Advances in Operations ResearchVolume 2011 (2011), Article ID 301205, 23 pagesdoi:10.1155/2011/301205

Research Article

On the Accuracy of Fluid Approximations to a Class of Inventory-Level-Dependent EOQ and EPQ Models

Alexey Piunovskiy and Yi Zhang

Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, UK

Received 29 September 2010; Accepted 18 January 2011

Academic Editor: Viliam Makis

Copyright © 2011 Alexey Piunovskiy and Yi Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Deterministic Economic Order Quantity (EOQ) models have been studied intensively in the literature, where the demand process is described by an ordinary differential equation, and the objective is to obtain an EOQ, which minimizes the total cost per unit time. The total cost per unit time consists of a “discrete” part, the setup cost, which is incurred at the time of ordering, and a “continuous” part, the holding cost, which is continuously accumulated over time. Quite formally, such deterministic EOQ models can be viewed as fluid approximations to the corresponding stochastic EOQ models, where the demand process is taken as a stochastic jump process. Suppose now an EOQ is obtained from a deterministic model. The question is how well does this quantity work in the corresponding stochastic model. In the present paper we justify a translation of EOQs obtained from deterministic models, under which the resulting order quantities are asymptotically optimal for the stochastic models, by showing that the difference between the performance measures and the optimal values converges to zero with respect to a scaling parameter. Moreover, we provide an estimate for the rate of convergence. The same issue regarding specific Economic Production Quantity (EPQ) models is studied, too.

1. Introduction

Consider an inventory item which is demanded. So the inventory level gradually decreases and is backed up by ordering new inventories from time to time. There are two costs to consider: a positive inventory level results in a holding cost, and every order induces a setup cost. The objective is to determine an order quantity that minimizes the total cost per unit time (). Such a minimizer is known as an Economic Order Quantity (EOQ), and the model itself is known as an EOQ model.

Arguably the simplest EOQ model (sometimes referred to as the classic EOQ model) is based on the following assumptions: (i) the instantaneous holding cost rate is constant; (ii) the setup cost is constant; (iii) the demand comes in a deterministic and continuous process at a constant rate; (iv) no backlogging is allowed so that at inventory level zero all the arrived demand is rejected; (v) the inventory item is homogeneous and nonperishable so that only the demand reduces the inventory level; (vi) the inventory level is reviewed continuously so that it can be described by an ordinary differential equation; (vii) the replenishment takes place instantaneously after ordering. A rigorous description of the classic EOQ model is shortly given in Section 2.1. Amongst the efforts of generalizing the classic EOQ model, a great deal have been made on relaxing assumption (iii). For instance, the EOQ model considered in [1] (see also the references therein) assumes the demand rate to be inventory level dependent, despite the dependence is of a specific form. This is a response to: “At times, the presence of inventory has a motivational effect on the people around it. It is a common belief that large piles of goods displayed in a supermarket will lead the customer to buy more [2].” Generalizations regarding other assumptions include allowing backlogging and periodic reviews in [3], accounting for perishable goods in [4–7], and so on. A comprehensive review of the literature on EOQ models is available in [8]. Note that the aforementioned works mainly focus on EOQ models where the demand process is deterministic. Quite formally these deterministic models can be viewed as the fluid approximations to the corresponding stochastic EOQ models, where the demand comes in a stochastic jump process. (For this reason, in what follows we call the deterministic models also fluid models, and EOQs derived from fluid models are referred to as fluid EOQs.)

Suppose now that a fluid EOQ is obtained and the corresponding stochastic EOQ model is appropriately formulated. Then the issue of interest is how to translate the fluid EOQ into an order quantity for the stochastic model, where the expected total cost per unit time () is (nearly) minimal. The fluid approximation would be justified if such a translation is obtained. The formal justification of the fluid approximations to various jump Markov optimization models with local transitions has been addressed by numerous authors, see [9–12], all of which focus on queueing networks. In greater detail the optimization problems considered in [9, 12] are with a discounted criterion over an infinite horizon, the one in [11] is with an expected total cost criterion over a fixed finite time horizon, while the performance measure for the queueing network in [10] is the expected total cost up to the first moment the system gets empty. Here we emphasize that in fluid (resp., stochastic) EOQ models the performance measures are tcus (resp., TCUs), which are long run averages. All of those works compare the optimal value for the fluid model with the performance measure for the (scaled) stochastic model under the translated policy and show that the difference converges to zero as the scaling parameter increases to infinity. However, none of them reveal the rate of convergence, which measures the accuracy of the fluid approximations and the efficiency of the underlying translations. Consequently the more recent development on this topic aims at obtaining the rate of convergence, see [13–15]. In particular, in [14] the author proposes a translation of the fluid EOQ and shows that it results in an asymptotically fluid optimal (AFO) and asymptotically optimal (AO) order quantity for the corresponding stochastic model. (The accurate definitions of “AFO” and “AO” are postponed to Sections 2 and 3.) However, fairly strong conditions on the system parameters are assumed there, restricting the applicability of the obtained results.

