IPA Derivatives for Make-to-Stock Production-Inventory Systems with Backorders

Transcription

1 Methool Comput Appl Probab (2006) 8: DOI /s IPA Derivatives for Make-to-Stock Prouction-Inventory Systems with Backorers Yao Zhao & Benjamin Melame Receive: 11 August 2005 / Revise: 11 August 2005 / Accepte: 28 December 2005 # Springer Science + Business Meia, LLC 2006 Abstract A single-stage Make-to-Stock (MTS) prouction-inventory system consists of a prouction facility couple to an inventory facility, an is subject to a policy that aims to maintain a prescribe inventory level (calle base stock) by moulating prouction capacity. This paper consiers a class of single-stage, singleprouct MTS systems with backorers, riven by ranom eman an prouction capacity, an subject to a continuous-review base-stock policy. A moel from this class is formulate as a stochastic flui moel (SFM), where all flows are escribe by stochastic rate processes with piecewise< constant sample paths, subject to very mil regularity assumptions that merely preclue accumulation points of jumps with probability 1. Other than that, the MTS moel in SFM setting is nonparametric in that it assumes no specific form for the unerlying probability law, an as such is quite general. The paper procees to erive formulas for the (stochastic) IPA (Infinitesimal Perturbation Analysis) erivatives of the sample-path time averages of the inventory level an backorers level with respect to the base-stock level an a parameter of the prouction rate. These formulas are comprehensive in that they are exhibite for any initial conition of the system, an inclue right an left erivatives (when they o not coincie). The erivatives erive are then shown to be unbiase an their formulas are seen to be amenable to fast computation. The generality of the moel an comprehensiveness of the IPA erivative formulas hol out the promise of graient-base applications. More specifically, since the basestock level an prouction rate are the key control parameters of MTS systems, the results provie the theoretical unerpinnings for optimizing the esign of MTS systems an for evising prospective on-line aaptive control algorithms that employ IPA erivatives. The paper conclues with a iscussion of those issues. Y. Zhao (*) : B. Melame Department of MSIS, Rutgers Business School Newark an New Brunswick, Rutgers University, 94 Rockafeller R., Piscataway, NJ 08854, USA yaozhao@anromea.rutgers.eu B. Melame melame@rbs.rutgers.eu

2 192 Methool Comput Appl Probab (2006) 8: Keywors Infinitesimal Perturbation Analysis. IPA. IPA erivatives. IPA graients. Make-to-Stock. Prouction-inventory systems. Stochastic flui moels. SFM 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. In simple MTS systems, a stage is comprise of a single-prouct replenishment flow between a pair of couple prouction-inventory facilities, such that prouction is moulate by inventory state information. An important instance of prouction-inventory 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 is concerne with MTS systems with backorers (see Section 2), while a forthcoming paper will treat MTS systems with lost sales. Economic consierations in supply chains call for effective control of inventory levels an prouction rates, in orer to optimize some prescribe performance metric. This motivates on-line algorithms that can aaptively control such systems over time with the objective of minimizing the 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 LðÞ be a ranom variable, parameterize by a generic real-value parameter chosen from a close an boune set Q. The IPA erivative of LðÞ with respect to is the ranom variable LðÞ, provie that it exists almost surely. An IPA erivative is sai to be unbiase, if the expectation an ifferentiation operators commute, namely, E½ LðÞŠ ¼ E½LðÞŠ; otherwise, it is sai to be biase. Sufficient conitions for unbiase IPA erivatives are given in the following result. FACT 1 (see Rubinstein an Shapiro (1993), Lemma A2, p. 70) An IPA erivative LðÞ is unbiase, if (a) For each 2 Q, the IPA erivatives LðÞ exist w.p.1 (with probability 1). (b) W.p.1, LðÞ is Lipschitz continuous in Q, an the (ranom) Lipschitz constant has a finite first moment. For IPA-base applications 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 particular, if a left-erivative oes not coincie with its right-erivative counterpart, then both shoul be exhibite.

3 Methool Comput Appl Probab (2006) 8: For statistical accuracy, they shoul be unbiase. 3. For generality, they shoul be nonparametric in the sense that they are solely computable 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. 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 the 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 et al. (1991) for a 2-stage prouction system which is governe by the 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 (2002) 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, 2002). 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 fluiflow 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 1 For simplicity we aress only open networks in this iscussion.

4 194 Methool Comput Appl Probab (2006) 8: 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 Babruptly among noes following a service time, while in the continuous queueing paraigm, transaction workloa moves Bgraually (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 Bmolecules 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 Blimiting 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 BDirac 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 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. (2002) erive IPA erivatives in SFM setting; we henceforth refer to this approach as IPA-over-SFM. The paper 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. (2004) treate multi-stage MTS prouction-inventory systems with backorers in SFM setting. Assuming that inventory at each stage is controlle by a continuous-time base-stock 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 optimal base-stock levels at each stage. Zhao an Melame (2004) applie the IPA-over-SFM approach to a class of single-prouct, single-stage MTS systems with backorers. Using a ifferent proof methoology from that of Paschaliis et al. (2004), this paper erive the IPA formulas of the time average inventory level an backorer with respect to

