 1 by assumption, ((t) (t))t = S. Differentiating the above yiels [( 1 ) ( 1)](=) 1 [( S) (S)](=)S =1. By (3.2) an (3.3), (=) R (t; )t = ( S )=( S ). Since (t; ) = on fr 1 t Q 2g, part 2) hols by [6, Proposition 3.1]. For part 3, let < for simplicity, an write L (;) = (1= )[ (t; )t (t; )t]. By efinition of ;I(t; ) = I(t; 1) for t 2 [; ) for sufficiently small 1 >. Furthermore, (t; ) = on f t Q 2 g, if =R1; similarly, (t; ) = on f t Q1g, if <R 1. hus, (= ) (t; )t = an part 3) hols by [6, Proposition 3.1]. For parts 4) an 5), consier 1. Because I = S > S( 1), the system starts in overage operational moe. Parts 4) an 5) follow by the proof of event B of part 1), because heorem 1 shows that in these cases, plays a role analogous to S. heorem 3 ner Assumptions 1 3, the sie IPA erivatives with respect to the base-stock level (= 6 )L I(;) an (= 6 )L (;) are unbiase for all > an 2 2. Proof Part 3) of Assumption 3 ensures that for all >, [5, Assumption C4 of Lemma A2, p. 7] hols for both L I (;) an L (;). Because ( S )=( S ) < 1 an (( )=( )) < 1 in heorems 1 an 2, it follows that (= 6 )L I(;) 1 by heorem 1. Furthermore, by heorem 2, (= 6 )L (;) 1N [; ] =, where E[N [; ] ] is finite because N [; ] is finite w.p.1 by part 2) of Assumption 1. Since the sample performance functions are continuous an piecewise ifferentiable, the one-sie erivatives exist for every. All IPA erivatives are boune. Hence, [5, Assumption C3 of Lemma A2] is in force, which completes the proof. IV. IPA DERIVAIVES WIH RESPEC O A PRODCION RAE PARAMEER his section erives sie IPA erivatives for the inventory time average L I (;) an the lost-sales time average L (;), both with respect to a prouction rate parameter of f(t; )g. Assumption 4 1) (=) (t; ) =1, where t 2 [;] an ) he process f(t)g an the base-stock level S are inepenent of. 3) For each 2 2, the sie erivatives of L I (;) an L (;) exist w.p.1. We point out that unlike [6], [4], an [7], Assumption 4 amits the possibility that sie IPA erivatives o not coincie. Inee, this coul happen on events of the form fi(t; ) = Sg f(t) = (t; )g an fi(t; ) = g f(t) =(t; )g. hese are generally not rare events, an in practice, their probabilities may well not vanish, because I(t; ) =S or I(t; ) =coul hol for an extene perio of time, an by part 1) of Assumption 1, f(t)g an f(t; )g have sample paths that are piecewise-constant w.p.1. In this section, we may assume without loss of generality that I S, since the prouction facility suspens replenishment in overage operational moe, so that the value of has no effect on the state of the system until it enters normal operational moe. Define ( m;v m );m = 1;...;M, to be the orere extremal subintervals of [; 1) such that for all t 2 ( m;v m ), either I(t; ) = S hols or both I(t; ) = an (t) > (t; ) hol. Define further ( n ;V n );n =1;...;N, to be the orere extremal subintervals of [; 1) such that for all t 2 ( n ;V n ), either I(t; ) = hols or both I(t; ) = S an (t) < (t; ) hol. By convention, if any of the aforementione enpoints oes not exist, then it is set to 1. For notational convenience, efine V =V =. By the continuity of fi(t; )g in t an part 3) of Assumption 1, 1 <V 1 < 2 <V 2 < 111 < M <V M an 1 <V 1 < 2 <V 2 < 111 < N <V. N We will nee the following horizon-epenent ranom inices. he restriction of M to a finite time horizon [;] is M I (;) = maxfm 