Therefore, the main contribution of this paper is to provide a refinement of the results obtained in [14] by relaxing the conditions assumed therein, and thus enlarging the applicability to cover a broader class of EOQ models. In greater detail, by taking the stochastic model as a continuous-time Markov chain, we justify the translation of fluid EOQ proposed in [14] and obtain its efficiency in the form of a rate of convergence. The Markov property of our stochastic model is a result of assuming exponentially distributed interarrival times in the demand process, which is standard in the current literature on inventory systems, see [7, 16–19]. In our models, the demand and holding cost rates are of a rather general inventory-level dependence, and thus with broad applicabilities. In particular, our results are applicable to the important case of discontinuous demand and holding cost rates (compared to that globally Lipschitz continuous rates are assumed in [14]), see more discussions on this in Section 4. Moreover, in this paper results similar to those for EOQ models are derived for Economic Production Quantity (EPQ) models, too.

The rest of this paper is organized as follows. In Sections 2 and 3 we formulate EOQ and EPQ models and state the main results. In Section 4 some comments are given on the issues of possible applicabilities of the obtained results and thus illustrate the contribution of this paper. We finish this paper with conclusions. The proofs of the main statements are postponed to the appendix.

2. Economic Order Quantity Models

In what follows, the trivial case of an order quantity taking zero is excluded from consideration, and the context should always make it clear when stands for the function taking the largest integer part of its argument.

2.1. Description of Mathematical Models

(a) Fluid ModelSuppose some order quantity is fixed. Then let be the inventory level process subject to the dynamics when and , where reaches state zero at . Here, is the demand rate, and the impulsive jump of at reflects the instantaneous replenishment assumption. Let be the time duration between two consecutive jumps of the inventory level process, the instantaneous holding cost rate, and the setup cost, incurred immediately whenever an order is made. We are interested in minimizing the long run average cost given by Let us call the EOQ for the fluid model, so that . In particular, if and (i.e., the demand rate and the holding cost rate per unit of inventory are constants ), then we have the so called classic EOQ model.

(b) (Scaled) Stochastic ModelIn the corresponding stochastic model, the inventory item is measured in small units, so that a scaling parameter is present to indicate the units. The intuitive meaning of this scaling parameter is explained in Remark 2.1 below. In greater detail, fixing some integer order quantity , the inventory level process is modelled as a continuous-time Markov chain with the state space and the transition rates given by and in the case of where is the instantaneous demand rate and is the parameter of the exponentially distributed lead time between the ordering and the corresponding replenishment. In other words, the time between two consecutive demand arrivals is exponentially distributed with mean when the current inventory level is , and new inventories are ordered when the inventory level hits zero.

Remark 2.1. The scaled stochastic model can be linked to the fluid model by taking , where is the order quantity for the fluid model (Here for big enugh . So below we assume that , that is, we consider large enough .). The above-described scaling is often referred to as a fluid scaling, and its intuitive meaning can be understood as follows. Clearly, when , the stochastic model is a corresponding version of the fluid model. As we increase , the demand comes in smaller units, and inventories are measured more accurately. Take as an example. Suppose that corresponds to the unit of (kg) so that on average (kg) units of demand come per time unit. In the case of , the unit will be (g): on average (g) units of demand come per time unit. Meanwhile, the cost rate is not amplified by multiplying because it costs the same to hold either (g) or (kg) of inventories.

We are interested in minimizing the performance measure given by where denotes the expectation operator with the initial inventory level (immediately after the replenishment) . Let us call the EOQ for the (scaled) stochastic model, so that .

We say that order quantity for the stochastic model is AFO if and AO if In what follows, the EOQs for both fluid and stochastic models are assumed to be unique, and the similar assumption applies to the EPQ models.

2.2. Main Results

Condition 1. (a) There exist constants and such that and ; here functions and are measurable and both defined on .(b) There exist finite intervals with such that on each of them and are Lipschitz continuous with a common Lipschitz constant and , respectively.