5 Methool Comput Appl Probab (2006) 8: the base-stock level, as well as a parameter of the prouction rate process. The goal of this paper is to erive IPA erivatives for Make-to-Stock systems with backorers, an to show them to be unbiase. The key contributions of this paper are two-fol. The first contribution is the erivation of IPA erivative formulas with respect to the base-stock level for all initial inventory states, incluing those that lie above the base-stock level. In contrast, the above-cite references consier only a subset of initial inventory states; for example, Zhao an Melame (2004) restricts such systems to start with a base-stock level of inventory, while Paschaliis et al. (2004) consiers initial inventory states that lie only below the base-stock level. In fact, we show in this paper that transient IPA erivatives epen strongly on the initial inventory state, an in some cases, only sie IPA erivatives exist. The importance of our contribution stems from potential applications of IPA erivatives to on-line control of MTS systems. Clearly, on-line control applications manate the computation of IPA erivatives for all initial inventory states, as well as all sie erivatives, 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. For example, consier the situation where an IPAbase control action sets the base-stock level to coincie with the current inventory level (this coul happen in applications where inventory levels are iscrete), in which case this paper shows that the sie erivatives exist but are not equal. These sie erivatives woul be neee in ue time to ecie on the next control action, where a base-stock level lowering action woul call for left IPA erivatives, while a base-stock level raising action woul call for right IPA erivatives; note that the inventory level just after each control action is consiere to be the new initial inventory state for the purpose of computing the new IPA erivatives. We point out that Zhao an Melame (2004) also consiers IPA erivatives from an initial inventory state that coincies with the base-stock level, but the initial inventory state an base-stock level are require there to vary together, which simplifies the analysis, but oes not amit on-line control applications. The secon contribution of this paper is the erivation of IPA erivative formulas with respect to a prouction-rate parameter, which moels the prouction capacity that replenishes the inventory system. Here, our results generalize Wari et al. (2002) an Zhao an Melame (2004), which only consier the case where the left an right IPA erivatives coincie. In contrast, this paper rops this restriction an erives all sie IPA erivative formulas, thereby extening the applicability of IPA-base on-line control. The computation of the general IPA erivatives in this paper requires major extensions of the results in the open literature, culminating in more elaborate formulas. We show that long-run IPA erivatives with respect to the base-stock level parameter are simpler, an in fact, coincie with publishe results for a subset of initial inventory states (e.g., Paschaliis et al. (2004), Zhao an Melame (2004)). However, as note above, IPA-base on-line control applications cannot rely on long-run IPA erivatives, but must generally utilize their transient counterparts. We mention that while this paper focuses on the IPA erivatives in MTS systems, our ultimate goal is to use erivative information for on-line control an optimization of supply chains, which will be the subject of further research.

6 196 Methool Comput Appl Probab (2006) 8: Throughout the paper, we use the following notational conventions an terminology. The inicator function of set A is enote by 1 A an x þ ¼ maxfx; 0g, whereas f ðxþþ an f ðx Þ enote the right an left limits of f at x an x f ðxþ þ an x f ðxþ enote, respectively, the right an left erivatives of f at x. A function f ðxþ 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. The rest of the paper is organize as follows. Section 2 presents the prouctioninventory moel 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 iscusses the results, their significance an their use in prospective esign an control applications. 2 The Make-to-Stock Moel 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 > 0. More specifically, the inventory an prouction facilities are couple, an have two operational moes as follows: Normal 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 also 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 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, so prouction is temporarily suspene until the inventory level reaches or crosses S from above, at which point normal operation is resume. We also refer to this operational moe as overage 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 backorer rule as follows: Any shortage of inventory is backorere from the prouction facility, an the eman waits in a FCFS buffer at the inventory facility until the prouction facility replenishes the inventory facility with the shortage amount. Thus, the system s overall actions aim to move the inventory level to the base-stock level, S.

7 Methool Comput Appl Probab (2006) 8: Mapping MTS Systems to SFM Versions We next procee to map the traitional iscrete MTS system with backorers into an SFM version, as epicte in Fig. 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, fiðtþg, where IðtÞ is the (flui) volume of inventory on-han at time t. Backorers level. The traitional jump process of the level of backorers at the inventory facility is mappe to a flui-level counterpart, fbðtþg, where BðtÞ is the (flui) volume of backorers 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, fxðtþg, where XðtÞ is the (flui) volume of outstaning orers at time t. Traffic-relate stochastic processes in Fig. 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, fðtþg, where ðtþ is the rate of incoming emans at time t. Prouction rate. The traitional service (prouction) process of iscrete prouct at the prouction facility is mappe to a flui-flow stochastic process, fðtþg, where ðtþ is the prouction rate 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 fluiflow stochastic process, fðtþg, where ðtþ is the rate of incoming outstaning orers at time t. Replenishment rate. The traitional replenishment process of iscrete replenishe prouct from the prouction facility to the inventory facility is mappe to a flui-flow stochastic process, fðtþg, where ðtþ is the replenishment rate of prouct at time t. Fig. 1 The Make-to-Stock prouction-inventory system with backorers

8 198 Methool Comput Appl Probab (2006) 8: We now procee to exhibit the formal efinitions of all flui-moel components of the MTS system with backorers. During overage operation, the inventory process is governe by the one-sie stochastic ifferential equation an IðtÞ ¼ ðtþ; tþ ð2:1þ BðtÞ¼0; XðtÞ¼0 ð2:2þ 1ðtÞ¼0; ðtþ¼0: ð2:3þ During normal operation, the moel satisfies the conservation relation, XðtÞþIðtÞ BðtÞ ¼S; ð2:4þ where IðtÞ ¼½S XðtÞŠ þ ; ð2:5þ BðtÞ ¼½XðtÞ SŠ þ ; ð2:6þ an the outstaning orers process is governe by the sie stochastic ifferential equation, 0; if XðtÞ ¼0 an ðtþ ðtþ; XðtÞ ¼ ð2:7þ tþ ðtþ ðtþ; otherwise; The arrival-rate process of outstaning orers is given by 0; if IðtÞ > S ðtþ ¼ ðtþ; if IðtÞ S: an the replenishment-rate process is given by ðtþ; if XðtÞ > 0 ðtþ ¼ minfðtþ;ðtþg; if XðtÞ ¼0: ð2:8þ ð2:9þ 2.2 Performance Metrics an Parameters Let ½0; TŠ be a given finite time interval, uring which system performances are evaluate before a control action regaring the inventory policy an/or prouction rate is taken. One 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 metrics.

9 Methool Comput Appl Probab (2006) 8: Inventory time average. The time average of flui volume of inventory on-han over the interval ½0; TŠ, given by L I ðtþ ¼ 1 IðtÞ t: ð2:10þ T 0 Backorer time average. The time average of flui volume of backorers over the interval ½0; TŠ, given by L B ðtþ ¼ 1 T Z T Z T 0 BðtÞ t: ð2:11þ Observe that the metrics L I ðtþ an L B ðtþ are ranom variables for each T. Let 2 Q enote a generic parameter of interest with a close an boune omain Q. We write SðÞ, ðt;þ, L I ðt;þ; L B ðt;þ 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 L IðT;Þ an L BðT;Þ 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, SðÞ ¼; 2 Q: ð2:12þ Prouction rate parameter. A parameter of the prouction rate process, such that ðt;þ¼1; t 2½0; TŠ; 2 Q; ð2:13þ interprete as an aitive 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. (2002), we efine two types of sample path events: Exogenous events. An exogenous event occurs whenever a jump occurs in the sample path of fðtþg or fðtþg. Enogenous events. An enogenous event occurs whenever a time interval is inaugurate, in which XðtÞ ¼0orXðtÞ ¼S. Throughout this paper, we assume the following mil regularity conitions [cf. Wari et al. (2002)]. ASSUMPTION 1 (a) The eman rate process, fðtþg, an the prouction rate process, fðtþg, have right-continuous sample paths that are piecewise-constant w.p.1.

