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

Transcription

1 IPA Derivatives for Make-to-Stock Prouction-Inventory Systems With Lost Sales Yao Zhao Benjamin Melame Rutgers University Rutgers Business School Newark an New Brunswick Department of MSIS 94 Rockafeller R. Piscataway, NJ Abstract A stochastic flui moel (SFM) is a queueing moel in which workloa flow is moele as flui flow. More specifically, the traitional iscrete arrival, service an eparture stochastic processes are replace by corresponing stochastic flui-flow rate processes in an SFM. This paper applies the SFM paraigm to a class of single-stage, single-prouct Make-to-Stock (MTS) prouctioninventory systems with stochastic eman an ranom prouction capacity, where the finishegoos inventory is controlle by a continuous-time base-stock policy an unsatisfie eman is lost. The paper erives formulas for IPA (Infinitesimal Perturbation Analysis) 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. These formulas are comprehensive in that they are exhibite for any initial inventory state, an inclue right an left erivatives (when they iffer). The formulas are obtaine via sample path analysis uner very mil 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 very fast to compute, thereby proviing the theoretical basis for on-line aaptive control of MTS prouction-inventory systems. Keywors an Phrases: Infinitesimal Perturbation Analysis, IPA, IPA erivatives, IPA graients, Lost Sales, Make-to-Stock, prouction-inventory systems, stochastic flui moels, SFM.

2 1 Introuction Prouction-inventory systems consist of prouction facilities that fee replenishment prouct to inventory facilities, riven by ranom eman an possibly ranom prouction processes, as well as feeback information from inventory to prouction facilities. An important instance of prouctioninventory systems is the Make-to-Stock (MTS) class, where the inventory facility sens its state information to the prouction facility as a control signal, which moulates prouction with the aim of maintaining the inventory level at a prescribe level, calle base-stock level. Such systems can amit backorers when stock is eplete, or suffer lost sales. This paper treats MTS systems with lost sales (see Section 2), an is a sequel to Zhao an Melame (25), which treats MTS systems with backorers. Economic consierations in supply chains call for effective control of inventory levels an prouction rates, in orer to optimize some prescribe performance metrics. In many real worl applications, the unerlying eman an prouction processes may be subject to time varying probability laws. This motivates on-line algorithms that can aaptively control such systems over time with the objective of minimizing inventory on-han without compromising customer service metrics. To this en, we propose to use IPA (Infinitesimal Perturbation Analysis) erivatives of selecte ranom variables [for comprehensive iscussions of IPA erivatives an their applications, refer to Glasserman (1991), Ho an Cao (1991) an Fu (1994a, 1994b)]. 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 on-line control algorithms. Specifically, let L(θ) be a ranom variable, parameterize by a generic real-value parameter θ chosen from a close an boune set Θ. The IPA erivative (graient) of L(θ) with respect to θ is the ranom variable L (θ) = θ L(θ), provie that it exists almost surely. Furthermore, L (θ) issaitobeunbiase, if the expectation an ifferentiation operators commute, namely, E[ θ L(θ)] = E[L(θ)]; otherwise, it is sai to be biase. Sufficient conitions for unbiase IPA erivatives are given in the θ following lemma Fact 1 (see Rubinstein an Shapiro (1993), Lemma A2, p. 7) An IPA erivative L (θ) is unbiase, if (a) For each θ Θ, the IPA erivatives L (θ) exist w.p.1 (with probability 1). (b) W.p.1, L(θ) is Lipschitz continuous in Θ, an the (ranom) Lipschitz constants have finite first moments. Comprehensive iscussions of IPA erivatives an their applications can be foun in Glasserman (1991), Ho an Cao (1991) an Fu (1994). Most papers on prouction-inventory systems (an MTS systems in particular) postulate specific probability laws that govern the unerlying stochastic processes (e.g., Poisson eman arrivals an exponential service times). For simple systems, such as the one-stage MTS variety, close-form formulas of key performance metrics (e.g., statistics of inventory levels an lost sales or backorers) have been erive as functions of control parameters. For example, Zipkin (1986) an Karmarkar (1987) obtain the optimal control of these systems with respect to batch sizes an re-orer points by stanar optimization techniques. For more complex MTS systems, such as the multi-stage serial variety, close-form formulas are not available. A sample path analysis is carrie out by Buzacott, Price an Shanthikumar (1991) for a 2-stage prouction system which is governe by the 1

3 continuous-time base-stock policy. Diffusion moels an eterministic flui moels have been propose in orer to mitigate the analytical an computational complexity of performance evaluation an optimal control. For example, Wein (1992) use a iffusion process to moel a multi-prouct, single-server MTS system, while Veatch (22) iscusse iffusion an flui-flow moels of serial MTS systems. Note, however, that iffusion moels require a heavy traffic conition in orer to be vali approximations (Wein 1992). In a similar vein, while eterministic flui-flow moels provie valuable insights into the control rules of such systems, eterministic moeling may well result in substantial numerical errors (Veatch 22). Simulation has been wiely use to stuy the performances of complex prouction-inventory systems uner uncertainty. Glasserman an Tayur (1995)consiere a class of prouction-inventory systems uner the so-calle perioic-review, moifie base-stock policy, an estimate its performance metrics an IPA erivatives using simulation. While perioic-review policies evaluate system performance at iscrete review times, iscrete-event simulation, in contrast, can track system performance continuously, but this can be overly time consuming for large-scale systems, ue to the large number of events that nee to be processe (e.g., arrivals an service completions). All in all, most papers on stochastic prouction-inventory systems postulate a specific unerlying probability law, an focus on off-line control an optimization algorithms. Recent work has sought to aress these shortcomings in the context of flui-flow queueing systems, an especially, the stochastic flui moel (SFM) setting, where transactions carry flui workloa, ranom iscrete arrivals become ranom arrival rates an ranom iscrete services become ranom service rates. SFM-like settings represent an alternative (continuous or flui-flow) queueing paraigm, which iffers from the traitional (iscrete) queueing paraigm in the way workloa is transporte in the system 1. Both paraigms are set in a network of noes, each of which houses a server an a buffer, where network sources an sinks are viewe as exogenous noes, an all others as enogenous noes. Transactions representing parcels of workloa arrive at the network from some source, traverse the network accoring to some itinerary, an then epart the network at some sink. The two queueing paraigms iffer, however, in the way workloa moves in the system. In the iscrete queueing paraigm, transaction workloa moves abruptly among noes following a service time, while in the continuous queueing paraigm, transaction workloa moves graually (i.e., flows like flui) for the uration of its service time. A heuristic moeling rationale unerlying SFM systems is the assumption that iniviual transactions carry miniscule workloa as compare to the entire transaction flow, so the effect of iniviual transactions is infinitesimal an akin to molecules in a flui flow. Furthermore, in many cases, a transaction workloa oes move graually from one noe to another, rather than abruptly (e.g., a conveyor belt carrying bulk material, loaing an unloaing a truck, train, etc.) In fact, iscrete queueing systems can be abstracte as limiting cases of continuous queueing systems, where the flow rate is zero when a transaction is still, but at the moment of motion the flow rate becomes momentarily infinite; in other wors, the flow rate is akin to a Dirac function. Pursuing this line of reasoning, the Dirac pulses of flow rates in a iscrete queueing system can be approximate by high flow rates of short uration in a continuous queueing system. Whichever reasoning is use, the moeler can often choose to moel a queueing system using either paraigm on equal footing. Finally, we point out that ceteris paribus, SFM systems enjoy an important avantage over their iscrete counterparts: IPA erivatives in SFM setting are unbiase, while their counterparts in iscrete queueing systems are by an large biase (Heielberger et al. 1988). Thus, the local shape of sample paths in the flui-flow paraigm confers technical avantages on 1 For simplicity we aress only open networks in this iscussion. 2