Note that Condition 1 implies that for any fixed , there exist (possibly -dependent) finite intervals such that on each interval and are Lipschitz continuous with Lipschitz constants and , respectively. For simplicity, we define .

Proposition 2.2. Under Condition 1, for any fixed order quantity in the fluid model

In particular, Proposition 2.2 implies that , that is, is AFO.

The same calculations as in [14, page 406] result in the next lemma.

Lemma 2.3. Suppose that Condition 1 is satisfied and , then and satisfies . In particular, .

Corollary 2.4. Under Condition 1, if (see Lemma 2.3), then one has , and , that is, is AO.

Corollary 2.5. Under Condition 1, suppose in addition that is -independent and on . Then the following statements hold. (a) For any (see (2.8)) and any large enough , where is a positive integer number satisfying , Here the -independent goes to zero as fast as in the sense of , whose meaning is that with some (nonnegative) constant . (b) For any large enough , Here we recall that and are the EOQ for the (scaled) stochastic model and fluid model, respectively.

Corollary 2.5 refines Corollary 2.4 for certain cases by providing an estimate for the rate of convergence.

Fix some order quantity for the fluid model and scaling parameter , and let be the stationary distribution of the inventory process and the invariant density in the fluid model of the underlying dynamics of . Then the following proposition shows that the fluid model can also be used to provide approximations to stationary distributions of the inventory level process in the (scaled) stochastic model.

Proposition 2.6. Suppose that Condition 1 is satisfied and some order quantity for the fluid model is fixed. Then one has and for

2.3. A Comparison with Section 4 of [14]

The fluid approximations of the EOQ models are also briefly considered in [14, Section 4]. Therefore, we mention the main difference of this paper from that one in this subsection.

Firstly, the present paper is based on a weaker condition. Indeed, instead of Condition 1, the following stronger condition is assumed in [14].

Condition 2. (a) Condition 1(a) holds.(b) Functions and are globally Lipschitz continuous.

The global Lipschitz property is essential to the corresponding proof in [14].

Secondly, the approach in [14] is based on the explicit expression of obtained by solving the associated Poisson equations. Instead of doing that, the present paper employs the recent results of [15]. One advantage of this approach lies in the weaker condition required (only piecewise Lipschitz continuity is needed). Another advantage is that it allows one to study EPQ models, because the Poisson equations for EPQ models are much more difficult to solve compared to those for EOQ models. That is why EPQ models are not considered in [14].

3. Economic Production Quantity Models

In EOQ models the inventory is backed up at once by ordering new inventory items from external suppliers. In this section we consider the situation where the inventory is gradually backed up by producing new items. In greater detail, the inventory level decreases gradually to meet the demand, and when it hits zero, the production is switched on and new inventory items are being produced to back up the inventory. The production is switched off as soon as the inventory is backed up to a predetermined level. Here we have to account for the cost incurred from switching on the production as well as from holding the inventory items. The aim is to obtain an Economic Inventory Backup Level (EIBL) that minimizes , and the resulting model is called an EPQ model. Similar to the previous section, below we justify the fluid approximations to stochastic EPQ models, whose rigorous description is shortly given.

3.1. Description of Mathematical Models

(a) Fluid ModelSuppose that we fix some real inventory backup level , meaning that the production is always on until the inventory reaches the level . Let represent the inventory level process in the fluid model with state space , and instantaneous demand and production rates and , respectively. Then the inventory level process is subject to the dynamics where the production-off phase and production-on phase, superseding each other, are triggered by and , respectively. In words, without any delay, once the inventory level reaches zero, production is switched on till it reaches the inventory backup level . Let be the holding cost rate, the setup cost incurred with switching on the production, and the time duration between two consecutive production switching-offs. So Let us denote by the EIBL for the fluid model, so that .

(b) (Scaled) Stochastic ModelSuppose that we fix some inventory backup level positive integer , meaning that the production is always on until the inventory reaches level . Let represent the inventory level process. We model it as a continuous-time Markov chain with the state space , where indicates that the inventory level is and the production is off and the denotation of can be understood in the same way. Its transition rates are given by for all and finally for all : where and stand for the instantaneous production and demand rates, is the parameter of the exponentially distributed lead time between the switching and the actual production-on, and we have ignored all the cases when the transition rates take zero. So we have with the holding cost and setup cost . Let us denote by the EIBL for the (scaled) stochastic model, so that .The concept of AFO and AO inventory backup level can be understood in the same manner as introduced at the end of Section 2.1.