10 200 Methool Comput Appl Probab (2006) 8: (b) (c) Each of the processes, fðtþg an fðtþg, 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. No multiple events occur simultaneously w.p.1. While parts (a) an (c) above are mil regularity assumptions, part (b) merits aitional motivation as follows. It makes sense to moel the eman arrival-rate process, fðtþg, as exogenous to the system, an as such we assume it to be inepenent of any parameter of interest, an this is true in particular of its iscontinuity points. The prouction-rate process, fðtþg, may epen on a scaling parameter, (see equation (2.13)), but its iscontinuity points are assume inepenent of. Note that such iscontinuities moel a change in prouction capacity which o not epen on scaling of the prouction rate. The following observations follow from Assumption 1. OSERVATION 1 1. W.p.1, there exists a finite integer N 0 an a sequence of (ranom) time points 0 ¼ T 0 < T 1 < < T N < T Nþ1 ¼ T, such that the process fðtþ ðtþg is constant over each interval ðt n ; T nþ1 Þ, n ¼ 0; ; N, an each time point T n, 1 n N, is a jump point of the process. 2. The process fðtþ ðtþg is constant over each time interval ðt n ; T nþ1 Þ,n¼ 0; ; N. Proof: To prove the first observation, note that the finiteness of N follows from part (b) of Assumption 1, while the strict inequalities are a consequence of part (c) of Assumption 1. The secon observation follows irectly from the first one. Finally, we shall be intereste in pairs of systems, the original system (inexe by ) an a perturbe system (inexe by D), 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 an initial backorer level o not epen on, namely, Ið0;Þ¼Ið0Þ an Bð0;Þ¼Bð0Þ, for all 2 Q. 3 Variational Bouns In this section, we erive variational bouns for various parameterize stochastic processes an performance metrics in the MTS moel with backorers. These results will be use in subsequent sections to simplify the erivation of IPA erivatives an to establish their unbiaseness. OSERVATION 2 For an MTS system with the backorer rule, the stochastic ifferential equation (2.7) governing the outstaning orer process fxðtþg in normal operation is a special case of the SFM queue in Wari an Melame (2001), where the buffer has unlimite capacity.

11 Methool Comput Appl Probab (2006) 8: Proof: Follows from the fact that we can ientify the eman arrival rate process an prouction rate process, respectively, with the inflow rate process an service rate process in an infinite-capacity SFM queue from Wari an Melame (2001). For notational convenience, we efine two auxiliary processes. The extene inventory process, fwðtþg, is efine by WðtÞ ¼IðtÞ BðtÞ ¼ IðtÞ; if IðtÞ > 0 BðtÞ; if IðtÞ ¼0: ð3:1þ Thus, WðtÞ etermines both IðtÞ an BðtÞ (an vice versa). The extene outstaning orers process, fyðtþg, is efine by YðtÞ ¼ S IðtÞ; if IðtÞ > S (overage operation) XðtÞ; if IðtÞ S (normal operation): ð3:2þ Observe that YðtÞ is negative uring overage operation an non-negative uring normal operation, an its time erivative satisfies 8 YðtÞ < ðtþ; if YðtÞ < 0 t þ ¼ 0; if YðtÞ ¼0 an ðtþ ðtþ ð3:3þ : ðtþ ðtþ; otherwise. Furthermore, equation (2.4) implies the conservation relation WðtÞ þ YðtÞ ¼ S; t 0; ð3:4þ vali for each operational moe (overage an normal). The variational bouns will be shown to hol with respect to control parameters of interest at each time point, starting from an arbitrary Wð0Þ. In the remainer of the paper the tile symbol will always inicate a realization of a ranom variable or a stochastic process. Accoringly, let f ~ yðt;;yþg enote a sample path realization of fyðt;þg that starts at initial conition y, so ~ yð0;;yþ ¼y. Furthermore, when we write f ~ yðt; 1 ; y 1 Þg an f ~ yðt; 2 ; y 2 Þg, we mean two sample path realizations for which the following conitions hol. 1. f ~ yðt; 1 ; y 1 Þg an f ~ yðt; 2 ; y 2 Þg are riven by the same realization, f ~ ðtþg, of the process fðtþg. 2. If the process fðtþg oes not epen on, then f ~ yðt; 1 ; y 1 Þg an f ~ yðt; 2 ; y 2 Þg are riven by the same realization, f ~ ðtþg, of the process fðtþg. Otherwise, f ~ yðt; 1 ; y 1 Þg an f ~ yðt; 2 ; y 2 Þg are riven by corresponing realizations, f ~ ðt; 1 Þg an f ~ ðt; 2 Þg of the process fðtþg, relate by ~ ðt; 1 Þ ~ ðt; 2 Þ¼ 1 2,in accorance with equation (2.13). Intuitively, the two realizations have ifferent IPA parameters an start from ifferent initial states, Yð0Þ, but are otherwise riven by the same Branomness in arrivals an prouction. In a similar vein, let f ~ {ðt;;yþg an f ~ bðt;;yþg enote the realizations of the processes fiðt;þg an fbðt;þg, respectively, both associate with f ~ yðt;;yþg.