4 them. IPA erivatives, erive in SFM setting, can provie important information an insights for their iscrete counterparts, by applying erivative formulas obtaine in SFM setting to queueing systems that have been traitionally viewe as belonging to the iscrete queueing paraigm. While preliminary unpublishe work by one of the authors suggests that this approach is viable, more work is neee to establish its broa applicability. Motivate by the consierations above, Wari et al. (22) erive IPA erivatives in SFM setting; we henceforth refer to this approach as IPA-over-SFM. Wari et al. (22) consiere two performance metrics: loss volume an buffer-workloa time average; each of these metrics was ifferentiate with respect to buffer size, a parameter of the arrival rate process an a parameter of the service rate process. The paper showe the IPA erivatives to be unbiase, easily computable an nonparametric. Consequently, these erivatives can be compute in simulations, or in the fiel, an the values can have potential applications to on-line control an stochastic optimization. Paschaliis et al. (24) treate multi-stage MTS prouction-inventory systems with backorers in SFM setting. Assuming that inventory at each stage is controlle by a continuous-time basestock policy, the paper compute the right IPA erivatives of the time average inventory level an service level with respect to base-stock levels, an use them to etermine the optimal base-stock levels at each stage. Zhao an Melame (24) applie the IPA-over-SFM approach to a class of single-prouct, single-stage MTS systems with backorers, an erive IPA formulas for the time averages of inventory level an backorer level with respect to the base-stock level, as well as a parameter of the prouction rate process. It shoul be pointe out that Wari et al. (22), Paschaliis et al. (24) an Zhao an Melame (24) assume that systems start with certain initial inventory states, an only consier cases where the left an right IPA erivatives coincie. In contrast, Zhao an Melame (25) consiere any initial inventory state an erive sie IPA erivative formulas where neee, thereby proviing the theoretical basis for IPA-base on-line control of MTS systems with backorers. The goal of this paper is to erive IPA erivatives for MTS systems with lost-sales, an to show them to be unbiase. The paper makes the following contributions. First, we erive IPA erivative formulas for two metrics, the inventory-level time average an lost-sales time average, with respect to the base-stock level for all initial inventory states, incluing sie erivatives when they iffer. We are only aware of one paper [Wari et al. (22)] aressing IPA-over-SFM queues with finite buffers, which can be use to moel MTS systems with lost sales. But unlike the current paper, Wari et al. (22) limits the initial conition to an empty buffer. Secon, we erive IPA erivative formulas for the aforementione metrics with respect to a prouction rate parameter, incluing sie erivatives when they iffer. In contrast, Wari et al. (22) only consiers cases where the left an right IPA erivatives coincie (in fact, the left an right IPA erivatives may iffer). The computation of the general IPA erivatives for any initial inventory state an for cases where the left an right IPA erivatives may iffer requires major extensions of the current results in the literature. As will become evient in the sequel, MTS systems with lost-sales are also analytically more challenging than MTS systems with backorers, a fact that results in more elaborate formulas. The merit of our contribution stems from potential applications of IPA erivatives to on-line control of MTS systems. Clearly, IPA-base on-line control applications manate the computation of IPA erivatives for all initial inventory states, as well as all sie erivatives when they iffer, since a control action can change system parameters at a variety of system states (which are then consiere as new initial states). Moreover, it obviously makes little or no sense to wait for the system to return to selecte inventory states for which IPA erivatives are known, as this coul suspen control actions over extene perios of time. To summarize, for IPA-base applications 3

5 to be general an efficacious, it is necessary that the requisite IPA erivative formulas satisfy the following requirements: 1. For usability, they shoul be comprehensive in the sense that they are vali for any initial conition of the system. In aition, if a left-erivative oes not coincie with its righterivative counterpart, then both shoul be exhibite. 2. For statistical accuracy, they shoul be unbiase. 3. For generality, they shoul be nonparametric inthe sense that they aresolelycomputable from the sample path observe without making any istributional assumptions on the unerlying probability law. 4. To enable on-line applications, they shoul be fast to compute. To this en, this paper erives all sie IPA erivatives for MTS systems with lost-sales for any initial inventory state. It further shows these IPA erivatives to be unbiase, nonparametric, an easy to compute, which facilitates on-line control applications. Throughout the paper, we use the following notational conventions an terminology. The inicator function of set A is enote by 1 A,anx + =max{x, }. A function f(x) issaito be locally ifferentiable at x if it is ifferentiable in a neighborhoo of x; itissaitobelocally inepenent of x if is constant in a neighborhoo of x. The rest of the paper is organize as follows. Section 2 presents the prouction-inventory moels uner stuy. Section 3 provies variational bouns for system metrics. Section 4 erives IPA erivative formulas an shows them to be unbiase. Finally, Section 5 conclues the paper. 2 The Make-to-Stock Moel With Lost Sales Consier the traitional single-stage, single-prouct MTS system, consisting of a prouction facility an an inventory facility. The two facilities interact: the latter sens back orers to the former, while the former prouces stock to replenish the latter. The prouction facility is comprise of a queue that houses a prouction server (a single machine, a group of machines or a prouction line), 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. The inventory facility satisfies incoming emans on a first come first serve (FCFS) basis, an is controlle by a continuous-time base-stock policy with some base-stock level S > (the case S = correspons to a just-in-time system an its treatment is a simple special case.) More specifically, the inventory an prouction facilities are couple, an operate in two moes as follows: Normal operational moe. While the inventory level oes not excee S, the inventory facility places the orers of incoming emans as iscrete prouction jobs in the prouction facility s buffer accoring to some operational rule (to be etaile below). The prouction facility fills these outstaning orers an replenishes the inventory facility back to its base-stock level, but no higher. We refer to this operational moe as normal operation, because the system strives to reach an inventory level S, an in so oing, it maintains an inventory level not exceeing S. Overage operational moe. While the inventory level excees S (this coul happen, for example, as a result of a control action that lowere S), the prouction facility buffer is empty, 4