3.2. Main Results

Condition 3. (a) There exist some constants , , , and measurable functions , and defined on such that , , , and .(b) There exist finite intervals with such that on each of them , and are Lipschitz continuous with Lipschitz constants ,, and , respectively.

Note that Condition 3 implies that for any fixed , there exists an integer (possibly -dependent) and intervals , such that on each interval , , and are Lipschitz continuous with Lipschitz constant , , and , respectively, and at the same time on each of these intervals belonging to , functions (with respect to ) and are Lipschitz continuous with Lipschitz constants and , respectively. Let us now denote the common Lipschitz constant by .

Proposition 3.1. Under Condition 3, for any fixed (fluid) inventory backup level , where one puts and and the -independent terms are given by (A.2) in the appendix.

As in the case of EOQ models, we observe from Proposition 3.1 that is AFO.

Corollary 3.2. Under Condition 3, the following statements hold. (a)If , then . (b)If , then and , that is, is AO. (c)If is -independent and , then the following two substatements hold.(c1)For and big enough so that , Here is -independent, and One has (c2).

4. Example and Comments

In this section we firstly verify our results by considering a specific EPQ model, where and can be analytically computed. Then we comment on the applications of our results.

Example 4.1. As for EOQ models, one may refer to [14] for an example. Hence we study the following classic setting for the (scaled) stochastic EPQ model: assume constant demand and production rates and , linear holding cost with a constant , constant setup cost , and finally no lead time between “switching” and “actual production-on,” corresponding to if we take . Therefore, if we consider the underlying continuous time Markov chain , state will be excluded. The transition rates are modified accordingly. Clearly, Condition 3 is satisfied with this classic setting. The following lemma gives the explicit formula for , where for simplicity we have put instead of for the inventory backup level.

Proposition 4.2. For the stochastic EPQ model described above,

The proof of this proposition is based on solving (quite tediously) the associated Poisson equation for and is omitted here.

The corresponding deterministic EPQ model can be solved easily, and we have

Clearly, if we put in the expression for , where is the inventory backup level for the fluid model, then one can easily see that which agree with Proposition 3.1. For the means of illustration, we put , , , . Then the graphs of and are plotted in Figure 1. Note that when , the curves of and nearly coincide.

Figure 1: The dotted line (resp., dashed line, solid line) corresponds to (resp., , ).

Figure 2: The illustrative graph of .

Secondly, if increases, then by inspecting the numerator (especially the first two terms) and the denominator of the expression for , we see that if does not increase as fast as , will blow up to (it can be easily checked that the expression ). Therefore, the condition of in Corollary 3.2 is satisfied.

CommentsLet us comment on the applicability issues of our main results (Propositions 2.2 and 3.1). We mainly focus on EOQ models, as absolutely similar comments can be made on EPQ models in the same manner.Although we assume the ordering point to be always zero, our results are still applicable when it is set to be some fixed positive level, because Lemma A.1 holds if we put another absorbing state instead of zero. In particular, if one allows the state taking negative values, by putting some negative state absorbing, our results also impound the case of backlogging. This flexibility regarding the ordering point together with the fact that is unrestricted in signs enriches the applications of our results, in that although we require to be separated from zero, when profit rather than solely operational cost is counted, the ordering point is most likely positive, meaning in cases of , , as in [20] and , , as in [1], will be essentially separated from zero, validating our results.The state-dependence given in Conditions 1 and 3 is fairly general. In particular, that functions , and being bounded is not restrictive, because once some EOQ for the fluid model is fixed, to validate Propositions 2.2 and 3.1, they are only required to be bounded on bounded intervals. Note that in addition to the demand rate, some authors such as those of [5, 6] also include a state-dependent deteriorating rate, to indicate that the underlying goods are perishable. Our results are also applicable to such cases: one only needs explain as the total reduction rate of the inventory level.Finally, Propositions 2.2 and 3.1 are significant extensions of the relevant results in [14], where the author only focuses on EOQ models and requires global Lipschitz continuity of and . However, from the modelling point of view, the case of discontinuous functions is interesting and important as demonstrated by [1, 3, 5, 21, 22], where [1] considers a piecewise constant function and the others consider discontinuous taking either a constant value or according to , ,. The results in [14] are derived based on the closed form of the solution to a Poisson equation satisfied by , which is tremendously difficult to get explicitly in the case of (stochastic) EPQ models.