12 202 Methool Comput Appl Probab (2006) 8: For each realization f ~ yðt;;yþg of an MTS with the backorer rule, equation (3.3) inuces a partition of the interval ½0; TŠ, Rð; yþ ¼fR 1 ð; yþ; R 2 ð; yþ; R 3 ð; yþg; ð3:5þ where each region, R k ð; yþ, 1 k 3, is efine by the conition in the corresponing k-th line of equation (3.3), namely, R 1 ð; yþ ¼ ft 2½0; TŠ : ~ yðt;;yþ 0g; R 2 ð; yþ ¼ ft 2½0; TŠ : ~ yðt;;yþ ¼0 an ~ ðtþ ~ ðtþg; R 3 ð; yþ ¼ ft 2½0; TŠ : ½ ~ yðt;;yþ ¼0 an ~ ðtþ > ~ ðtþš or ~ yðt;;yþ > 0g: 3.1 Variational Bouns With Respect to the Base-Stock Level In this section, the IPA parameter of interest is SðÞ ¼ for 2 Q, so the initial state of the extene outstaning orers process is Yð0;Þ¼SðÞ Wð0Þ. LEMMA 1 For an MTS system uner the backorer rule, let y 1 y 2. Then for each 2 Q, 0 ~ yðt;;y 2 Þ ~ yðt;;y 1 Þy 2 y 1 ; t 2½0; TŠ: ð3:6þ Proof: We show that for each, the ifference realization, f ~ yðt;;y 2 Þ ~ yðt;;y 1 Þg, is non-increasing in t from the initial value of y 2 y 1 0, without changing sign. We first prove the lefthan inequality of (3.6). By assumption, ~ yð0;;y2 Þ ~ yð0;;y 1 Þ¼y 2 y 1 0: ð3:7þ Observe that if ~ yðt ;;y 2 Þ¼ ~ yðt ;;y 1 Þ for some time point t, then ~ yðt;;y 2 Þ¼ ~ yðt;;y1 Þ for all t t. To see that, note that equation (3.3) implies that once the realizations synchronize, they remain synchronize thereafter. Finally, since the ifference realization is continuous, it cannot change signs. We next prove the righthan inequality of (3.6) by noting that in view of equation (3.7), it suffices to show that the erivative of the ifference realization satisfies t ½~ þ yðt;;y2 Þ ~ yðt;;y 1 ÞŠ 0 for all t 0. To this en, we examine the behavior of ~ t þ yðt;;y1 Þ an ~ t þ yðt;;y2 Þ in the three regions of the partition (3.5). Informally, the proof computes t ½~ þ yðt;;y2 Þ ~ yðt;;y 1 ÞŠ for all pairs of regions in the partitions associate with each initial state, such that ~ yðt;;y 1 Þ is in one region an ~ yðt;;y 2 Þ is in the other. More formally, the computation covers t in all intersections of the form R i ð; y 1 Þ T R j ð; y 2 Þ,1i; j 3. However, the following two observations reuce substantially the number of region-pair cases to be checke. First, there is no nee to check for pairs of regions with the same subscript, i ¼ j, since in their intersection ½~ yðt;;y2 Þ ~ yðt;;y 1 ÞŠ ¼ 0, trivially. Secon, in view of the lefthan sie of (3.6), it t þ suffices to consier only region pairs in which ~ yðt;;y 1 Þ < ~ yðt;;y 2 Þ, since obviously, ½~ yðt;;y2 Þ ~ yðt;;y 1 ÞŠ ¼ 0when ~ yðt;;y 2 Þ¼ ~ yðt;;y 1 Þ. t þ

13 Methool Comput Appl Probab (2006) 8: Consequently, we nee only check the following list of cases. Case 1: t 2R 1 ð; y 1 Þ T R 2 ð; y 2 Þ. In this case, t ½~ þ yðt;;y2 Þ ~ yðt;;y 1 ÞŠ ¼ ðtþ ~ 0. Case 2: t 2R 1 ð; y 1 Þ T R 3 ð; y 2 Þ. In this case, t ½~ þ yðt;;y2 Þ ~ yðt;;y 1 ÞŠ ¼ ðtþ ~ 0. Case 3: t 2R 2 ð; y 1 Þ T R 3 ð; y 2 Þ. In this case, t ½~ þ yðt;;y2 Þ ~ yðt;;y 1 ÞŠ ¼ ðtþ ~ ðtþ ~ 0, where the inequality follows from the efinition of the intersection of R 2 ð; y 1 Þ an R 3 ð; y 2 Þ. The proof is complete. We next erive a variational boun for an MTS system uner the backorer rule, starting from an arbitrary value, w 0, of the initial extene inventory, Wð0Þ. PROPOSITION 1 For an MTS system with the backorer rule, let y 1 ¼ Sð 1 Þ w 0 an y 2 ¼ Sð 2 Þ w 0, where 1 ; 2 2 Q, an w 0 is arbitrary. Then maxfj ~ yðt; 1 ; y 1 Þ ~ yðt;2 ; y 2 Þj:t 2½0; TŠgj 1 2 j. Proof: Without loss of generality, assume Sð 1 Þ¼ 1 2 ¼ Sð 2 Þ, from which it follows that 0 y 2 y 1 ¼ 2 1. Applying Lemma 1 to ~ yðt; 1 ; y 1 Þ an ~ yðt; 1 ; y 2 Þ yiels 0 ~ yðt; 1 ; y 2 Þ ~ yðt; 1 ; y 1 Þ 2 1 t 2½0; TŠ: ð3:8þ The proposition follows immeiately from (3.8), by realizing that the realization f ~ yðt; 1 ; y 2 Þg is ientical to f ~ yðt; 2 ; y 2 Þg, since they start with the same initial state, an their ynamics are inepenent of by equation (3.3). COROLLARY 1 For an MTS system with the backorer rule, let y 1 ¼ Sð 1 Þ w 0 an y 2 ¼ Sð 2 Þ w 0, for any w 0 an 1 ; 2 2 Q. Then, j ~ { ðt; 1 ; y 1 Þ ~ {ðt; 2 ; y 2 Þj 2 j 1 2 j; t 2½0; TŠ ð3:9þ j ~ bðt; 1 ; y 1 Þ ~ bðt; 2 ; y 2 Þj 2 j 1 2 j; t 2½0; TŠ: ð3:10þ Proof: Equations (3.1) an (3.4) an Proposition 1 imply that j ~ {ðt; 1 ; y 1 Þ ~ {ðt; 2 ; y 2 Þj ¼ j½sð 1 Þ ~ yðt; 1 ; y 1 ÞŠ þ ½Sð 2 Þ ~ yðt; 2 ; y 2 ÞŠ þ j2 j 1 2 j: A similar proof applies to the backorer process. 3.2 Variational Bouns With Respect to the Prouction Rate Parameter In this section, the IPA parameter of interest is a parameter,, of the prouction rate process, fðt;þg, satisfying equation (2.13).

