The meanfield computation in a supermarket model with server multiple vacations


 Roy Robinson
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1 DOI.7/s The meanfiel computation in a supermaret moel with server multiple vacations QuanLin Li Guirong Dai John C. S. Lui Yang Wang Receive: November / Accepte: 8 October 3 SpringerScienceBusinessMeiaNewYor3 Abstract While vacation processes are consiere to be orinary behavior for servers, the stuy of queueing networs with server vacations is limite, interesting, challenging. In this paper, we provie a unifie effective metho of functional analysis for the stuy of a supermaret moel with server multiple vacations. Firstly, we analyze a supermaret moel of N ientical servers with server multiple vacations, set up an infiniteimensional system of ifferential (or meanfiel equations, which is satisfie by the expecte fraction vector, in terms of a technique of taile equations. Seconly, as N we use the operator semigroup to provie a meanfiel limit for the sequence of Marov processes, which asymptotically approaches a single trajectory ientifie by the unique global solution to the infiniteimensional system of limiting ifferential equations. Thirly, we provie an effective algorithm for computing the fixe point of the infiniteimensional system of limiting ifferential equations, use the fixe point to give performance analysis of this supermaret moel, incluing the mean of stationary queue length in any server the expecte sojourn time that any arriving customer spens in this system. Finally, we use some numerical examples to analyze how the performance measures epen on some crucial factors of this supermaret moel. Note that the metho of this paper will be useful effective for performance analysis of Q.L. Li (B G. Dai School of Economics Management Sciences, Yanshan University, Qinhuangao 664, People s Republic of China J. C. S. Lui Department of Computer Science & Engineering, The Chinese University of Hong Kong, Shatin, NT, Hong Kong Y. Wang Institute of Networ Computing & Information Systems, Peing University, Beijing 87, People s Republic of China
2 complicate supermaret moels with respect to resource management in practical areas such as computer networs, manufacturing systems transportation networs. Keywors Supermaret moel Romize loa balancing Server vacation Join the shortest queue Expecte fraction vector Operator semigroup Meanfiel limit Fixe point Performance analysis Introuction During the last three ecaes consierable attention has been pai to stuying queueing systems with server vacations. Queues with server vacations are always useful in moeling many real life situations such as igital communication, computer networs, prouction/inventory systems, transportation networs business systems. Various queueing moels with server vacations have been extensively reporte by a number of authors, for example, basic vacation policies inclue server multiple vacations, server single vacations, server woring vacations, Npolicy, D policy Tpolicy. Reaer may refer to Taagi (99, Dshalalow (995, 997 Tian Zhang (6 for more etails. In the stuy of queueing systems with server vacations, an important result is stochastic ecompositions of stationary queue length of stationary waiting time. For singleserver queues, the stochastic ecompositions in the M/G/ queue with server vacations were first establishe by Fuhrmann Cooper (985; while in multipleserver queues, the conitional stochastic ecompositions for the M/M/c queues with server vacations were first analyze in Tian et al. (999. Up to now, extensive research on the singleserver (or multipleserver queueing systems with server vacations has been wellocumente, such as, by three survey papers of Doshi (986, 99 Alfa(3, by two boos of Taagi (99TianZhang(6. Until now, the available results of queueing networs with server vacations has been very limite. Note that the supermaret moels are an important class of queueing networs play a ey role in the area of networing resource management, thus the supermaret moel with server vacations is very interesting in the stuy of queueing networs with server vacations, it can also provie some new unersting valuable highlight for the orinary queues with server vacations which are escribe in Taagi (99 Tian Zhang(6. For queueing networs with server vacations, Vveensaya Suhov (5 firstiscussea supermaret moel with server On/Off vacations, analyze the stationary queue length istribution by means of the fixe point. However, the On/Off vacation iscipline is not accurate for unersting the vacation processes, because it is not clear why to begin a vacation how to en this vacation. This motivates us in this paper to further consier a supermaret moel with server multiple vacations, while for other cases such as server single vacations server woring vacations, we can similarly give performance analysis. Note that the results given in Vveensaya Suhov (5 is very interesting, it also inspires us to further provie an effective algorithm to compute the fixe point with respect to the choice number 3,which
3 have not been given a complete solution in the literature up to now. Note that the choice constant N, wheren is the number of servers in the supermaret moel. Dynamic romize loa balancing is often referre to as the supermaret moel. Recently, some supermaret moels have been analyze by means of queueing methos as well as Marov processes. For the simplest supermaret moel (that is, Poisson arrivals exponential service times, Vveensaya et al. (996 applie the operator semigroups of Marov processes to analyze the stationary istribution obtaine an important result: Superexponential ecay tail. The superexponential solution is a substantial improvement of system performance over that in the orinary M/M/ queue. At nearly the same time, Mitzenmacher (996 also analyze the same supermaret moel in terms of the ensityepenent jump Marov processes,e.g., see Kurtz (98.Later, Turner (998 provie a martingale approach to further iscuss this supermaret moel. The path space evolution of the supermaret moel was stuie by Graham (a, b, 4 whoshowethat starting from inepenent initial states, as N the queues of the limiting process evolve inepenently. Lucza Norris (5 provie a strong approximation for the supermaret moel, Lucza McDiarmi (6, 7 showe that the length of the longest queue scales as (log log N/ log O(. The positive Harris recurrence of the Marov processes unerlying some supermaret moels was iscusse in Foss Chernova (998 Bramson(8,. Certain generalization of the supermaret moel has been explore in stuying various variations, for example, moeling more crucial factors by Mitzenmacher (999, Jacquet Vveensaya (998, Jacquet et al. (999 VveensayaSuhov (5; analyzing nonexponential server times or nonpoisson input by Bramson et al. (,,, Vveensaya Suhov (997, Mitzenmacher et al. (, Li et al. (,, Li Lui ( Li(; fast Jacson networs by Martin Suhov (999, Martin ( SuhovVveensaya(. Up to now, there have been three excellent survey papers by Turner (996, Vveensaya Suhov (997 Mitzenmacheretal.