5. Conclusions

To sum up, in this work we formally justified a general class of inventory level-dependent deterministic EOQ and EPQ models, regarded as the fluid approximations to their stochastic versions, by showing a translation of the fluid EOQ (EIBL) to provide an order quantity (inventory backup level) asymptotically achieving some optimality for the stochastic model. The efficiency of the translation was obtained, as distinguished from the majority of the works on fluid approximations. The class of inventory models are quite broad so that to various extent, the obtained results are directly applicable to the existing works such as [1, 5, 6, 8, 20–22]. The present work is a significant extension of the relevant results in [14].

Appendix

To aid our proof, firstly, let us consider the following one-dimensional birth-and-death process with state space and birth and death rates and , respectively, where nonnegative measurable functions and are defined on and indicates the current state of the process. In addition, , where the equality holds only at 0, meaning that state zero is absorbing. Let denote the expectation of any underlying functional of the process with the initial state . Let a real measurable function defined on be fixed with . Now we are in the position to state the following condition.

Condition 1. (a) There exist constants , , , and such that , , , . Here if , then can be arbitrary.(b ) There exist finite intervals , with such that on each of them, is a Lipschitz continuous function with a common Lipschitz constant .

Note that Condition 1(b) implies that for any fixed there exists an integer (possibly -dependent) and finite intervals such that on each interval, function is Lipschitz continuous with a common Lipschitz constant .

The following lemma is a slightly stronger version of [15, Theorem 2] and will play an important role in our proof.

Lemma A.1. Suppose that Condition 1 is satisfied. Then for each where regarding the second integral the underlying dynamics is given by , and and are given by

Proof. It can be easily checked in the proof of [14, Theorem 2] that our Condition 1, weaker than the original conditions imposed therein, is sufficient for the statement. See also [23].

Proofs of Proposition 2.2, Corollaries 2.4 and 2.5, and Proposition 2.6For both the fluid model and (scaled) stochastic model let us call the time duration between two consecutive replenishments a cycle and denote them by and , respectively. Here for simplicity, we do not explicitly indicate the -dependence (resp., -dependence) of (resp., ). Clearly is a regenerative process [24, page 425] in that it probabilistically repeats itself from one cycle to the next. It then follows from [25, Theorem 1.1, Proposition 131] (see also [24, Proposition 7.3]) that as far as the long-run average is concerned, it suffices to consider the inventory level process and the cost incurred with it over only one cycle. For simplicity, we always consider the cycle starting at time with the initial position . Let us denote by and the total cost incurred over the cycle in the stochastic and fluid model, respectively. Then the following lemma indicates that the difference between and and the one between and cannot be too big.

Lemma A.2. Under Condition 1 the following two inequalities hold with nonnegative and given by (A.2): (a); (b).

Proof. (a) Let us denote by the time duration from the starting point up to the point when firstly reaches state zero. Obviously we have Then , where the second term on the right hand side is the expected lead time. Now observe firstly that Condition 1 is a specific version of Condition 1: one can take to be arbitrary, and put functions (Here it does no matter to put .) , and ; and secondly that the inventory level process from up to is a pure death process. Therefore, one can refer to Lemma A.1 for .(b) Let us denote by the cost incurred during the interval , so that , where the second term on the right hand side corresponds to the cost incurred over the lead time. In the same way as in part (a), comparing with first, and then adding results in the statement. Remember that the setup cost cancels out.

Proof of Proposition 2.2. Under Condition 1 we have where the last inequality follows from the facts that , , and Lemma A.2.Now let us easily observe that and both decrease with . It follows that , , and thus the above-derived expression all decrease with , where we recall that can be an arbitrary number on the interval , see Condition 1. This implies that

Proof of Corollary 2.4. For any fixed , let us denote . We do the proof in two parts.Part 1. We consider the case of a convergent sequence . Suppose now that as , does not go to but ; here we allow to be from the extended real line. In particular, for big enough , is separated from zero. According to Proposition 2.2, we have . But we also have , since is continuous in . This gives . However, it follows from Proposition 2.2 that . This indicates that at least for big enough , , which is a desired contradiction. Hence , and consequently, , as required.Part 2. Now consider the case of a divergent sequence . One only needs consider the following two situations: either it has a bounded subsequence, which by Bolzano-Weierstrass theorem further has a convergent subsequence; or it does not have a bounded subsequence, which means that it has a subsequence blowing up to . However, by taking the corresponding subsequences, we find that both situations have been essentially covered in Part 1. Part 2 is thus proved.

Proof of Corollary 2.5. (a) Under the conditions of the statement we have (Here we use the fact thatdecreases withand.) (a) is now clear.(b) According to Lemma 2.3, , and satisfies . Therefore, according to part (a), for we have in one direction and in the other direction. Combining both directions results in the statement.