14 204 Methool Comput Appl Probab (2006) 8: PROPOSITION 2 Let 1 ; 2 2 Q, an assume that Yð0; 1 Þ¼Yð0; 2 Þ. Then, maxfjyðt; 1 Þ Yðt; 2 Þj : t 2½0; TŠg T j 1 2 j: Proof: In view of the fact that fyðt; 1 Þg an fyðt; 2 Þg coincie uring overage operation, it suffices to consier the case Yð0; 1 Þ¼Yð0; 2 Þ0. The proposition follows immeiately from Proposition 3.2 of Wari an Melame (2001), because the proof there is inepenent of the initial state. COROLLARY 2 For any 1 ; 2 2 Q, an jiðt; 1 Þ Iðt; 2 Þj T j 1 2 j; jbðt; 1 Þ Bðt; 2 Þj T j 1 2 j; t 2½0; TŠ t 2½0; TŠ: Proof: Follows from equations (3.1) an (3.4) an Proposition 2 by a proof similar to that of Corollary 1. 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. Let ðq j ðþ; R j ðþþ; j ¼ 1;...; JðÞ be the orere extremal subintervals of ½0; 1Þ, such that Yðt;Þ > 0 for all t 2ðQ j ; R j Þ; that is, the enpoints, Q j ðþ an R j ðþ, are obtaine via inf an sup functions, respectively. By convention, if any of these enpoints oes not exist, then it is set to 1. OBSERVATION 3 Q 1 ðþ < R 1 ðþ < Q 2 ðþ < R 2 ðþ <...< Q JðÞ ðþ < R JðÞ ðþ; w:p:1: ð4:1þ Proof: The strict inequalities will follow in equation (4.1) if we show that the equalities Q j ðþ ¼R j ðþ an R j ðþ ¼Q jþ1 ðþ are impossible. The first equality is rule out because the intervals ðq j ðþ; R j ðþþ are extremal by efinition. The secon equality is rule out by part (c) of Assumption IPA Derivatives with Respect to the Base-Stock Level In this section we erive IPA erivatives for the performance metrics L I ðt;þ an L B ðt;þ with respect to the base-stock level, ¼ SðÞ, from any initial inventory state. The approach is to first erive IPA erivatives for the inventory process, fiðt;þg, an backorer process, fbðt;þg, an then use the results to erive the IPA erivatives of the requisite performance metrics. ASSUMPTION 3 (a) SðÞ ¼, where 2 Q. (b) The processes fðtþg an fðtþg are inepenent of the parameter.

15 Methool Comput Appl Probab (2006) 8: The following lemma ientifies the time points after which the epenence of fyðt;þg on the parameter ceases. LEMMA 2 For every j ¼ 1;...; JðÞ, the process fyðt;þ : t > R j ðþg is locally inepenent of, an consequently, Yðt;Þ¼0 on the event ft > R jðþg. Proof: It suffices to prove the lemma for j ¼ 1 only, but as the proof remains unchange, we prove it for any j. We first show that R j ðþ is locally ifferentiable with respect to for each j ¼ 1;...; JðÞ. To this en, note that R j ðþ is not locally ifferentiable only when the following two simultaneous sample path events occur at time R j ðþ: the first is a jump in either of fðtþg or fðtþg at R j ðþ, an the secon correspons to IðR j ðþ;þ¼sðþ (equivalently, YðR j ðþ;þ¼0). However, part (c) of Assumption 1 rules out such simultaneous sample path events. We next prove the lemma on the event fr j ðþ < t Q jþ1 ðþg. Since R j ðþ < Q jþ1 ðþ by Observation 3, it follows that Q jþ1 ðþ is a jump point of fðtþ ðtþg, such that ðq jþ1 ðþ Þ ðq jþ1 ðþ Þ while ðq jþ1 ðþþ >ðq jþ1 ðþþ. Consequently, for each j 1, Q jþ1 ðþ is locally inepenent of an Yðt;Þ is locally inepenent of on the event fr j ðþ < t Q jþ1 ðþg an vanishes there. It remains to prove the lemma on the event ft > Q jþ1 ðþg. But this follows from the fact that YðQ jþ1 ðþ;þ is locally inepenent of as shown above, an its erivative values in equation (3.3) involve only ðtþ an ðtþ, which are inepenent of. The proof is complete. In the next two lemmas we make use of the hitting time, T S ðþ, efine by T S ðþ ¼ min ft 2½0; 1Þ : Iðt;Þ¼SðÞg; if the minimum exists 1; otherwise: ð4:2þ LEMMA 3 Consier an MTS system with the backorer rule on the event fwð0þ < SðÞg (that is, the system starts in normal operation with backorers or partial inventory). Then, for any t 0 an 2 Q, (a) On the event AðÞ ¼fWð0Þ < SðÞg T ft < T S ðþg, Iðt;Þ¼ Bðt;Þ¼0: (b) On the event BðÞ ¼fWð0Þ < SðÞg T ft > T S ðþg T fiðt;þ > 0g, Iðt;Þ¼1; Bðt;Þ¼0: (c) On the event CðÞ ¼fWð0Þ < SðÞg T ft > T S ðþg T fbðt;þ > 0g, Iðt;Þ¼0; Bðt;Þ¼ 1: Proof: By Observation 3, 0 ¼ Q 1 ðþ < T S ðþ ¼R 1 ðþ < Q 2 ðþ on fwð0þ < SðÞg; ð4:3þ an this hols for all cases of this lemma.