1 V m g, if it exists, an zero, otherwise. he restriction of N to a finite time horizon [;] is N I (;) = maxfn 1 V n g, if it exists, an zero, otherwise. heorem 4 W.p.1, the IPA erivatives of the inventory time average with respect to the prouction rate parameter are given for all > an 2 2 as follows LI(;)= 1 M (;) [minf 2 m1;gv m ] 2 m= (4.4) L I(;)= 1 N (;) [minf 2 n1;gv n ] 2 n= (4.5) Proof We only prove (4.4) since the proof of (4.5) is analogous. By part 3) of Assumption 4 an Leibniz s rule, (= 6 )L I(;) = (1= )(= 6 ) I(t; ) t = (1= ) (=6 )I(t; ) t. We next compute (= )I(t; ). By efinition of ( m;v m ); one has I(t; ) = I(t; 1) on the events f m < t < V m g;m = 1;...;M, for sufficiently small 1. On the events fv m < t < m1 g;m = ; 1;...;M 1; one has I(t; ) = I(V m ;) t V [(; ) ( )]. By stanar arguments (see [2], [6], an [8]), V m an, therefore, I(V m ;) are each locally inepenent of in a right neighborhoo of. Consequently, (= )I(t; ) =[(V m ;) (V m )] (= )V m t V = t V m. A moicum of algebra completes the proof. o erive IPA erivatives for L (;), we will nee the following horizon-epenent ranom inices M (;) = maxfm 1 m g, if it exists, an zero, otherwise, as well as N (;) = maxfn 1 n g, if it exists, an zero, otherwise. Let 8(;) be the set of all inices m 2f1; 2;...;M (;)g such that I(t; ) =an (t) >(t; ) on the event f m <t<v m g. In a similar vein, let 9(;) be the set of all inices n 2f1; 2;...;N (;)g such that I(t; ) = on the event f n <t<v n g.

5 IEEE RANSACIONS ON AOMAIC CONROL, VOL. 52, NO. 8, AGS heorem 5 W.p.1, the IPA erivatives of the lost-sales time average with respect to the prouction rate parameter are given for all > an 2 2 as follows L (;)= 1 m28(;) [ m ;gv m1] (4.6) L (;)= 1 n29(;) [ n ;gv n1] (4.7) Furthermore, letting st enote stochastic orering, the sie IPA erivatives satisfy L (;) st L (;); > ; 2 2 (4.8) Stochastic equality hols above, provie fi(t; ) =Sg f(t) < (t; )g hols for all t 2 [;] an fi(t; ) =gf(t) >(t; )g hols for all t 2 [;]. Proof We only prove (4.6) since the proof of (4.7) is analogous. By stanar arguments (see [2], [6], an [8]), the set 8(;) is locally inepenent of in a right neighborhoo of, an m is locally ifferentiable with respect to in a right neighborhoo of. hus L (;)= 1 m28(;) By part 1) of Assumption 4, for each m 2 8(;) ; g (t; ) t (4.9) ; g (t; ) t = [( m)( m;)] 2 m ; g t (4.1) o compute the term associate with (= ) m, consier [(t) (t; )] t for m 2 8(;). A moicum V of algebra shows that [(t) (t; )] t is locally V inepenent of. aking the right erivative of this integral yiels [(V m1 ) (V m1 ;)](= )V m1 [( m) ( m;)](= ) m t =, V because the first term vanishes, resulting in [( m) ( m;)](= ) m = t. Finally, substituting the previous equation into (4.8) yiels (= ; g V ) ; g (t; ) t = t = [ V m ;gv m1 ]. Equation (4.6) now follows by substituting the previous equation into (4.1). Finally, inequality (4.8) follows irectly from (4.6) an (4.7) an the efinition of V m an Vn by stanar arguments. heorem 6 ner Assumptions 1 an 4, the IPA erivatives with respect to the prouction rate parameter, (= 6 )L I (;) an (= 6 )L (;), are unbiase for all > an 2 2. Proof Part 3) of Assumption 4 ensures that for all >, [5, Assumption C4 of Lemma A2] hols for both L I (;) an L (;). Moreover, heorems 4 an 5 imply that (= 6 )L I(;) =2 an j(= 6 )L (;)j 1. Since the one-sie