(, one boo by Mitzenmacher Upfal (5. The meanfiel equations meanfiel limits play an important role in the stuy of supermaret moels. Reaers may refer to recent publications for the meanfiel moels, among which are Sznitman (989, Vveensaya Suhov (997, Le Bouec et al. (7, Benaim Le Bouec (8, Borenave et al. (9, Gast Gaujal (9,, Gast et al. ( TsitsilisXu(. This paper provies a clear picture for illustrating how to use meanfiel moels to numerically analyze performance measures of complicate supermaret moels, is organize into three ey parts: (Part one setting up system of ifferential equations, see Section. (Part two theoretical support, see Sections 3 4. In Section 3, we use the operator semigroup to give some strict proofs for the meanfiel limit (or propagation of chaos, which shows the asymptotic inepenence of queues in the supermaret moel with server vacations. Section 4 is a necessary supplementary part of the meanfiel limit, in which the Lipschitzian conition is establishe for guaranteeing the existence uniqueness of solution to the system of limiting ifferential equations. (Part three performance analysis, Sections 5 6 provie a novel meanfiel metho for being able to numerically analyze performance
4 measures of this supermaret moel after the basic preparation given in Sections 3 4. Although analysis of the supermaret moel with a finitely big N is very ifficult, we use the meanfiel limit to be able to numerically analyze performance measures of one queue, the information of which will help us to unerst the total behavior of this supermaret moel as N.Itisworthwhiletonotethat some simulations in Bramson et al. (,, inicate that the asymptotic inepenence of queues can be forme well when N. Therefore, the metho of this paper is effective for performance analysis of complicate supermaret moels. The main contributions of this paper are threefol. The first one is to provie aunifieeffectivemethoforsettingupaninfiniteimensionalsystemof ifferential (or meanfiel equations, which is satisfie by the expecte fraction vector in terms of a technique of taile equations. Specifically, we erive an important relation: the invariance of environment factor. Notethattheinvarianceof environment factor plays a ey role in our later stuy with respect to this supermaret moel. The secon contribution is the evelopment of a useful technique for establishing the Lipschitzian conition for the infiniteimensional fraction vector function f : R C (R for the general choice number. Note that the choice number was always assume in several important references, e.g., see Vveensaya Suhov (997, 5 Mitzenmacher et al. (. As seen in this paper, the case with has a special structure in the system of nonlinear equations satisfie by the fixe point, which is easily ealt with from some simple computation; while for the case with 3, this paper gives some new interesting results when establishing the the Lipschitzian conition, which leas to the strict proofs for the meanfiel limit. The thir contribution of this paper is to provie an effective algorithm for computing the fixe point, also to provie performance analysis of this supermaret moel. Note that our algorithm has a ey which has the ability to etermine the bounary probabilities in the system of nonlinear equations satisfie by the fixe point. The remainer of this paper is organize as follows. In Section, weescribe a supermaret moel of N ientical servers with server multiple vacations, introuce the sequence of fraction vectors which express the supermaret moel as infiniteimensional Marov processes, set up an infiniteimensional system of ifferential equations satisfie by the expecte fraction vector in terms of a technique of taile equations. In Section 3, we use the operator semigroup to provie a meanfiel limit for the sequence of Marov processes, which asymptotically approaches a single trajectory ientifie by the unique global solution to the infiniteimensional system of limiting ifferential equations. In Section 4, we provie a unifie effective metho for organizing the Lipschitzian conition for the infiniteimensional fraction vector function f : R C (R. Then we apply the Lipschitzian conition the Picar approximation to show that the limiting expecte fraction vector is the unique global solution to the system of limiting ifferential equations. In Section 5, weprovieaneffectivealgorithmtocompute the fixe point of the infiniteimensional system of limiting ifferential equations. In Section 6, we use the fixe point to give performance analysis of this supermaret moel, incluing the mean of the stationary queue length in any server the expecte sojourn time that any arriving customer spens in this system. Furthermore, we use some numerical examples to analyze how the performance measures epen
5 on some crucial factors of this supermaret moel. Some concluing remars are given in the final section. A supermaret moel with server multiple vacations In this section, we first escribe a supermaret moel of N ientical servers with server multiple vacations. Then we introuce the sequence of fraction vectors, which are use to express the supermaret moel as infiniteimensional Marov processes. Finally, we provie a unifie effective metho to set up an infiniteimensional system of ifferential equations satisfie by the expecte fraction vector of the supermaret moel in terms of a technique of taile equations. The supermaret moel consists of N ientical servers, where each server has an infinite buffer. The service times of each server are i.i.. with an exponential istribution of service rate µ. The vacation process of each server is base on the multiple vacation policy: When there is not any customer at one server its buffer, it immeiately taes a vacation eeps taing vacations until it fins at least one customer waiting in the server or its buffer at the vacation completion instant. The vacation time istribution of each server is exponential with vacation rate θ >. The common input flow is Poisson with arrival rate Nλ for λ >. Uponarrival,each customer chooses servers from the N servers inepenently uniformly at rom, joins the one whose queue length is the shortest. If there is a tie, servers with the shortest queue length are chosen romly. All customers in any server will be serve in the firstcomefirstserve (FCFS manner, the arrival, service vacation processes are inepenent of each other. Figure provies a physical illustration for the supermaret moel of N ientical servers with server multiple vacations. Lemma The supermaret moel of N ientical servers with server multiple vacations is stable if < λ <. Fig. Asupermaretmoel with each customer choosing the loaing of servers Server µ, θ Possion input Nλ Server µ, θ Each customer probes servers Server 3 µ, θ Server N µ, θ