6 Figure 1: The Make-to-Stock prouction-inventory system with lost sales so prouction is temporarily suspene until the inventory level reaches or crosses S from above, at which point normal operation is resume. We refer to this operational moe as overage operation, because it allows the system to aapt to a lower base-stock level, S, aiming to enter normal operation. The eman process consists of an interarrival-time process of emans an their ranom magnitue. Demans arrive at the inventory facility an are satisfie from inventory on han (if available). Otherwise, when an inventory shortage is encountere, the behavior of the MTS queue is governe by the lost-sales rule as follows: The incoming eman is satisfie by the amount of inventory on han, an any shortage of inventory becomes a lost sale. Thus, the system s overall actions aim to move the inventory level to the base-stock level, S. 2.1 Mapping MTS Systems to SFM Versions We next procee to map the traitional iscrete MTS system with lost sales into an SFM version, as epicte in Figure 1. Level-relate stochastic processes are mappe into flui versions of their traitional counterparts in a natural way, as follows: Inventory level. The traitional jump process of the level of inventory on han at the inventory facility is mappe to a flui-level counterpart, {I(t)}, where I(t) is the (flui) volume of inventory on-han at time t. Outstaning orers. The traitional jump process of the level of outstaning orers in the buffer of the prouction facility is mappe to a flui-level counterpart, {X(t)}, wherex(t) isthe (flui) volume of outstaning orers at time t. Traffic-relate stochastic processes in Figure 1 are mappe into flui versions of their traitional counterparts, as follows: Arrival rate. The traitional arrival process of iscrete emans at the inventory facility is mappe to a flui-flow stochastic process, {α(t)}, where α(t) is the rate of incoming emans at time t. 5

7 Prouction rate. The traitional service (prouction) process of iscrete prouct at the prouction facility is mappe to a flui-flow stochastic process, {µ(t)}, where µ(t) is the prouction rate at time t. Loss rate. The traitional loss process of iscrete sales at the inventory facility is mappe to a flui-flow stochastic process, {ζ(t)}, where ζ(t) is the (flui) loss rate of sales at time t. Outstaning orer rate. The traitional arrival process of signals for placing iscrete outstaning orers at the prouction facility is mappe to a flui-flow stochastic process, {λ(t)}, where λ(t) is the rate of incoming outstaning orers at time t. Replenishment rate. The traitional traffic process of iscrete replenishe prouct from the prouction facility to the inventory facility is mappe to a flui-flow stochastic process, {ρ(t)}, where ρ(t) is the traffic rate of prouct at time t. We now procee to exhibit the formal efinitions of all flui-moel components of the MTS system with lost sales. During overage operation, the inventory process is governe by the one-sie stochastic ifferential equation I(t) = α(t), (2.1) t + an ζ(t) =, (2.2) X(t) =. (2.3) During normal operation, the moel satisfies the conservation relation, X(t)+I(t) =S. (2.4) The outstaning orers process is governe by the sie stochastic ifferential equation,, X(t) = an α(t) µ(t) X(t) t + =, X(t) =S an α(t) µ(t) α(t) µ(t), otherwise (2.5) The lost-sales rate process is given by ζ(t) = [α(t) µ(t)] 1 {I(t)=,α(t)>µ(t)}, t. (2.6) The arrival-rate process of outstaning orers is given by, if I(t) > S λ(t) = µ(t), if I(t) = an α(t) >µ(t) α(t), otherwise (2.7) an the replenishment-rate process is given by ρ(t) = { µ(t), if X(t) > min{µ(t),λ(t)}, if X(t) =. (2.8) 6

8 2.2 Performance Metrics an Parameters Let [,T] be a finite time interval, where T is pre-efine constant, etermines the time perio uring which system performances are evaluate before a control action regaring the inventory policy an/or prouction rate is taken. We shoul not confuse T with the review perio of a perioic-review inventory policy. In this paper, we will be intereste in the following ranom variables, to be henceforth referre to as performance ranom variables or simply metrics. Inventory time average. The time average of the inventory on-han (flui volume) over the interval [,T], given by L I (T )= 1 T I(t) t. (2.9) T Lost-sales time average. The time average of flui rate of lost sales over the interval [,T], given by L ζ (T )= 1 T ζ(t) t. (2.1) T Observe that the metrics L I (T )anl ζ (T ) are ranom variables for each T. Let θ Θ enotes a generic parameter of interest with a close an boune omain Θ. We write S(θ), µ(t, θ), L I (T,θ),L ζ (T,θ) an so on to explicitly isplay the epenence of a performance ranom variable on its parameter of interest. Our objective is to erive formulas for the IPA erivatives θ L I(T,θ), an θ L ζ(t,θ) in the SFM setting, using sample path analysis, an to show them to be unbiase. The parameters of interest in this section are liste below: Base-stock level. The base-stock level of the inventory facility, S(θ) =θ, θ Θ. (2.11) Prouction rate parameter. A parameter of the prouction rate process, such that µ(t, θ) =1, t [,T], θ Θ, (2.12) θ interprete as a scaling parameter of the prouction rate. 2.3 Assumptions The 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 Wari et al. (22), we efine two types of sample path events: Exogenous events. An exogenous event occurs either whenever a jump occurs in the sample path of {α(t)} or {µ(t)}, or when the time horizon T, is reache. Enogenous events. An enogenous event occurs whenever a time interval is inaugurate, in which X(t)=orX(t)=S. 7

9 Throughout this paper, we assume the following mil regularity conitions (cf. Wari et al. (22)). Assumption 1 (a) (b) The eman rate process, {α(t)}, an the prouction rate process, {µ(t)}, have right-continuous sample paths that are piecewise-constant w.p.1. Each of the processes, {α(t)} an {µ(t)}, 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. (c) No multiple events occur simultaneously w.p.1. The following observations follow from Assumption 1. Observation 1 1. W.p.1, there exists a finite integer N an a sequence of (ranom) time points =T < T 1 < <T N <T N+1 = T, such that the process {α(t) µ(t)} is constant over each interval (T n,t n+1 ), n =,,N,aneachtimepointT n, 1 n N, is a jump point of the process. 2. The process {α(t) µ(t)} is constant over each time interval (T n,t n+1 ), n =,,N. Proof. See Observation 1 in Zhao an Melame (25). Finally, we shall be intereste in pairs of systems, the original system (inexe by θ) ana perturbe system (inexe by θ ± θ), both starting at the same initial conitions. To simplify the notation in the sequel, we shall also make the following assumption, without any loss of practical generality. Assumption 2 The initial inventory level oes not epen on θ, namely,i(,θ)=i() for all θ Θ. 3 Variational Bouns In this section, we erive variational bouns for various parameterize stochastic processes an performance metrics in the MTS moel with lost sales. These results will be use in subsequent sections to simplify the erivation of IPA erivatives an to establish their unbiaseness. The variational bouns will be shown to hol with respect to the control parameters of interest at each time point, starting from an arbitrary initial inventory level, I(). It follows from Eqs. (2.1), (2.4) an (2.5) that the time erivative of I(t) satisfies I(t) t + = α(t), if I(t) >S, if I(t) =S an α(t) µ(t), if I(t) =anα(t) µ(t) µ(t) α(t), otherwise. (3.1) 8