erivatives exist for every an all IPA erivatives are boune, [5, Assumption C3 of Lemma A2] is in force, which completes the proof. V. DISCSSION Reference [8] an the current note can provie a theoretical basis for new online control algorithms of prouction-inventory systems, incluing those where the unerlying stochastic processes (i.e., eman an prouction capacity processes) may be subject to nonstationary probability laws. One irection of future research is the extension of the current results to more general supply chains with multiple proucts, such as assemble-to-orer systems (e.g., such as those implemente by Dell Computer Corporation, Roun Rock, X), where eman patterns fluctuate consierably over time. REFERENCES [1] C. G. Cassanras an S. Lafortune, Introuction to iscrete event systems. Boston, MA Kluwer, [2] C. G. Cassanras, Y. Wari, B. Melame, G. Sun, an C. G. Panayiotou, Perturbation analysis for on-line control an optimization of stochastic flui moels, IEEE rans. Autom. Control, vol. 47, no. 8, pp , Aug. 22. [3] M. C. Fu an. Q. Hu, Conitional Monte Carlo Graient Estimation an Optimization Applications. Boston, MA Kluwer, [4] I. Paschaliis, Y. Liu, C. G. Cassanras, an C. Panayiotou, Inventory control for supply chains with service level constraints A synergy between large eviations an perturbation analysis, Ann. Oper. Res., vol. 126, pp , 24. [5] R. Y. Rubinstein an A. Shapiro, Discrete Event Systems Sensitivity Analysis an Stochastic Optimization by the Score Function Metho. New York Wiley, [6] Y. Wari, B. Melame, C. G. Cassanras, an C. G. Panayiotou, Online IPA graient estimators in stochastic continuous flui moels,. Optim. heory Appl., vol. 115, no. 2, pp , 22. [7] Y. Zhao an B. Melame, Make-to-stock systems with backorers IPA graients, presente at the Winter Simulation Conf. (WSC), Washington, D.C., Dec. 5 8, 24. [8] Y. Zhao an B. Melame, IPA erivatives for make-to-stock prouction-inventory systems with backorers, Methoology Comput. Appl. Probability, vol. 8, no. 2, pp , 25. Minimal Communication for Essential ransitions in a Distribute Discrete-Event System Feng Lin, Karen Ruie, an Stéphane Lafortune Abstract In a istribute iscrete-event system with ecentralize information, agents at the various sites (e.g., controllers or iagnosers) may be require to communicate in orer to correctly perform some prescribe tasks. Banwith, power, or security constraints motivate the esign of Manuscript receive December 13, 26; revise March 26, 27. Recommene by Associate Eitor S. Haar. his work was supporte in part by the Natural Sciences an Engineering Research Council of Canaa (NSERC) uner Grant , the National Science Founation (NSF) uner Grants CCR-82784, CCR , an IN , the National Aeronautics an Space Aministration (NASA) uner Grant NAG2-1279, the.s. Office of Naval Research (ONR) uner Grant N , an the National Institute of Health (NIH) uner Grant 1 R21 EB1529-1A1. F. Lin is with the Department of Electrical an Computer Engineering, Wayne State niversity, Detroit, MI 4822 SA ( flin@ece.eng.wayne.eu). K. Ruie is with the Department of Electrical an Computer Engineering, Queen s niversity, Kingston, ON K7L 3N6 Canaa ( karen.ruie@queensu.ca). S. Lafortune is with the Department of Electrical Engineering an Computer Science, niversity of Michigan, Ann Arbor, MI SA ( stephane@eecs.umich.eu) /$ IEEE