6 Proof If, thenthissupermaretmoelofn ientical servers with server multiple vacations is equivalent to a system of N inepenent M/M/ queues with server multiple vacations. From Chapter of Tian Zhang (6, it is seen that the M/M/ queue with server multiple vacations is stable if ρ λ/µ λ <.Usinga coupling metho, as given in Theorems 4 5 of Martin Suhov (999, it is easy to see that for a fixe number N,, 3,...,thissupermaretmoelofN ientical servers is stable if ρ λ <. This completes the proof.. An infiniteimensional Marov process For this supermaret moel, let L (N be the number of woring servers with at least customers (the serving customer is also taen into account at time t, M (N l the number of vacation servers with at least l customers at time t. We write U (N V (N l L(N N,, M(N l N, l. Clearly, U (N for V (N l for l are the fractions of these woring servers with at least customers at time t the fractions of these vacation servers with at least l customers at time t, respectively.set U (N (U (N, U (N, U (N 3,... V (N (V (N, V (N, V (N,.... It is easy to see that for any given N, U (N V (N are all rom vectors. Base on the exponential or Poisson assumptions of the arrival, service vacation processes, {( U (N, V (N, t } is an infiniteimensional Marov process whose state space E N is given by E N {(,, 3,...;,,,... : 3, 3, N Note that M (N l N l are nonnegative integers for l M (N l for l t, it is obvious that. Similarly, the fact that L (N L (N V (N V (N V (N for t can yiel that U (N U (N U (N.Furthermore, since the two rom variables U (N V (N l tae values in the set {, /N, /N,...,(N /N, } for, l t, this gives that for t, there exist two positive integers K L such that U (N U (N U (N (N K >, U for K ; }.
7 V (N V (N V (N L >, V(N l for l L. To analyze the infiniteimensional Marov process {( U (N, V (N, t } on state space E N,wewrite E U (N,, Let l E V (N l, l. (,, 3,... (,,,.... It is easy to see that with 3. In the remainer of this section, we set up an infiniteimensional system of ifferential equations whose purpose is to be able to etermine the expecte fraction vector (,.. The system of ifferential equations To etermine the expecte fraction vector (,,thissubsectionprovies aunifieeffectivemethotosetupaninfiniteimensionalsystemofifferential equations satisfie by the expecte fraction vector in terms of a technique of taile equations. To that en, we first provie an example with to inicate how to erive these ifferential equations. In the supermaret moel of N ientical servers, we nee to etermine the expecte change in the number of servers with at least customers over a small time perio, t, that is, we shall compute the rate that any arriving customer selects servers from the N servers inepenently uniformly at rom joins the one whose queue length is the shortest. From Figs. 3, itisseenthatanyarriving customer joins either server wors or server vacations among the selecte servers, thus we nee to consier the following two cases: Case one: Entering one woring server. Inthiscase,theratethatanyarriving customer joins a woring server with the queue length the
8 Vacation Vacation Vacation Vacation 3 Woring Woring Woring 3 Service process Arrival process Vacation process Fig. The state transition relation in the M/M/ queue with server vacations m woring servers one with the shortest queue m vacation servers one with the shortest queue Woring server area Vacation server area Vacation server area Woring server area m vacation servers (a m woring servers (b Fig. 3 Two ifferent cases when joining a woring server or a vacation server
9 queue lengths of the other selecte servers are not shorter than is given by where Nλ W (N (u, u ; v,v ; t u(n W (u, u ; v,v ; t t, ( m m m C j m j m C m m m m C m r r m u(n m u(n m j m m Cm m v(n m m j u(n r m r. m m It is necessary to provie a etaile interpretation for how to erive Eq.. Fromthejoiningprocessexpresseby(ainFig.3 from the set ecomposition of all possible events inicate in Fig. 4, itisseen that the probability W (N (u, u ; v,v ; t given in Eq. contain the following three parts. Part I: Neither of the selecte servers is taing a vacation, that is, each of the selecte servers is woring for service. In this case, the probability that any arriving customer joins a woring server with the queue length the queue Fig. 4 Set ecomposition of all possible events when joining a woring server Each of the selecte servers is woring for service, there is at least one woring server with the shortest queue length . (Part I In the selecte servers, there is at least one woring server with the shortest queue length , there exists at least one vacation server while the queue length of each vacation server is more than customers. (Part II In the selecte servers, there are at least one woring server with the shortest queue length  at least one vacation server with the shortest queue length . (Part III
10 Part II: lengths of the other selecte woring servers are not shorter than is given by m m m u(n u(n m m m u(n, where C m!/ m! ( m! is a binomial coefficient, m u(n is the probability that any arriving customer who can only choose one queue maes m inepenent selections uring the m selecte woring servers with the queue length at time t. For the selecte servers, there is at least one woring server with customers, there exist at least one vacation server while the queue length of each vacation server is more than customers. In this case, the probability that any arriving customer joins a woring server with the shortest queue length ; for the other selecte servers, the queue lengths of the selecte woring servers are not shorter than, thereexistatleastonevacationserverwhilethe queue length of each vacation server is more than customers, is given by C m m m C j m j m u(n u(n m C j m j m j C m m j m j j. m u(n Part III: For the selecte servers, there are at least one woring server with customers at least one vacation server with customers. In this case, if there are the selecte m servers with the shortest queue length where there are m woring servers m m vacation servers, then the probability that any arriving customer joins a woring server is equal to m /m. Therefore, the probability that any arriving customer joins a woring server with the queue length,
11 the queue lengths of the other selecte servers are not shorter than, thereareatleastoneworingserver with customers at least one vacation server with customers, is given by m C m m m m C m r r m m m m Cm m u(n m C m r r m u(n r m m m Cm m m m v(n m r m u(n r m r. m m v(n Using the above three parts, Eq. can be obtaine immeiately. Besies the above analysis for the arrival process, in what follows we consier the service vacation processes. The rate that a customer leaves a server queue by customers is given by N t. ( The rate that a server queue by at least customers completes its vacation is given by Using Eqs., 3, weobtain E L (N Nλ N this gives t u(n λ by means of E Nθ t. (3 u(n W (N (u, u ; v,v ; t t t Nθ t, u(n W (N (u, u ; v,v ; t θ (4 L (N /N.