10 3.1 Variational Bouns With Respect to the Base-Stock Parameter In this section, the IPA parameter of interest is S(θ) =θ for θ Θ. Let {I(t, θ)} be the inventory level process in an MTS system with lost-sales, where θ ΘanI(,θ)=I(). Then, Eq. (3.1) inuces a (ranom) partition of the interval [,T], given by where each region, R k (θ), 1 k 4, is efine as follows, R(θ) ={R 1 (θ), R 2 (θ), R 3 (θ), R 4 (θ)}, (3.2) R 1 (θ) = {t [,T]: I(t, θ) =anα(t) µ(t)}, R 2 (θ) = {t [,T]: [I(t, θ) =anα(t) <µ(t)] or [I(t, θ) =S(θ) anα(t) >µ(t)] or <I(t, θ) <S(θ)}, R 3 (θ) = {t [,T]: I(t, θ) =S(θ) anα(t) µ(t)}, R 4 (θ) = {t [,T]: I(t, θ) >S(θ)}. We first prove the variational bouns for the inventory level process, {I(t, θ)}. Proposition 1 For an MTS system with the lost-sales rule, let θ 1,θ 2 Θ. Then, Proof. Recall that by Assumption 2, I(t, θ 1 ) I(t, θ 2 ) θ 1 θ 2, t [,T]. (3.3) I(,θ 1 ) = I(,θ 2 )=I(). (3.4) Clearly, Eqs. (3.4) an (3.1) imply the result trivially for θ 1 = θ 2. It remains to show the result for the case θ 1 θ 2. Without loss of generality, we assume that θ 1 <θ 2,anshowthat I(t, θ 2 ) I(t, θ 1 ) θ 2 θ 1, t [,T]. (3.5) To this en, we first prove the lefthan sie of inequality (3.5) by showing that whenever I(t, θ 1 )= I(t, θ 2 ) for any t [,T], one has t + [I(t, θ 2) I(t, θ 1 )]. An examination of Eq. (3.1) reveals that the equality I(t, θ 1 )=I(t, θ 2 ) can take place only in the following cases. Case 1: t R k (θ 1 ) R k (θ 2 )forsome1 k 4. In this case, we immeiately have I(t, θ 1 )] =. t + [I(t, θ 2) Case 2: t R 3 (θ 1 ) R 2 (θ 2 ). In this case, I(t, θ 1 )=I(t, θ 2 )=S(θ 1 )=θ 1 an t + [I(t, θ 2) I(t, θ 1 )] = µ(t) α(t), where the inequality follows from the efinition of R 3 (θ 1 ). Case 3: t R 4 (θ 1 ) R 2 (θ 2 ). In this case, R 2 (θ 2 )anr 4 (θ 1 ). Case 4: t R 4 (θ 1 ) R 3 (θ 2 ). In this case, R 3 (θ 2 )anr 4 (θ 1 ). t + [I(t, θ 2) I(t, θ 1 )] = µ(t) by the efinition of t + [I(t, θ 2) I(t, θ 1 )] = α(t) by the efinition of 9

11 The lefthan sie of inequality (3.5) follows from Eq. (3.4) an by the continuity of the realizations of the inventory level process. To prove the righthan sie of inequality (3.5), we examine the behavior of {I(t, θ 2 ) I(t, θ 1 )} in the four regions of the partition (3.2). Informally, the proof characterizes {I(t, θ 2 ) I(t, θ 1 )} for all pairs of regions in the partitions associate with each θ, such that I(t, θ 1 ) is in one region an I(t, θ 2 ) is in the other. More formally, the characterization covers t in all intersections of the form R i (θ 1 ) R j (θ 2 ), 1 i, j 4. Note that the intersections partition the interval [,T]antheir number is finite w.p.1 by Part (b) of Assumption 1. The proof procees in two steps. In the first step, we consier the extremal open set (a, b) of any such intersection. We then show that if then I(a, θ 2 ) I(a, θ 1 ) θ 2 θ 1, (3.6) I(t, θ 2 ) I(t, θ 1 ) θ 2 θ 1, a<t<b. (3.7) By continuity of the inventory level process, the inequality (3.7) will then exten to the interval [a, b]. In the secon step, we orer the intervals (a k,b k ) contiguously, an prove the inequality (3.5) throughout [,T], by a straightforwar inuction on k, where the inuction basis hols by Eq.(3.4), an the inuction step is immeiate from the contiguity of the orere intersections. The following observation reuces substantially the number of region-pair cases (intersections) to be checke. There is no nee to check for pairs of regions with the same subscript, i = j, since in their intersection t + [I(t, θ 2) I(t, θ 1 )] = trivially, which implies that I(t, θ 2 ) I(t, θ 1 )is constant in the intersection. It remains to check the following list of cases. Case 1: t R 1 (θ 1 ) R 2 (θ 2 ). In this case, t + [I(t, θ 2) I(t, θ 1 )] = µ(t) α(t), where the inequality follows from the efinition of R 1 (θ 1 ). In view of (3.6), inequality (3.7) immeiately follows. Case 2: t R 1 (θ 1 ) R 3 (θ 2 ). In this case, t + [I(t, θ 2) I(t, θ 1 )] =. Case 3: t R 3 (θ 1 ) R 2 (θ 2 ). In this case, I(t, θ 2 ) θ 2 an I(t, θ 1 )=θ 1, which implies that I(t, θ 2 ) I(t, θ 1 ) θ 2 θ 1. Case 4: t R 4 (θ 1 ) R 2 (θ 2 ). In this case, I(t, θ 2 ) θ 2 an θ 1 <I(t, θ 1 ), which implies that I(t, θ 2 ) I(t, θ 1 ) θ 2 θ 1. Case 5: t R 4 (θ 1 ) R 3 (θ 2 ). In this case, I(t, θ 2 )=θ 2 an θ 1 <I(t, θ 1 ), which implies that I(t, θ 2 ) I(t, θ 1 ) θ 2 θ 1. The proof is complete. We next erive variational bouns for the time average of lost sales. To this en, we efine K(T,θ) to be the number of extremal subintervals of [,T]inwhichI(t, θ) =. Proposition 2 For an MTS system with the lost-sales rule, let θ 1,θ 2 Θ. Then, T ζ(t, θ 1 ) ζ(t, θ 2 ) t max{k(t,θ 1 ),K(T,θ 2 )} θ 1 θ 2. (3.8) 1