Proof of Proposition 2.6. Now and can be easily computed as done in [14]. So we have Here we put for convenience. Then , and (Here we recall (a) of Lemma A.2.) Recall that in the above derived expression, can be any number from . After passing to the limit , we eventually end up with as required.

Proofs of Proposition 3.1 and Corollary 3.2Let us call a cycle the time duration between two consecutive moments when the inventory is fully backed up. Arguing similarly as for EOQ models, it suffices to consider the inventory level process and the cost incurred over one complete cycle, for which we put the starting time of . Let us denote by , and , the duration of a cycle and the cost incurred over a cycle in the fluid and scaled stochastic model, respectively. Notice additionally that a cycle is always constituted to by two phases corresponding to the on and off of the production. This raises another set of denotations: let ,  (, ), and , (, ) be the total cost incurred during the production-on (off) phases in the fluid and (scaled) stochastic model, respectively. We agree on that in both fluid and (scaled) stochastic model the setup cost is accounted for in and . Then obviously we have , and , . Here and below, for convenience we omit the subscript of the expectation operator.

Lemma A.3. Under Condition 3 the following inequalities hold: (a), ; (b) with nonnegative and given by (A.2),

Proof. (a) Let us concentrate on the inventory level process over the production-on phase. In the fluid model, it appears convenient to reflect the trajectory (corresponding to the solid curve in Figure 2) about the horizontal -axis first, and then shift the resulting trajectory (corresponding to the curve of crosses in Figure 2) upwards by units, and finally further shift the resulting trajectory to the left by shifting the time by units to the right to get (corresponding to the curve of solid boxes in Figure 2). Note now, for with the roles of production and demand have switched over: each produced unit reduces by one unit, and each demanded unit increases by one unit. More precisely, let us define the following functions: Then the dynamics of with for is of our interest, because we can write .Absolutely similar arguments are applicable to the (scaled) stochastic model. Consequently, we can consider the inventory level process during a production-on phase as a birth-and-death process with initial condition , state space , birth and death rates given by and when the current state is , and the cost rate given by . By recognizing and that Condition 3 is a specific version of Condition 1, we can refer to Lemma A.1 for . Arguing similarly as above (see also the proof of Lemma A.2), we have . (a) is now clear.(b) The production-off phase has already been covered when analyzing EOQ models. Therefore, one can directly refer to Lemma A.2 for the statement.

Proof of Proposition 3.1. Lemma A.3 implies that and similarly . Then according to (A.4) and the facts of , and (recall and ), we have

Proof of Corollary 3.2. (a) Suppose that the statement does not hold. That is, for some subsequence with as , in that . Under we have where by [26, Theorem 1, page 175] the term inside the first curry bracket corresponds to and the second (last) sum corresponds to . Here we agree on that when , the term in the second curry bracket reduces to . This gives where , where the last inequality follows from the fact . Clearly, the right hand side expression of the above inequality goes to infinity as , because and are both bounded and separated from zero, and by supposition. On the other hand, obviously there exists some with , which according to Proposition 3.1 leads to that at least for big enough , outperforms , which is a desired contradiction. Part (a) is thus proved.(b) The proof of this part coincides with the one of Corollary 2.4, and is thus omitted.(c1) Let us notice first of all that under the conditions of the statement, we have that the expression is positive and decreases with . Here and come from replacing by in and . Indeed, as for the positivity part, one only needs to see the denominator if is subject to the given condition. The decreasing (with respect to ) part follows from whenever . Remember that and are both -dependent, and is -independent.Now let us prove (c1) of the corollary. Observe that under the conditions of the statement, Proposition 3.1 implies that For , one can bound from the above the right hand side expression in (A.18) by substituting in it, which leads to (c1).(c2) Let us notice that , and for big enough , . Indeed, due to (b), to justify the second inequality, we only need verify , which is done as follows. For the fluid model, clearly we have where the numerator corresponds to and the denominator corresponds to , and which is negative if . But the latter inequality holds if , because and . This means .Now with the help of (c1) and Proposition 3.1, (c2) can be proved in the same way as for (b) of Corollary 2.5.

Acknowledgments

The authors are grateful to the reviewers for their valuable comments. This research is partially supported by the Alliance: Franco-British Research Partnership Programme, project “Impulsive Control with Delays and Application to Traffic Control in the Internet” (PN08.021). Y. Zhang thanks Mr. Daniel S. Morrison for his comments which improve the English presentation of this paper.

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