12 Case two: Entering one vacation server. In this case, the rate that any arriving customer joins a vacation server with the queue length the queue lengths of the other selecte servers are not shorter than is given by Nλ v(n V (N (u, u ; v,v ; t t, (5 where V (N (u, u ; v,v ; t m m m C j m j m C m m m m C m r r m v(n m v(n m j m m Cm m u(n m m j v(n r m r. m m Note that Eq. 5 can be erive similarly to that in Case one by means of (b in Figs Using a similar analysis to Eq. 4, it follows from Eq. 5 that t v(n λ v(n V (N (u, u ; v,v ; t θ. (6 (u ; v,v ; t, The following theorem simplifies expressions for V (N V (N (u, u ; v,v ; t W (N (u, u ; v,v ; t for. Note that the simplifie expressions will be a ey in our later stuy. Fig. 5 Set ecomposition of all possible events when joining a vacation server Each of the selecte servers is at vacation, there is at least one vacation server with the shortest queue length . (Part I In the selecte servers, there is at least one vacation server with the shortest queue length , there exists at least one woring server while the queue length of each woring server is more than customers. (Part II In the selecte servers, there are at least one vacation server with the shortest queue length  at least one woring server with the shortest queue length . (Part III
13 Theorem V (N (u ; v,v ; t m m m, for W (N (u, u ; v,v ; t m u(n m m v(n V (N (u, u ; v,v ; t m v(n m. m u(n Hence, for we have W (N (u, u ; v,v ; t V (N (u, u ; v,v ; t. Proof It is easy to see that V (N (u ; v,v ; t m m m m C m r r m j j m m. For, weobtain W (N (u, u ; v,v ; t C m m u(n m u(n m u(n m
14 m C j m j m C m m m m C m r C r m m C j m j v(n m j m m Cm m m m r u(n m C m m m C C j j m m C j m j v(n j m u(n m r m u(n m j m m Cm m m C m m m C C j j v(n j m m m C m r r j m u(n m u(n m j m m Cm m j j m m m C m r r j r m u(n j r
15 m C m m m C m m m v(n m C m m m m m m m Cm m m m Cm m v(n m m m Cm m v(n m m m C m r r m u(n m u(n m m m C m r r m m m u(n m m m m m v(n similarly, we have V (N (u, u ; v,v ; t m m r r m C m m m u(n m u(n v(n, v(n m. m u(n This completes the proof. Set L (N (u ; v,v ; t V (N (u ; v,v ; t for L (N (u, u ; v,v ; t W (N (u, u ; v,v ; t V (N (u, u ; v,v ; t. The sequence: L (N (u ; v,v ; t L (N (u, u ; v,v ; t for, is calle the invariance of environment factor, which will play a ey role in our later stuy with respect to how to set up the system of ifferential equations.
16 Using some similar analysis to Eqs. 4 6,weobtainaninfiniteimensionalsystem of ifferential equations satisfie by the expecte fraction vector (, as follows: t v(n θ, (7 for t v(n t u(n λ t v(n λ L (N (u ; v,v ; t θ, (8 λ u(n with the bounary conition v(n L (N L (N (u, u ; v,v ; t θ (9 (u, u ; v,v ; t θ (, t, ( with the initial conitions { u (N ( g,, where l ( h l, l. g g g 3, h h h, h g. ( Remar If, thenw (N (u, u ; v,v ; t for V (N (u l, u l ; v l,v l ; t for l. Inthiscase,wehave λ v(n t v(n { λ θ. l l u(n l u(n } λ u(n l v(n v(n v(n l θ u(n Therefore, the system of ifferential equations (7 to( is the same as those in Vveensaya Suhov (5..3 A useful probabilistic interpretation In this subsection, we provie a useful probabilistic interpretation for the invariance of environment factor L (N (u ; v,v ; t L (N (u, u ; v,v ; t for, this will help us to further unerst the system of ifferential equations (7to(.