12 Proof. Thecaseofθ 1 = θ 2 is trivial, so it remains to prove the case θ 1 θ 2, an assume θ 1 <θ 2 without loss of generality. We prove the following inequality, T [ζ(t, θ 1 ) ζ(t, θ 2 )] t K(T,θ 1 )[θ 2 θ 1 ], (3.9) which is stronger than the requisite result. Since θ 1 <θ 2, the proof of Proposition 1 implies that I(t, θ 1 ) I(t, θ 2 ) for all t [,T]. Consequently, ζ(t, θ 2 ) ζ(t, θ 1 ) for all t [,T], which establishes the lefthan sie of (3.9). We next prove the righthan sie of inequality (3.9). In view of the inequality I(t, θ 1 ) I(t, θ 2 ) for all t [,T], it suffices to show that for any extremal subinterval [U, V ]of[,t]inwhich I(t, θ 1 ) =, one has t U [ζ(τ,θ 1 ) ζ(τ,θ 2 )] τ I(U, θ 2 ) I(U, θ 1 ), t [U, V ]. (3.1) Define W [U, V ] to be the first time point at which I(t, θ 2 ) =, if it exists; otherwise, efine W = V. Since I(t, θ 1 ) = I(t, θ 2 ) = for t [W, V ), it follows from Eq. (2.6) that V [ζ(τ,θ 1 ) ζ(τ,θ 2 )] τ =, so it remains to consier the interval [U, W ). But for any t [U, W ), W ζ(t, θ 2 )=anζ(t, θ 1 )=α(t) µ(t). Hence, for every t [U, W ), t U [ζ(τ,θ 1 ) ζ(τ,θ 2 )] τ = t U [α(τ) µ(τ)] τ. We conclue that for every t [U, W ), t U [α(τ) µ(τ)] τ = I(U, θ 2 ) I(t, θ 2 ) I(U, θ 2 ) I(U, θ 1 ) θ 2 θ 1, where the equality is ue to the ynamics of Eq. (3.1), the first inequality follows from the relation I(t, θ 2 ) I(U, θ 1 ) =, an the secon inequality follows from (3.5). The result now follows by applying inequality (3.1) to all extremal subintervals of the form [U, V ] an summing the corresponing inequalities. 3.2 Variational Bouns With Respect to a Prouction Rate Parameter In this section, the IPA parameter of interest is a parameter, θ, of the prouction rate process, {µ(t, θ)}, satisfying Eq. (2.12). Our results buil upon prior results in Wari an Melame (21), which assume a special initial conition for the workloa. Observation 2 For an MTS system with the lost-sales rule, the stochastic ifferential equations governing the outstaning orers process {X(t)} in normal operation, the loss-rate process, {ζ(t)}, an the replenishment rate process, {ρ(t)}, are ientical to those governing the buffer workloa, overflow an outflow processes, respectively, in the SFM queuing system stuie in Wari et al. (21). Proof. Follows from the fact that we can ientify the eman arrival rate process, prouction rate process, an base-stock level parameter, respectively, with the inflow rate process, service rate process an buffer capacity parameter in Wari an Melame (21). 11

13 For notational convenience, we efine an auxiliary process, calle the extene outstaning orers process, {Y (t, θ)}, by Y (t, θ) = { S I(t, θ), if I(t, θ) >S (overage operation) X(t, θ), if I(t, θ) S (normal operation) (3.11) Observe that Y (t) is negative uring overage operation an non-negative uring normal operation. Furthermore, Eq. (2.4) implies the conservation relation, vali for each operational moe (overage an normal). I(t, θ)+y (t, θ) = S, t, (3.12) Proposition 3 For an MTS system with the lost-sales rule, let θ 1,θ 2 Θ. Then, max{ Y (t, θ 1 ) Y (t, θ 2 ) : t [,T]} T θ 1 θ 2 an T ζ(t, θ 1 ) ζ(t, θ 2 ) t 2 T θ 1 θ 2. Proof. By Assumption 2 an Eq. (3.12), Y (,θ 1 )=Y (,θ 2 ). In view of the fact that {Y (t, θ 1 )} an {Y (t, θ 2 )} coincie uring overage operation, it suffices to assume that the system starts in normal operation, namely, Y (,θ 1 )=Y (,θ 2 ). The results follow immeiately from Proposition 3.2 of Wari an Melame (21), since the proof there is reaily seen to hol for any initial state in normal operation. Corollary 1 For an MTS system with the lost-sales rule, let θ 1,θ 2 Θ. Then, I(t, θ 1 ) I(t, θ 2 ) T θ 1 θ 2, t [,T]. Proof. Eq. (3.12) an Proposition 3 imply that I(t, θ 1 ) I(t, θ 2 ) = [S Y (t, θ 1 )] [S Y (t, θ 2 )] T θ 1 θ 2. 4 IPA Derivatives We are now in a position to erive IPA erivatives for various parameterize stochastic processes an performance metrics in the MTS moel subject to lost sales rule. We mention in passing that such systems are analytically more challenging than MTS systems with backorers, because the inventory state of the former has an extra bounary. More specifically, while the inventory state of both systems is boune from above by S, that of MTS systems with lost sales is also boune from below by. Let (Q j (θ),r j (θ)), j =1,...,J(θ) be the orere extremal subintervals of [, ), such that I(t, θ) <Sfor all t (Q j,r j ), that is,the enpoints, Q j (θ) anr j (θ), are obtaine via inf an sup functions, respectively. By convention, if any of these enpoints oes not exist, then it is set to. Furthermore, we let Z j (θ) (Q j (θ),r j (θ)) be the first time point in this interval at which I(t, θ) = if such a point exists; otherwise, let Z j (θ) =R j (θ). 12

14 Observation 3 Q 1 (θ) <R 1 (θ) <Q 2 (θ) <R 2 (θ) <...<Q J(θ) (θ) <R J(θ) (θ). (4.1) Proof. See Observation 3 in [19]. 4.1 IPA Derivatives with Respect to the Base-Stock Level This section treats IPA erivatives (incluing sie ones) for the inventory time average, L I (T,θ), an the lost-sales time average, L ζ (T,θ), both with respect to the base-stock level, S, an exhibits their formulas for any initial inventory state. We first prove a number of useful lemmas that simplify the proofs of the main results later in this section. We then procee to obtain the IPA erivatives for L I (T,θ) by first obtaining those for the inventory process, {I(t, θ)}, following which we obtain the IPA erivatives for L ζ (T,θ). Finally, we establish the unbiaseness of all the IPA erivatives above. Assumption 3 (a) S(θ) =θ, whereθ Θ. (b) The processes {α(t)} an {µ(t)} are inepenent of the parameter θ. The following lemma provies basic properties for the inventory level process. Lemma 1 (a) For every j 1, (b) For every j 1, I(t, θ) =1, θ t I(t, θ) =, θ t [R j(θ),z j+1 (θ)). (Z j(θ),r j (θ)). Proof. To prove part (a), note that each R j (θ), j 1 is locally ifferentiable with respect to θ by part (c) of Assumption 1. By Observation 3, R j (θ) <Q j+1 (θ), where Q j+1 (θ) isajumppointof {α(t) µ(t)}, an therefore Q j+1 (θ) is locally inepenent of θ. Consequently, I(t, θ) =S(θ) for t (R j (θ),q j+1 (θ)] an t I(t, θ) =S(θ)+ [µ(t) α(t)] t, Q j+1 (θ) t (Q j+1 (θ),z j+1 (θ)). Part (a) now follows by ifferentiating I(t, θ) with respect to θ for t (R j (θ),z j+1 (θ)]. To prove part (b), consier first the extremal time interval [Z j (θ), Z j (θ)] in which I(t, θ) =. By part (c) of Assumption 1, Z j (θ) is locally ifferentiable with respect to θ, Z j (θ) isajumppoint of {α(t) µ(t)} an Z j (θ) < Z j (θ). Therefore, Zj (θ) is locally inepenent of θ an I(t, θ) = is locally inepenent of θ for t (Z j (θ), Z j (θ)]. Finally, note that by Eq. (3.1), I(t, θ) is t + inepenent of θ for t ( Z j (θ),r j (θ)). The proof is now complete. The following lemma provies basic properties for the time average of lost sales. 13