16 206 Methool Comput Appl Probab (2006) 8: To prove part (a), note that by equations (3.1) an (3.3), AðÞ. It follows that Yðt;Þ¼Yð0;Þþ 0 Yðt;Þ t þ ½ðÞ ðþš on AðÞ: Substituting the above into equation (3.4) implies that Wðt;Þ¼SðÞ Yðt;Þ¼Wð0Þ 0 ½ðÞ ðþš ¼ ðtþ ðtþ on is inepenent of on AðÞ. It follows from equation (3.1) that Iðt;Þ¼WðtÞ þ an Bðt;Þ¼½ WðtÞŠ þ are also inepenent of on AðÞ, from which part (a) follows. To prove parts (b) an (c), we apply Lemma 2 to T S ðþ ¼R 1 ðþ an conclue that Yðt;Þ¼0on fwð0þ < SðÞg \ ft > T S ðþg: ð4:4þ Next, equation (3.4) implies that on BðÞ one has Iðt;Þ¼SðÞ Yðt;Þ an Bðt;Þ¼0, while on CðÞ one has Iðt;Þ¼0anBðt;Þ¼Yðt;Þ SðÞ. Parts (b) an (c) follow by ifferentiating these relations with respect to an substituting equation (4.4). LEMMA 4 Consier an MTS system with the backorer rule on the event fwð0þ > SðÞg (that is, the system starts in overage operation). Then, for any t 0 an 2 Q, (a) (b) On the event AðÞ ¼fWð0Þ > SðÞg T ft < T S ðþg, Iðt;Þ¼ Bðt;Þ¼0: On either of the events B 1 ðþ ¼fWð0Þ > SðÞg T ft S ðþ < Q 1 ðþg T T ft > T s ðþg fiðt;þ > 0g or B2 ðþ ¼fWð0Þ > SðÞg T ft S ðþ ¼Q 1 ðþg T T ft > R 1 ðþg fiðt;þ > 0g, Iðt;Þ¼1; Bðt;Þ¼0: (c) On either of the events C 1 ðþ ¼fWð0Þ > SðÞg T ft S ðþ < Q 1 ðþg T ft > T S ðþg T fbðt;þ > 0g or C 2 ðþ ¼fWð0Þ > SðÞg T ft S ðþ ¼Q 1 ðþg T ft > R 1 ðþg T fbðt;þ > 0g, () (e) Iðt;Þ¼0; Bðt;Þ¼ 1: On the event DðÞ ¼fWð0Þ > SðÞg T ft S ðþ ¼Q 1 ðþg T fq 1 ðþ < t < R 1 ðþg T fiðt;þ > 0g, Iðt;Þ¼ðQ 1ðÞÞ ðq 1 ðþþ ; Bðt;Þ¼0: On the event EðÞ ¼fWð0Þ > SðÞg T ft S ðþ ¼Q 1 ðþg T fq 1 ðþ < t < R 1 ðþg T fbðt;þ > 0g, Iðt;Þ¼0; Bðt;Þ¼ ðq 1ðÞÞ ðq 1 ðþþ :

17 Methool Comput Appl Probab (2006) 8: Proof: To prove part (a), note that by equations (2.2) an(3.1), Wðt;Þ¼Iðt;Þ on AðÞ. It follows from equation (2.1) that tþwðt;þ¼ ðtþ on AðÞ. Therefore, Wðt;Þ¼Wð0Þ 0 ðþ is inepenent of on AðÞ. From equation (3.1) it follows that Iðt;Þ¼WðtÞ þ an Bðt;Þ¼½ WðtÞŠ þ are also inepenent of on AðÞ, from which part (a) follows. To prove part (b) an (c) on the event B 1 ðþ S C 1 ðþ, note that by Observation 3, T S ðþ < Q 1 ðþ < R 1 ðþ on B 1 ðþ [ C 1 ðþ: ð4:5þ Clearly, T S ðþ is locally ifferentiable with respect to. By a proof similar to that of Lemma 2, we conclue that Yðt;Þ is locally inepenent of on the event fwð0þ > SðÞg T ft S ðþ < tg. Consequently, parts (b) an (c) on each of the events B 1 ðþ an C 1 ðþ follow from equation (4.5) by the proof of parts (b) an (c) in Lemma 3. To prove part (b) an (c) on the event B 2 ðþ S C 2 ðþ, note that by Observation 3, T S ðþ ¼Q 1 ðþ < R 1 ðþ < Q 2 ðþ on B 2 ðþ [ C 2 ðþ: ð4:6þ Applying Lemma 2 to R 1 ðþ yiels that Yðt;Þ is locally inepenent of on the event fwð0þ > SðÞg T fr 1 ðþ < tg. Parts (b) an (c) on each of the events B 2 ðþ an C 2 ðþ follow from equation (4.6) by the proof of parts (b) an (c) in Lemma 3. To prove () an (e), note first that Z Q1 ðþ 0 ðþ ¼ Wð0Þ SðÞ ¼Wð0Þ ; on fwð0þ > SðÞg \ ft S ðþ ¼Q 1 ðþg: Differentiating the above with respect to yiels after some manipulation, Q 1 1ðÞ ¼ ðq 1 ðþþ ; on fwð0þ > SðÞg \ ft S ðþ ¼Q 1 ðþg: ð4:7þ We next show that both Q 1 ðþ an R 1 ðþ epen Z on on fwð0þ > SðÞg T t ft S ðþ ¼ Q 1 ðþg. To this en, we write Yðt;Þ¼ ½ðÞ ðþš on the event Q 1 ðþ DðÞ S EðÞ, an then ifferentiate it with respect to. Itfollowsthat Yðt;Þ¼ ½ðQ 1ðÞÞ ðq 1 ðþþš on DðÞ [ EðÞ; Q 1ðÞ ¼ ðq 1ðÞÞ ðq 1 ðþþ ; ðq 1 ðþþ ð4:8þ where the secon equality is obtaine by substituting equation (4.7), an noting the inequalities ðq 1 ðþþ >ðq 1 ðþþ 0 on the event fwð0þ > SðÞg T fq 1 ðþ ¼T S ðþg. Finally, equation (3.4) implies that on DðÞ one has Iðt;Þ¼SðÞ Yðt;Þ an Bðt;Þ¼0, while on EðÞ one has Iðt;Þ¼0anBðt;Þ¼Yðt;Þ SðÞ. Parts () an (e) now follow by ifferentiating these relations with respect to an substituting into equation (4.8.)