17 Using Theorem, it is easy to chec that V (N (u ; v,v ; t, for v(n L (N (u, u ; v,v ; t { v(n u(n L (N (u, u ; v,v ; t { v(n v(n L (N (u, u ; v,v ; t v,v ; t for, we introuce some notation v(n v(n } u(n v(n } To give the probabilistic interpretation for u(n. u(n u(n V (N (u ; v,v ; t, L (N (u, u ; W V are the events in which any arriving customer is reirecte to a woring server or a vacation server with the queue length,respectively. X enotes the number of times in the romize loa balancing policy we choose a server with the exactly queue length, where we o not istinguish woring vacation servers. Y enotes the number of times in the romize loa balancing policy we choose a server with whose queue length is not shorter than. Now, we compute the two probabilities P {W } P {V } for. Using the law of total probability, we obtain P {W } P {W X m, Y m} P {X m, Y m}. m We can compute the conitional probability P {W X m, Y m} u(n v(n u(n, (3 which is inepenent of the number m. Infact,theconitionalprobabilityiseasyto compute, e.g., by thining in terms of an urn moel with blac white balls, from which one raws m balls, blac ones with probability q u(n v(n u(n
18 white ones with probability q. Then, once m balls are extrace, one raws at rom one ball from the m ones. The probability of having chosen a blac ball is equal to q. By means of Mitzenmacher (996, we obtain P {X m, Y m} m v(n Base on Eqs. 3 4,wehavethefollowingprobabilisticsetting u(n L (N (u, u ; v,v ; t P {W }. Similarly, we have Thus we obtain v(n L (N (u, u ; v,v ; t P {V }. P {W } P {W X m, Y m} P {V } P {V X m, Y m}. (4 P {X m, Y m} m P {X m, Y m}. m 3Ameanfiellimit In this section, we use the operator semigroup to provie a meanfiel limit for the sequence {( U (N, V (N, t } of infiniteimensional Marov processes for N,, 3,...,showthatthissequenceofMarovprocessesasymptotically approaches a single trajectory ientifie by the unique global solution to the infiniteimensional system of limiting ifferential equations. For the two vectors we write N (,, 3,... { (, : 3, (,,,...,, N N l are nonnegative integers for l } N {(, : (, N e e < }, where e is a column vector of ones with a suitable imension in the context. For the two vectors u (u, u, u 3,... v (v,v,v,v 3,...,set {(u, v : u u u 3, v v v v 3 }
19 { (u, v : (u, v ue ve < }. Obviously, N N N. In the vector space, wetaeametric ρ ( (u, v, (u, v { { u u sup max, v v }} (5 for (u, v, (u, v. Note that uner the metric ρ ( (u, v, (u, v, the vector space is separable compact. For (g, h N,wewrite for L (g ; h, h L (g, g ; h, h C m (h h m (h g m, m C m (g g h h m (g h m. m Now, we consier the infiniteimensional Marov process { ( U (N, V (N, t } on state space N (or N in a similar analysis for N,, 3,... Note that the stochastic evolution of this supermaret moel of N ientical servers is escribe as the Marov process {( U (N, V (N, t },where ( U (N, V (N A N f ( U (N, V (N, t where A N acting on functions f : N R is the generating operator of the Marov process {( U (N, V (N, t }, A N A In N AOut N, (6 for (g, h N A In N f (g, h λn (g g L (g, g ; h, h f (g e N, h f (g, h λn (h h L (g ; h, h f (g, h e N f (g, h λn (h h L (g, g ; h, h f (g, h e N f (g, h (7
20 A Out N N θ N (g g f (g e N, h f (g, h h f (g e N, h e N f (g, h, (8 where e sts for a row vector with the th entry all others. For (g, h N, it follows from Eqs. 6 to 8 that A N f (g, h λn (g g L (g, g ; h, h f (g e N, h f (g, h λn (h h L (g ; h, h f (g, h e N f (g, h λn (h h L (g, g ; h, h f (g, h e N f (g, h N θ N (g g f (g e N, h f (g, h h f (g e N, h e N f (g, h The operator semigroup of the Marov process {( U (N, V (N, t } is efine as T N, whereif f : N C,thenfor ( g, h N t (9 T N f (g, h E f (U N, V N U N ( g, V N ( h. ( Note that A N is the generating operator of the operator semigroup T N, itiseasy to see that T N exp {A N t} for t. Definition Aoperatorsemigroup{S : t } on the Banach space L C( is sai to be strongly continuous if lim t S f f for every f L; itissaitobea contractive semigroup if S for t. Let L C( be the Banach space of continuous functions f : R with uniform metric f max f (u, similarly, let L N C( N. The inclusion u N inuces a contraction mapping N : L L N, N f (u f (u for f L u N. Now, we consier the limiting behavior of the sequence {(U (N, V (N, t } of Marov processes for N,, 3,... Two formal limits for the sequence {A N } of generating operators for the sequence {T N } of semigroups are expresse as A lim N A N T lim N T N for t, respectively. It follows from
21 Eq. 9 that as N A f (g, h λ We write for (g g L (g, g ; h, h g f (g, h λ (h h L (g ; h, h f (g, h h λn (h h L (g, g ; h, h f (g, h h (g g f (g, h θ g L (u ; v,v ; t h f (g, h g C m v v m v u m, m L (u, u ; v,v ; t m f (g, h. ( h u u v v m u v m. Let u lim N v lim N for t, where u lim N for v l lim N l for l. Baseonthelimiting generating operator A given in Eq., asn it follows from the system of ifferential equations (7 to( that(u, v is a solution to the following system of ifferential equations t v u u θv, ( for t v λ v v L (u ; v,v ; t θv, (3 t v λ v v L (u, u ; v,v ; t θv, (4 t u λ u u L (u, u ; v,v ; t u u θv, with the bounary conition (5 v u, t, (6