15 Lemma 2 Let u T be a time point, inepenent of θ. (a) For every j =1,...,J(θ), on the event {Z j (θ) =R j (θ)}, u ζ(t, θ) t =, for Q j (θ) <u R j (θ) (4.2) θ Q j (θ) (b) For every j =2,...,J(θ), on the event {Z j (θ) <R j (θ)}, u ζ(t, θ) t = θ Q j (θ) {, for Qj (θ) <u<z j (θ) 1, for Z j (θ) <u R j (θ) (4.3) (c) For every j =2,...,J(θ) an for u = Z j (θ), on the event {Z j (θ) <R j (θ)}, u θ + ζ(t, θ) t = (4.4) Q j (θ) u θ ζ(t, θ) t = 1 (4.5) Q j (θ) Proof. To prove part (a), we show that the integral in Eq. (4.2) is locally inepenent of θ. To see that, observe that {Z j (θ) =R j (θ)} = {I(t, θ) >,t [(Q j (θ),r j (θ)]} {ζ(t, θ) =,t [(Q j (θ),r j (θ)]} an each I(t, θ) is continuous in θ by Proposition 1. The result now follows for this part since the integral clearly vanishes. To prove part (b) for Q j (θ) <u<z j (θ), the proof of part (a) is applicable. To prove part (b) for Z j (θ) <u R j (θ), note that Eq. (2.6) implies u Q j (θ) ζ(t, θ) t = u Z j (θ) [α(t) µ(t)] 1 {I(t,θ)=,α(t)>µ(t)} t on {Z j (θ) <R j (θ)}. (4.6) By part (c) of Assumption 1, Z j (θ) is locally ifferentiable with respect to θ. Furthermore, from part (b) of Lemma 1, we conclue that {I(t, θ)} is locally inepenent of θ for t (Z j (θ),r j (θ)]. It follows from Leibnitz s rule that ifferentiating Eq. (4.6) with respect to θ yiels u ζ(t, θ) t = [α(z j (θ)) µ(z j (θ))] θ Z j (θ) θ Z j(θ). (4.7) To compute the right-han sie of Eq. (4.7), note first that the proof of part (a) of Lemma 1, implies that Q j (θ) is locally inepenent of θ for j 2. Since Zj (θ) [α(t) µ(t)] t = I(Q j (θ)) I(Z j (θ)) = S(θ), Q j (θ) ifferentiating this equation with respect to θ yiels [α(z j (θ)) µ(z j (θ))] θ Z j(θ) =1. The result now follows by substituting the above into Eq. (4.7). Part (c) follows from Eq. (4.3), by noting that u = Z j (θ) satisfiesz j (θ θ) <u<z j (θ θ). Remark. The event {Z j (θ) =u} often has probability, so the brief proof of part (c) above is inclue just for completeness. 14

16 Lemma 3 (a) For every j =1,...,J(θ), Rj (θ) ζ(t, θ) t =, θ Q j (θ) on {Z j (θ) =R j (θ)}. (4.8) (b) For every j =2,...,J(θ), Rj (θ) ζ(t, θ) t = 1 θ Q j (θ) on {Z j (θ) <R j (θ)}. (4.9) Proof. Part (a) follows by an argument similar to that in the proof of part (a) in Lemma 2. To prove part (b), note that by part (c) of Assumption 1, both Z j (θ) anr j (θ) is locally ifferentiable with respect to θ. Combining these facts with ζ(r j (θ),θ) =, it follows from Leibnitz s rule that Rj (θ) Q j (θ) ζ(t, θ) t = Rj (θ) Z j (θ) [α(t) µ(t)] 1 {I(t,θ)=,α(t)>µ(t)} t on {Z j (θ) <R j (θ)}. (4.1) The rest of the proof is similar to that of part (b) in Lemma 2. We first erive the IPA erivatives for the inventory process {I(t, θ)}. In the next two lemmas we make use of the hitting time, T S (θ), efine by { min{t [, ] : I(t, θ) =S(θ)}, if the minimum exists T S (θ) =, otherwise (4.11) Lemma 4 Consier an MTS system with the lost sales rule on the event {I() <S(θ)} (that is, the system starts in normal operation with partial inventory). Then, for any t an θ Θ, (a) On the event A(θ) ={I() <S(θ)} {t<t S (θ)}, I(t, θ) =. θ (b) On the events B j (θ) ={I() <S(θ)} {R j (θ) <t<z j+1 (θ)}, j 1, I(t, θ) =1. θ (c) On the events C j (θ) ={I() <S(θ)} {Z j (θ) <t<r j (θ)}, j 2, Proof. By Observation 3, I(t, θ) =. θ =Q 1 (θ) <T S (θ) = R 1 (θ) <Q 2 (θ) on {I() <S(θ)}, (4.12) an this hols for all cases of this lemma. 15