18 208 Methool Comput Appl Probab (2006) 8: On the event fsðþ ¼Wð0Þg, the situation is more complex, because the left an right erivatives of Iðt;Þ an Bðt;Þ with respect to o not coincie an must be compute separately. We first erive the right-erivatives, Iðt;Þ an þ Bðt;Þ, by borrowing from þ Lemma 3 an making use of the hitting time, T ðþ, given by 8 minft 2½0; Q 1 ðþþ: ðtþ>ðtþg; if the minimum exists on the event fq 1 ðþ>0g >< R T ðþ¼ 1 ðþ; if R 1 ðþ exists on the event fq 1 ðþ ¼0g S ½fQ 1 ðþ > 0g T fðtþ ¼ðtÞ; t 2½0; Q 1 ðþþgš >: 1; otherwise: ð4:9þ LEMMA 5 Consier an MTS system with the backorer rule on the event fwð0þ ¼SðÞg(that is, the system starts in normal operation with full inventory). Then, for any t 0 an 2 Q, (a) On the event AðÞ ¼fWð0Þ ¼SðÞg T ft < T ðþg, þ Iðt;Þ¼ þ Bðt;Þ¼0: (b) On the event BðÞ ¼fWð0Þ ¼SðÞg T ft > T ðþg T fiðt;þ > 0g, þ Iðt;Þ¼1; þ Bðt;Þ¼0: (c) On the event CðÞ ¼fWð0Þ ¼SðÞg T ft > T ðþg T fbðt;þ > 0g, þ Iðt;Þ¼0; þ Bðt;Þ¼ 1: Proof: Consier a perturbe system with Sð þ DÞ ¼ þ D, where D > 0. Since Wð0Þ ¼SðÞ < Sð þ DÞ, it follows that the perturbe system starts in normal operation. Denote DS ¼ Sð þ DÞ SðÞ. By Observation 3, 0 ¼ Q 1 ð þ DÞ T ðþ < R 1 ð þ DÞ < Q 2 ð þ DÞ on fwð0þ ¼SðÞg; ð4:10þ an this hols for all cases of this lemma. To prove part (a), note first that the event ft ðþ ¼0g can be preclue, since it implies AðÞ ¼;, where ; enotes the empty set. Otherwise, by the efinition of T ðþ an equations (3.1) an (3.3), W ðt;þ ¼ ðtþ ðtþ on AðÞ, so that t þ Wðt;Þ¼Wð0Þ 0 ½ðÞ ðþš on AðÞ: Furthermore, the efinition of T ðþ ensures that both fwðt;þg an fwðt;þ DÞg are boune from above by SðÞ on the event AðÞ, so that Wðt;þ DÞ ¼Wð0Þ 0 ½ðÞ ðþš on AðÞ: We conclue that Wðt;þ DÞ ¼Wðt;Þ are inepenent of on AðÞ. The rest of the proof of part (a) is ientical to that of part (a) in Lemma 3.

19 Methool Comput Appl Probab (2006) 8: To prove parts (b) an (c), observe that part (b) of Assumption 1 implies that there exists >0, such that for any D, DS R 1 ð þ DÞ ¼T ðþþ on fwð0þ ¼SðÞg; ð4:11þ ðt ðþþ ðt ðþþ where the inequality ðt ðþþ ðt ðþþ > 0 follows from the efinition of T ðþ. Clearly, T ðþ is locally ifferentiable with respect to. In view of equation (4.10), it follows by a proof similar to that of Lemma 2 that Yðt;þ DÞ ¼Yðt;Þ on the event fwð0þ¼ SðÞg T ft >R 1 ð þdþg. But because R 1 ð þ DÞ!T ðþ on fwð0þ ¼SðÞg as D! 0 by equation (4.11), we conclue that Yðt;Þ¼0 on the event BðÞg S þ CðÞ. The rest of the proof is similar to that of parts (b) an (c) in Lemma 3. We next erive the left-erivatives, Iðt;Þ an Bðt;Þ, by borrowing from Lemma 4, an making use of the hitting time, T, given by T ¼ minft 2½0; TŠ : ðtþ > 0g; if the minimum exists 1; otherwise: ð4:12þ Note that T is inepenent of. LEMMA 6 Consier an MTS system with the backorer rule on the event fwð0þ ¼ SðÞg (that is, the system starts in normal operation with full inventory). Then, for any t 0 an 2 Q, (a) On the event AðÞ ¼fWð0Þ ¼SðÞg T ft < T g, (b) (c) () Iðt;Þ¼ Bðt;Þ¼0: On either of the events B 1 ðþ ¼fWð0Þ ¼SðÞg T ft < Q 1 ðþg T ft > T g T fiðt;þ > 0g or B 2 ðþ ¼fWð0Þ ¼SðÞg T ft ¼ Q 1 ðþg T ft > R 1 ðþg T fiðt;þ > 0g, Iðt;Þ¼1; Bðt;Þ¼0: On either of the events C 1 ðþ ¼fWð0Þ ¼SðÞg T ft < Q 1 ðþg T ft > T g T fbðt;þ > 0g or C 2 ðþ ¼fWð0Þ ¼SðÞg T ft ¼ Q 1 ðþg T ft > R 1 ðþg T fbðt;þ > 0g, Iðt;Þ¼0; Bðt;Þ¼ 1: On the event DðÞ ¼fWð0Þ ¼SðÞg T ft ¼ Q 1 ðþg T fq 1 ðþ < t < R 1 ðþg T fiðt;þ > 0g, (e) Iðt;Þ¼ðQ 1ðÞÞ ðq 1 ðþþ ; Bðt;Þ¼0: On the event EðÞ ¼fWð0Þ ¼SðÞg T ft ¼ Q 1 ðþg T fq 1 ðþ < t < R 1 ðþg T fbðt;þ > 0g, Iðt;Þ¼0; Bðt;Þ¼ ðq 1ðÞÞ ðq 1 ðþþ :

20 210 Methool Comput Appl Probab (2006) 8: Proof: Consier a perturbe system with Sð DÞ ¼ D, where D >0. Since by assumption Wð0Þ ¼SðÞ > Sð DÞ, it follows that the perturbe system starts in overage operation. Denote DS ¼ SðÞ Sð DÞ. To prove part (a), note first that the event ft ¼ 0g can be preclue, since it implies AðÞ ¼;. Otherwise, on the event AðÞ, the perturbe system is in overage operation with no eman arrivals, so that Wðt; DÞ ¼Wð0Þ ¼Wðt;Þ on the event AðÞ. The rest of the proof follows from that of part (a) in Lemma 4. To prove parts (b) an (c), observe that part (b) of Assumption 1 implies that there exists >0, such that for any D, T S ð DÞ ¼T þ DS ðt Þ on fwð0þ ¼SðÞg; ð4:13þ where the inequality ðt Þ > 0 follows from the efinition of T. To prove part (b) an (c) on the event B 1 ðþ S C 1 ðþ, note that by Observation 3, T < T S ð DÞ < Q 1 ðþ < R 1 ðþ on B 1 ðþ [ C 1 ðþ: ð4:14þ Clearly, T is locally ifferentiable with respect to. In view of equation (4.14), it follows from a similar proof to that of Lemma 2 that Yðt; DÞ ¼Yðt;Þ on the event fwð0þ ¼SðÞg T ft > T S ð DÞg. But because T S ð DÞ!T on fwð0þ ¼ SðÞg as D! 0 by equation (4.13), we conclue that Yðt;Þ¼0 on the event B 1 ðþ S C 1 ðþ. The rest of the proof is similar to that of parts (b) an (c) in Lemma 4. We next prove the remaining cases, namely, parts (b) an (c) for the events B 2 ðþ an C 2 ðþ, as well as parts () an (e). In all these cases, by Observation 3, Q 1 ðþ ¼T < T S ð DÞ ¼Q 1 ð DÞ < R 1 ðþ on B 2 ðþ [ C 2 ðþ [ DðÞ [ EðÞ: ð4:15þ In other wors, the process fiðt;þg stays at SðÞ until time T, at which point the arrival rate jumps, such that ðt Þ >ðt Þ. It follows that WðT S ð DÞ;Þ¼SðÞ DS ðt Þ ½ðT Þ ðt ÞŠ on B 2 ðþ [ C 2 ðþ [ DðÞ [ EðÞ: ð4:16þ But over the interval ½T S ð DÞ; R 1 ð DÞŠ, both the original system an the perturbe system operate in their respective normal moe an are riven by ientical ynamics. Therefore, the ifference process fwðt;þ Wðt; DÞg is constant an positive over that interval. Consequently, by part (c) of Assumption 1, we can choose sufficiently small D >0, such that R 1 ð DÞ ¼R 1 ðþ DS ½ðT Þ ðt ÞŠ ðt Þ½ðR 1 ðþþ ðr 1 ðþþš on B 2 ðþ [ C 2 ðþ [ DðÞ [ EðÞ: ð4:17þ