22 with the initial conitions { u ( g,, v l ( h l, l. (7 Remar In the next section, we shall prove that the vector (u(t, g, h, v(t, g, h is the unique global solution to the system of ifferential equations (to(7for t,whereu(, g, h g v(, g, h h. We efine a mapping: ( g, h (u(t, g, h, v(t, g, h, where ( u(t, g, h, v(t, g, h is a solution to the system of ifferential equations ( to(7. For the operator semigroup T acts in the space L. If f L (g, h, then T f (g, h f (u(t, g, h, v(t, g, h. (8 From Eqs. 9, it is easy to see that the operator semigroups T N T are strongly continuous contractive, see, for example, Section. in Chapter one of Ethier Kurtz (986. We enote by D(A the omain of the generating operator A. ItfollowsfromEq.8 that if f is a function from L has the partial erivatives g i f (g, h for i f (g, h L for j, sup i, j h j { } g i f (g, h, h j f (g, h <, then f D(A. Let D be the set of all functions f L that have the partial erivatives g i g j f (g, h, such that f (g, h, { sup i,i g i g i j, j (g,h g i h j f (g, h h j g i f (g, h, h i h j f (g, h, there exists C C( f < { } sup f (g, h i, j g, f (g, h i h < C (9 j f (g, h, f (g, h g i h, } f (g, h j h j h < C. (3 j We call that f L epens only on the first K two imensional variables if for (g (, h (, (g (, h (, it follows from g ( i g ( i for i K h ( j h ( j for j K that f (g (, h ( f (g (, h (.Asimilarsimpleprooftothatin Proposition in Vveensaya et al. (996 can show that the set of functions from L that epens on the first finite two imensional variables is ense in L. The following lemma comes from Proposition in Vveensaya et al. (996. We restate it here for convenience of escription. Lemma Consier an inf initeimensional system of if ferential equations: For, z ( c z t z i a i, b, i
23 let ai, a, b b exp {bt}, c ϱ, b a < b. Then i z ϱ exp {at} b exp {bt} exp {at}. b a Definition Let A be a close linear operator on the Banach space L C(. A subspace D of D (A is sai to be a core for A if the closure of the restriction of A to D is equal to A,i.e.,A D A. We introuce some notations M 3 4 M M C m m m (, m C m ( m m m m ( (, C m (m m m m C m ( m( m m m 6 m 6 C m ( m(m m m m m (m ( m C m (m (m m 3 m m ( m( m, a M θ M a M θ M. The following lemma is a ey to prove that the set D is a core for the generating operator A. Lemma 3 Let (u, v be a solution to the system of if ferential equations ( to (7. Then { } u (t, g, h g, u (t, g, h i h exp {(M θ M t}, (3 j sup i, j sup i, j { } v (t, g, h g, v (t, g, h i h exp {(M θ M t}, (3 j
24 { u (t, g, h sup g i g, u (t, g, h i g i h, } u (t, g, h j h j h j i,i j, j M M 3 exp {at} exp {at} M θ M (33 sup i,i j, j { v (t, g, h g i g, v (t, g, h i g i h, } v (t, g, h j h j h j M M 3 M θ M exp { a t } exp { a t }. (34 Proof We only prove (3, while (3to(34canbeprovesimilarly. It is easy to verify that the solution (u, v to the system of ifferential equations (to(7possessesthepartialerivatives u (t, g, h g j, u (t, g, h h j, v (t, g, h g j, v (t, g, h h j, u (t, g, h g i g j, u (t, g, h g i h j, u (t, g, h h i h j, v (t, g, h g i g j, v (t, g, h g i h j, v (t, g, h h i h j. In what follows we only compute the two erivatives u(t,g,h g j v(t,g,h h j,while the other erivatives can be compute similarly. For simplicity of escription, we write that u u (t, g, h, u, j u(t,g,h v (t,g,h h j. It follows from Eqs., 5 6 that for all, j, t u, j u, j u, j θv, j g j or v, j u, j t ( ( λ u, j u, j L (u, u ; v,v u, j u, j θv, j λ (u u L, j (u, u ; v,v,
25 L, j (u, u ; v,v ; t C m ( m(u u v v m m (v u m ( v, j u, j C m (m (u u v v m m ( (v u m u, j u, j v, j v, j. Using Lemma, we obtain Inequalities (3with a M θ M, a M θ M, b, b, ϱ. This completes the proof. Lemma 4 The set D is a core for the generating operator A. Proof It is obvious that D is ense in L D D(A. Let D be the set of functions from D, which epen only on the first finite two imensional variables. It is easy to see that D is ense in L. Using Proposition 3.3 in Chapter of Ethier Kurtz (986, it can show that for any t, the operator semigroup T oes not bring D out of D. Select an arbitrary function ϕ D let f (g, h ϕ(u(t, g, h, v(t, g, h for (g, h. ItfollowsfromLemma3that f has the partial erivatives u (t, g, h g j, u (t, g, h h j, v (t, g, h g j, v (t, g, h h j, u (t, g, h g i g j, u (t, g, h g i h j, u (t, g, h h i h j, v (t, g, h g i g j, v (t, g, h g i h j, v (t, g, h h i h j. they satisfy the inequalities (3 to(34. Therefore f D. This completes the proof. The following theorem applies the operator semigroup to provie an meanfiel limit, which shows that the sequence {( U (N, V (N, t } of Marov processes asymptotically approaches a single trajectory ientifie by the unique global solution to the infiniteimensional system of ifferential equations (to(7. Theorem For any continuous function f : R t >, lim sup TN f (g, h f (u(t, g, h, v(t; g, h, (g,h N N the convergence is uniform in t within any boune interval.