17 To prove part (a), note that by Eq. (3.1) an Assumption 2, the time erivative I(t, θ) is t + locally inepenent of θ on A(θ). Consequently, I(t, θ) =I() + I(τ,θ) τ τ + is inepenent of θ on A(θ), which proves part (a). t on A(θ) Finally, part (b) follows immeiately from part (a) of Lemma 1, while part (c) follows immeiately from part (b) of Lemma 1. Lemma 5 Consier an MTS system with the lost-sales rule on the event {I() >S(θ)} (that is, the system starts in overage operation). Then, for any t an θ Θ, (a) On the event A(θ) ={I() >S(θ)} {t<t S (θ)}, I(t, θ) =. θ (b) On either of the events B 1 (θ) ={I() >S(θ)} {T S (θ) <Q 1 (θ)} {T S (θ) <t<z 1 (θ)} or B 2,j (θ) ={I() >S(θ)} {T S (θ) <Q 1 (θ)} {R j (θ) <t<z j+1 (θ)}, j 1 or B 3,j (θ) ={I() > S(θ)} {T S (θ) =Q 1 (θ)} {R j (θ) <t<z j+1 (θ)}, j 1, I(t, θ) =1. θ (c) On the events C j (θ) ={I() >S(θ)} {Z j <t<r j (θ)}, j 1, I(t, θ) =. θ () On the event D(θ) ={I() >S(θ)} {T S (θ) =Q 1 (θ)} {T S (θ) <t<z 1 (θ)}, θ I(t, θ) = µ(q 1(θ)) α(q 1 (θ)). Proof. To prove part (a), note that by Eq. (3.1) that t I(t, θ) =I() α(τ) τ is inepenent of θ on A(θ), whence the result follows. To prove part (b) on the event B 1 (θ), note that by Observation 3, I(t, θ)= α(t) ona(θ). Therefore, t + T S (θ) <Q 1 (θ) <R 1 (θ) on B 1 (θ). (4.13) Clearly, T S (θ) is locally ifferentiable with respect to θ. In view of Eq. (4.13), Q 1 (θ) correspons to a jump in {α(t) µ(t)}, an part (b) on the event B 1 (θ) follows by a proof similar to that of part (a) in Lemma 1. Part (b) on the events B 2,j an B 3,j (θ), j 1 follows immeiately from part (a) of Lemma 1. Part (c) follows immeiately from part (b) of Lemma 1. To prove part (), note that by Observation 3, T S (θ) = Q 1 (θ) <R 1 (θ) <Q 2 (θ) on D(θ). (4.14) 16

18 Furthermore, Q1 (θ) α(τ) τ = I() S(θ) =I() θ, Differentiating the above equation with respect to θ yiels, θ Q 1(θ) = on D(θ). 1, on D(θ). (4.15) α(q 1 (θ)) t next, write I(t, θ) = S(θ) + [µ(τ) α(τ)] τ on the event D(θ), an then ifferentiate it with Q 1 (θ) respect to θ, yieling the esire result θ I(t, θ) =1 [µ(q 1(θ)) α(q 1 (θ))] θ Q 1(θ) = µ(q 1(θ)) α(q 1 (θ)) on D(θ), (4.16) where the secon equality is obtaine by substituting Eq. (4.15), an noting the inequalities α(q 1 (θ)) >µ(q 1 (θ)) on the event {I() >S(θ)} {Q 1 (θ) =T S (θ)}. On the event {I() = S(θ)}, the situation is more complex, because the left an right erivatives of I(t, θ) with respect to θ o not coincie an must be compute separately. This event cannot be exclue because it may happen in applications where inventory levels are iscrete. We first erive the right-erivatives, I(t, θ) by borrowing from Lemma 4 an making use θ + of the hitting time T µ (θ), given by min{t [,Q 1 (θ)) : µ(t) >α(t)}, if the minimum exists on the event {Q 1 (θ) > } R T µ (θ) = 1 (θ), if R 1 (θ) exists on the event {Q 1 (θ) =} [{Q 1 (θ) > } {α(t) =µ(t), t [,Q 1 (θ))}], otherwise (4.17) In wors, T µ (θ) is a hitting time of {I(t, θ)}, which correspons to the first time that the inventory level changes in any perturbe process, {I(t, θ + θ)} for any θ >. Lemma 6 Consier an MTS system with the lost-sales rule on the event {I() = S(θ)} (that is, the system starts in normal operation with full inventory). Then, for any t an θ Θ, (a) On the event A(θ) ={I() = S(θ)} {t<t µ (θ)}, I(t, θ) =. θ + (b) On the events B 1 (θ) ={I() = S(θ)} {T µ (θ) <R 1 (θ)} {T µ (θ) <t<z 1 (θ)} or B 2,j (θ) = {I() = S(θ)} {T µ (θ) <R 1 (θ)} {R j (θ) <t<z j+1 (θ)}, j 1 or B 3,j (θ) ={I() = S(θ)} {T µ (θ) = R 1 (θ)} {R j (θ) <t<z j+1 (θ)}, j 1, I(t, θ) =1. θ + (c) On the events C 1,j (θ) ={I() = S(θ)} {T µ (θ) =R 1 (θ)} {Z j (θ) <t<r j (θ)}, j 2 or C 2,j (θ) ={I() = S(θ)} {T µ (θ) <R 1 (θ)} {Z j (θ) <t<r j (θ)}, j 1, I(t, θ) =. θ + 17

19 Proof. Consier a perturbe system with S(θ + θ) = θ + θ, where θ >. Since I() = S(θ) <S(θ + θ), it follows that the perturbe system starts in normal operation. Denote S = S(θ + θ) S(θ). By Observation 3, = Q 1 (θ + θ) T µ (θ) <R 1 (θ + θ) <Q 2 (θ + θ) on {I() = S(θ)}, (4.18) an this hols for all cases of this lemma. To prove part (a), note first that the event {T µ (θ) =} can be preclue, since it implies A(θ) =. Otherwise, the efinition of T µ (θ) an Eq. (3.1) imply that I(t, θ)= I(t, θ + θ) t + t + on A(θ). By Assumption 2, we conclue that I(t, θ + θ) =I(t, θ) are inepenent of θ on A(θ), an therefore, part (a) follows immeiately. To prove part (b), observe that part (b) of Assumption 1 implies that there exists ɛ>, such that for any θ ɛ, R 1 (θ + θ) = T µ (θ)+ S µ(t µ (θ)) α(t µ (θ)) on {I() = S(θ)}, (4.19) where the inequality µ(t µ (θ)) α(t µ (θ)) > follows from the efinition of T µ (θ). We now procee with the proof on the event B 1 (θ), by consiering two cases. Case 1: On the event B 1 (θ) {Z 1 (θ) <R 1 (θ)}, it follows from the efinition of T µ (θ) an part (c) of Assumption 1 that T µ (θ) <Q 1 (θ) anq 1 (θ) isajumppointof{α(θ) µ(θ)}. Therefore, Q 2 (θ + θ) =Q 1 (θ) for sufficiently small θ. Note that in this case, I(t, θ + θ) =I(t, θ) on { <t<t µ (θ)}, an then it increases to S(θ + θ) an stays there until Q 2 (θ + θ). Furthermore, over the interval [R 1 (θ + θ),z 1 (θ)] an for sufficiently small θ, both the original system an the perturbe system operate in normal moe an are riven by ientical ynamics. Consequently, the ifference process {I(t, θ + θ) I(t, θ)} is constant an equals θ, over that interval. By part (c) of Assumption 1, we can choose sufficiently small θ, such that Z 2 (θ + θ) = Z 1 (θ)+ S α(z 1 (θ)) µ(z 1 (θ)) on B 1 {Z1 (θ) <R 1 (θ)}, (4.2) where the inequality α(z 1 (θ)) µ(z 1 (θ)) > follows from the efinition of Z 1 (θ). But because in this case, R 1 (θ + θ) T µ (θ) anz 2 (θ + θ) Z 1 (θ) as θ by Eqs.(4.19) an (4.2), we conclue that θ + I(t, θ) = 1 on the event B 1(θ) {Z 1 (θ) <R 1 (θ)}. Case 2: on the event B 1 (θ) {Z 1 (θ) =R 1 (θ)} the proof is similar, except that Z 2 (θ + θ) nee not be consiere. Finally, Part (b) on events B 2,j (θ) anb 3,j (θ), j 1 follows immeiately from part (a) of Lemma 1, an part (c) follows immeiately from part (b) of Lemma 1. We next erive the left-erivatives, I(t, θ), by borrowing from Lemma 5, an making use θ of the hitting time T α,givenby T α = { min{t [,T]: α(t) > }, if the minimum exists, otherwise (4.21) Note that T α is inepenent of θ. 18