21 Methool Comput Appl Probab (2006) 8: Furthermore, by the efinition of T S ð DÞ an equation (4.16), Wðt;Þ DS ðtþ Wðt; DÞ ¼, t 2½T ðt Þ Sð DÞ; R 1 ð DÞŠ. Combining this with equation (3.4), we conclue that Yðt;Þ Yðt; DÞ ¼DS ðt Þ ðt Þ ðt Þ on ft S ð DÞ t R 1 ð DÞg: ð4:18þ Moreover, Yðt; DÞ ¼Yðt;Þ on ft > R 1 ðþg; ð4:19þ ue to part (c) of Assumption 1. Next, sen DS ¼ D! 0 in equations (4.13) an (4.17), yieling T S ð DÞ!T an R 1 ð DÞ!R 1 ðþ, respectively. The left erivative Yðt;Þ can now be reaily compute on events B 2 ðþ an C 2 ðþ from equation (4.19), an on events DðÞ an EðÞ from equation (4.18). Finally, the rest of the proof of parts (b) an (c) follows similarly to the proof of parts (b) an (c) of Lemma 4, while that of parts () an (e) follows similarly to the proof of parts () an (e) of Lemma 4. 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 > 0 an 2 Q as follows: (a) On the event fwð0þ < SðÞg, L IðT;Þ¼1 fts ðþ<tg Z 1 T 1 fiðt;þ>0g t: ð4:20þ T T S ðþ (b) On the event fwð0þ > SðÞg T ft S ðþ < Q 1 ðþg, L IðT;Þ¼1 fts ðþ<tg Z 1 T 1 fiðt;þ>0g t: ð4:21þ T T S ðþ (c) On the event fwð0þ > SðÞg T ft S ðþ ¼Q 1 ðþg, L 1 ðq1 ðþþ IðT;Þ¼1 fq1 ðþ<tg T ðq 1 ðþþ Q 1 ðþ Z T þ 1 fr1 ðþ<tg 1 fiðt;þ>0g t : R 1 ðþ Z minfr1 ðþ;tg 1 fiðt;þ>0g t ð4:22þ () On the event fwð0þ ¼SðÞg, þ L IðT;Þ¼1 ftðþ<tg Z 1 T 1 fiðt;þ>0g t: ð4:23þ T T ðþ (e) On the event fwð0þ ¼SðÞg T ft < Q 1 ðþg, L IðT;Þ¼1 ft <Tg Z 1 T 1 fiðt;þ>0g t: ð4:24þ T T

22 212 Methool Comput Appl Probab (2006) 8: (f) On the event fwð0þ ¼SðÞg T ft ¼ Q 1 ðþg, L IðT;Þ¼1 fq1 ðþ<tg " Z 1 ðq 1 ðþþ minfr1 ðþ;tg Z # T 1 fiðt;þ>0g t þ 1 fr1 ðþ<tg 1 fiðt;þ>0g t : T ðq 1 ðþþ R 1ðÞ Q 1 ðþ ð4:25þ Proof: We show that Leibniz s rule can be applie to equation (2.10) yieling Z T L IðT;Þ¼ 1 Iðt;Þt ¼ 1 Iðt;Þt: T 0 T 0 ð4:26þ To this en, note that Assumption 1 an Lemmas 3 6 ensure that w.p.1., the sie erivatives Iðt;Þ exist an are finite over the interval ½0; TŠ, except possibly for a finite number of time points. Furthermore, since the starting time an ening time in the integral of equation (4.26) are inepenent of, it follows from Corollary 1 that the ifferentiation an the integration operations commute there. The theorem now follows by substituting the values of the erivatives compute in Lemmas 3 6 into equation (4.26). Z T THEOREM 2 W.p.1, the IPA erivatives of the backorer time average with respect to the base-stock level are given for all T > 0 an 2 Q as follows: (a) (b) On the event fwð0þ < SðÞg, L BðT;Þ¼ 1 fts ðþ<tg On the event fwð0þ > SðÞg T ft S ðþ < Q 1 ðþg, Z 1 T 1 fbðt;þ>0g t: ð4:27þ T T S ðþ (c) L BðT;Þ¼ 1 fts ðþ<tg On the event fwð0þ > SðÞg T ft S ðþ ¼Q 1 ðþg, Z 1 T 1 fbðt;þ>0g t: ð4:28þ T T S ðþ () L 1 ðq1 ðþþ BðT;Þ¼ 1 fq1 ðþ<tg T ðq 1 ðþþ Q 1 ðþ Z T þ 1 fr1 ðþ<tg 1 fbðt;þ>0g t : On the event fwð0þ ¼SðÞg, R 1 ðþ Z minfr1 ðþ;tg 1 fbðt;þ>0g t ð4:29þ (e) þ L BðT;Þ¼ 1 ft ðþ<tg On the event fwð0þ ¼SðÞg T ft < Q 1 ðþg, Z 1 T 1 fbðt;þ>0g t: ð4:30þ T T ðþ L BðT;Þ¼ 1 ft <Tg Z 1 T 1 fbðt;þ>0g t: ð4:31þ T T