26 Proof This proof is to use the convergence of the operator semigroups as well as the convergence of their corresponing generating generators, e.g., see Theorem 6. in Chapter of Ethier Kurtz (986. Lemma 4 shows that the set D is a core for the generating operator A. For any function f D, wehave N f (g e N, h f (g, h f (g, h g N f (g, h e N f (g, h f (g, h h γ, (g f (g γ, (g e N where < γ i, (g, γ i, (h < for i,. Since γ, (g f (g γ, (g e N, h N C N g g N, h, γ, (h f (g, h γ, (h e N g N, γ, (h f (g, h γ, (h e N g N C N, we obtain A N f (g, h f (g, h C (g g L (g, g ; h, h N (h h L (g, g ; h, h (h h L (g ; h, h (g g θ h { C } M (g l g l (h h g θ h N l C M (g h g θ h. N Note that C, M h are all finite, it is clear that as N, lim sup A N f (g, h A f (g, h. (g,h N N This completes the proof. Remar 3 ( As iscusse in Ethier Kurtz (986, there have been at least three basic techniques: operator semigroup, martingale ensity epenent population process, for analyzing the wea approximation of the sequences of Marov processes. In fact, the three techniques have been applie to the meanfiel limit
27 of supermaret moels up to now, e.g., see the operator semigroup by Vveensaya et al. (996, the ensity epenent population process by Mitzenmacher (996 the martingale by Turner (996, 998. In this section, we use the operator semigroup to provie a meanfiel limit for the supermaret moel with server multiple vacations, show that the sequence of corresponing Marov processes asymptotically approaches a single trajectory ientifie by the unique global solution to the infiniteimensional system of meanfiel limit equations. ( The meanfiel limit over the finite time intervals has been generalize in the PhD thesis of Mitzenmacher (996 to the infiniteimensional case when the righth sie of the system of ifferential equations is Lipschitz. Therefore, using the generalize results by Mitzenmacher (996, we may significantly simplify some results in this section, also we can possibly obtain a stronger form of convergence some error bouns. Furthermore, reaers may refer to Graham (a, b, 4 Lucza McDiarmi (6, 7 forthe longest queue length, the asymptotic inepenence chaoticity on path space. 4Existenceuniqueness In this section, we first provie an effective technique to organize the Lipschitzian conition of the infiniteimensional fraction vector function F : R C (R. Note that the Lipschitzian conition for 3 is always ifficult in the literature. Then we apply the Lipschitzian conition, together with the Picar approximation, to show that the limiting expecte fraction vector is the unique global solution to the system of ifferential equations. For convenience of escription, we write that u u (t, g, h for v l v l (t, g, h for l. Usingv (t, g, h u (t, g, h, the system of ifferential equations (to(7 can be simplifie as an initial value problem as follows for, t v u u θv, (35 t v λ (v v L (u ; v,v ; t θv, (36 t v λ (v v L (u, u ; v,v ; t θv (37 t u λ (u u L (u, u ; v,v ; t (u u θv, (38 with the bounary conition with the initial conition v u (39 { u ( g,, v l ( h l, l. (4
28 4. A Lipschitzian conition To establish the Lipschitzian conition, we nee to use the erivative of the infiniteimensional function G : R C (R. Thus it is necessary to provie some useful notation efinitions of erivatives as follows. For the infiniteimensional function G : R C (R,wewritex (x, x, x 3,... G(x (G (x, G (x, G 3 (x,..., wherex G (x are scalar for. Then the matrix of partial erivatives of the infiniteimensional function G(x is efine as G (x x G (x DG(x x G (x x 3. G (x x G (x x G (x x 3. G 3 (x x G 3 (x x G 3 (x. (4 x 3. Now, we efine two classes of erivatives for the infiniteimensional function G(x. In fact, they are a irect minor generalization from the erivatives of finiteimensional functions, e.g., see Chapter of Taylor (996 Chapter 3 of Fleming (977 for more etails. Definition 3 Let the infiniteimensional function G : R C (R. ( If there exists a linear operator A : R R a scalar t R such that for any vector h R G (x ht G (x hat lim, t t then the function G (x is calle to be Gateaux ifferentiable at x R.Inthis case, we write the Gateaux erivative G G (x A. ( If there exists a linear operator B : R R such that for any vector h R G (x h G (x hb lim, h h then the function G (x is calle to be Frechet ifferentiable at x R.Inthis case, we write the Frechet erivative G F (x B. It is easy to chec that if the infiniteimensional function G (x is Frechet ifferentiable, then it is also Gateaux ifferentiable. At the same time, we have G G (x G F (x DG(x. (4
29 Let t (t, t, t 3,... with t for. Then we write DG(x t (y x G (x t (y x x G (x t (y x x G (x t (y x x 3. G (x t (y x x G (x t (y x x G (x t (y x x 3. G 3 (x t 3 (y x x G 3 (x t 3 (y x x G 3 (x t 3 (y x. x 3. The following lemma provies two useful properties for the Gateaux erivatives of the infiniteimensional functions. Obviously, the two useful properties also hol for the Frechet erivatives. Lemma 5 If the inf initeimensional function G : R C (R is Gateaux ifferentiable, then there exists a vector t (t, t, t 3,... with t for such that Furthermore, we have G (y G (x (y x DG(x t (y x. (43 G (y G (x sup DG(x t (y x y x. (44 t Proof For the function G (x,itiseasytochecthatthereexistsanumbert, such that G (y G (x Note that we obtain Since it follows i (y i x i G (x t (y x x i ( G (x t (y x (y x, G (x t (y x T,.... x x G (y G (x (G (y G (x, G (y G (x, G 3 (y G 3 (x,..., G (y G (x (y x DG(x t (y x. DG(x t (y x sup DG(x t (y x, t G (y G (x sup DG(x t (y x y x. t This completes the proof.
30 Note that v u, wesetthatx (u,v ; u,v ; u 3,v 3 ; u 4,v 4 ;... F(x (F (x, F (x, F 3 (x,..., where F (x (u u θv, λ(v v L (u ; v,v ; t θv (45 F (x (λ(u u L (u, u ; v,v ; t (u u θv, λ(v v L (u, u ; v,v ; t θv. (46 Then F (x is in C ( R, the system of ifferential vector equations (35to(4 is rewritten as with initial conition x F (x (47 t x ( (g, h ; g, h ; g 3, h 3 ; g 4, h 4 ;.... (48 In what follows we show that the infiniteimensional function F(x is Lipschitzian for t. From ( of Definition 3 Eq. 4, the matrix of partial erivatives of the function F(x is given by A (x B (x C (x A (x B 3 (x DF(x C (x A 3 (x B 4 (x, ( where A (x is a matrix of size for, thesizesofalltheotherscanbe etermine accoringly. Let for Then for A (x L L (u ; v,v ; t V (u ; v,v ; t L W (u, u ; v,v ; t V (u, u ; v,v ; t. A (x θ λ (v v L u λl λ (v v L v, θ λl λ (u u L u λ (v v L u λ (u u L v θ λv λ (v v L v, θ B (x λl λ (u u L u λ (u u L v λ (v v L u λv λ (v v L v,
31 C j (x (, j, where L u C m ( m v v m v u m, m for L u L v C m ( m v v m v u m m C m ( m v v m v u m, m C m ( m u u v v m u v m m C m ( m u u v v m u v m, m L v C m ( m u u v v m u v m m C m ( m u u v v m u v m, m L u C m (m u u v v m u v m, m L v C m (m u u v v m u v m. m Hence it follows from Eq. 49 that { e T DF(x max A (x B (x, sup e T C (x A (x B (x }. (5 Let M 5 m ( m.