20 Lemma 7 Consier an MTS system with the lost-sales rule on the event {I() = S(θ)} (that is, the system starts in normal operation with full inventory). Then, for any t an θ Θ, (a) On the event A(θ) ={I() = S(θ)} {t<t α }, I(t, θ) =. θ (b) On either of the events B 1 (θ) ={I() = S(θ)} {T α <Q 1 (θ)} {T α <t<z 1 (θ)} or B 2,j (θ) = {I() = S(θ)} {T α <Q 1 (θ)} {R j (θ) <t<z j+1 (θ)}, j 1 or B 3,j (θ) ={I() = S(θ)} {T α = Q 1 (θ)} {R j (θ) <t<z j+1 (θ)}, j 1, I(t, θ) =1. θ (c) On the events C j (θ) ={I() = S(θ)} {Z j (θ) <t<r j (θ)}, j 1, I(t, θ) =. θ () On the event D(θ) ={I() = S(θ)} {T α = Q 1 (θ)} {Q 1 (θ) <t<z 1 (θ)}, θ I(t, θ) = µ(t α) α(t α ). Proof. Consier a perturbe system with S(θ θ) =θ θ, where θ>. Since I() = S(θ) > S(θ θ) by assumption, it follows that the perturbe system starts in overage operation. Denote S = S(θ) S(θ θ). To prove part (a), note first that the event {T α =} can be preclue, since it implies A(θ) =. Otherwise, on the event A(θ), the perturbe system is in overage operation with no eman arrivals, so that I(t, θ θ) =I() = I(t, θ) on the event A(θ), an the result follows immeiately. To prove part (b), observe that part (b) of Assumption 1 implies that there exists ɛ>, such that for any θ ɛ, T S (θ θ) = T α + S α(t α ) on {I() = S(θ)}, (4.22) where the inequality α(t α ) > follows from the efinition of T α. Note that Observation 3 implies T α <T S (θ θ) <Q 1 (θ) <R 1 (θ) on B 1 (θ). (4.23) We now procee with the proof on the event B 1 (θ), by consiering two cases. Case b.1: On the event B 1 (θ) {Z 1 (θ) <R 1 (θ)}, Eq. (4.23) implies that Q 1 (θ) isajumppoint of {α(t) µ(t)}. It follows by a proof similar to that of part (a) in Lemma 1 that I(t, θ θ) = I(t, θ) θ on the event {I() = S(θ)} {T α <Q 1 (θ)} {T S (θ θ) <t<z 1 (θ θ)}, where Z 1 (θ θ) = Z 1 (θ) S α(z 1 (θ)) µ(z 1 (θ)) on B 1 (θ) {Z 1 (θ) <R 1 (θ)}, (4.24) an the inequality α(p 1 (θ)) µ(z 1 (θ)) > follows from the efinition of Z 1 (θ). To see that, observe that {I(t, θ θ)} starts in overage moe an hits its base-stock level, S(θ θ) attime T S (θ θ), an then stays there until time Q 1 (θ θ) =Q 1 (θ). But because T S (θ θ) T α 19

21 an Z 1 (θ θ) Z 1 (θ) onb 1 (θ) {Z 1 (θ) <R 1 (θ)} as θ by Eqs.(4.22) an (4.24), we conclue that I(t, θ) =1onthisevent. θ Case b.2: On the event B 1 (θ) {Z 1 (θ) =R 1 (θ)}, the proof is similar, except that except that Z 1 (θ θ) nee not be consiere. Part (b) on the remaining events, B 2,j an B 3,j, j 1, follows immeiately from part (a) of Lemma 1, while part (c) follows immeiately from part (b) of Lemma 1. Finally, we prove part () by consiering two separate cases. Here, the process {I(t, θ)} stays at S(θ) untiltimet α, at which point the arrival rate jumps, such that α(t α ) >µ(t α ). Case.1: Consier the event D(θ) {Z 1 (θ) <R 1 (θ)}. Then, Q 1 (θ) =T α <T S (θ θ) = Q 1 (θ θ) <R 1 (θ) on D(θ), (4.25) where the first inequality is a consequence of Eq. (4.22), an both the secon equality an secon inequality follow from part (b) of Assumption 1. It follows that I(T S (θ θ),θ) = S(θ) S α(t α ) [α(t α) µ(t α )] on D(θ). (4.26) But over the interval [T S (θ θ),z 1 (θ θ)], both the original system an the perturbe system operate in normal moe an are riven by ientical ynamics, given by Eq. (2.5). Therefore, over this interval, I(t, θ) I(t, θ θ) = Sµ(T α) α(t α ) on D(θ) {Z 1 (θ) <R 1 (θ)} (4.27) is constant, an by part (c) of Assumption 1, we can choose sufficiently small θ, such that Z 1 (θ θ) = Z 1 (θ) Sµ(T α ) α(t α )[µ(z 1 (θ)) α(z 1 (θ))] on D(θ) {Z 1 (θ) <R 1 (θ)}. (4.28) Next, sen S = θ in Eqs. (4.22) an (4.28), yieling respectively, T S (θ θ) T α an Z 1 (θ θ) Z 1 (θ). The requisite result on the event D(θ) {Z 1 (θ) <R 1 (θ)} now follows from Eq. (4.27). Case.2: Consier the event D(θ) {Z 1 (θ) =R 1 (θ)}. The proof for this case is ientical to the proof of part () in Lemma 6 of Zhao an Melame (24). In the next proof we shall make use of horizon-epenent ranom inices, J S (T,θ), which constitute restrictions of J(θ) to finite time horizons [,T] as follows, J S (T,θ)= { max{j 1: Rj (θ) T }, if it exists, otherwise. (4.29) We are now in a position to erive the IPA erivatives for the inventory time average, L I (T,θ). Theorem 1 W.p.1, the IPA erivatives of the inventory time average with respect to the base-stock level are given for all T> an θ Θ as follows: (a) On the event {I() <S(θ)}, θ L I(T,θ)= J S (T,θ) j=1 [min{z j+1 (θ),t} R j (θ)]. (4.3) 2