32 Base on the expression L (u ; v,v ; t L (u, u ; v,v ; t for, itis easy to chec that for L (u ; v,v ; t M L (u, u ; v,v ; t M. At the same time, we have for L u L u L v L v L u L v M 5, M 5, M 5. Note that e T A (x C (x ( L θ, λv λ (v v L θ, u v we obtain e T A (x B (x λm λm 5 θ. (5 Since for e T A (x B (x C (x ( L λ (u u u L u L v L θ, v L λ (v v L L L u u v v θ, we obtain e T A (x B (x C (x 4λM5 θ. (5 Then it follows form Eqs. 5, 5 5 that DF(x M, where M max (λm λm 5 θ, 4λM 5 θ.
33 Note that for (u, v, (ũ, ṽ, itfollowsfromeq.43 that F(u, v F(ũ, ṽ (u, v (ũ, ṽ DF((u, v t (u, v (ũ, ṽ from Eq. 44 that F(u, v F(ũ, ṽ sup DF((ũ, ṽ t (u, v (ũ, ṽ (u, v (ũ, ṽ t M (u, v (ũ, ṽ. (53 Therefore, the function F(u, v is Lipschitzian for (u, v. Remar 4 Let G : R m R n be continuously ifferentiable. Then Proposition 4.5 or Proposition 4.4 in Fleming (977 prove that the function G is locally Lipschitzian for t. Note that Eq. 53 extens the Lipschitzian conition to the infiniteimensional continuously ifferentiable function F : R R,sucha generalization is always necessary in the stuy of supermaret moels. Remar 5 The Lipschitzian conition for 3 is always ifficult in the literature, while few available results were organize in the supermaret moels with,e.g., see Vveensaya Suhov (997, 5Mitzenmacheretal.(. Therefore, here we provie a unique effective metho to compute the Lipschitzian conition for more complicate supermaret moels with. 4. The Picar approximation In this subsection, we apply the Lipschitzian conition, together with the Picar approximation, to show that the limiting expecte fraction vector is the unique global solution to the system of ifferential equations. Let v u x (u,v ; u,v ; u 3,v 3 ; u 4,v 4 ;...Wewrite {x : u u, v v v 3 } It follows from Eqs. 47 to 48 that for x this gives where x x ( t F(x (ss, x (g, h t F(x (ss, (54 ( g, h (g, h ; g, h ; g 3, h 3 ; g 4, h 4 ;.... Base on the integral equation (54, the following theorem inicates that there exists the unique global solution to the system of ifferential equations (35 to (4fort. Theorem 3 For (g, h, there exists the unique global solution to the Eq. 54 for t.
34 Proof We tae the Picar sequence as follows for n It follows from Eq. 53 that x, x n (g, h t F ( x n (s s. x n x n t F ( x n (s F ( x n (s s Mt x n x n (Mtn x x. (n! From the bounary conition: x x, itisclearthatif t /M, thenlim n (Mt n /(n!, whichleastothatasn,thepicar sequence {x n } is uniformly convergent for t, /M. Let x lim n x n for t, /M. Then x is a solution to Eq. 54 for t, /M. Let y is another solution to Eq. 54 for t, /M. Then it is easy to chec that this gives that for t, /M, x y (Mtn (n!, y lim n x n x. This shows that x is the unique solution to Eq. 54 for t, /M. We consier the following equation Tae the Picar sequence for n x x( t M F(x (ss. M x, x n x( t M F ( x n (s s. It is easy to show that x lim n x n is the unique solution to Eq. 54 for t /M, /M. We assume that for l, x is the unique solution to Eq. 54 for t /M,( /M. Then for l, we consier the following equation ( t x x M M M F(x (ss.
35 Tae the Picar sequence x for n x n x( t M M F(x n (ss. It is easy to inicate that x lim n x n is the unique solution to Eq. 54 for t ( /M,( /M. By inuction, it is easy to show that x lim n x n is the unique solution to Eq. 54 for t l/m,(l /M for l,,,... Note that,, M M, M M, 3, M thus, x is the unique global solution to Eq. 54 for t. This completes the proof. Remar 6 Comparing with the finiteimensional system of integral equations (e.g., see Chapter of Hale (98, Theorem 3 maes some necessary generalization of the Picar approximation in orer to eal with existence uniqueness of solution to the infiniteimensional system of integral equations. Note that such a generalization is always necessary in the stuy of supermaret moels. Remar 7 Note that the infiniteimensional system of ifferential equations for the supermaret moel with server multiple vacations is efine on a Banach space, the existence uniqueness of solution is immeiately obtaine ue to the existing results on Banach spaces given that the righth sie of the infiniteimensional system of ifferential equations is Lipschitzian. 5 The fixe point In this section, we analyze the fixe point of the infiniteimensional system of ifferential equations ( to(7, set up an infiniteimensional system of nonlinear equations satisfie by the fixe point. Base on this, we provie an effective algorithm for computing the fixe point. Note that the fixe point is a ey in performance analysis of this supermaret moel. Let π (π (W, π (W, π (W 3,...; π (V, π (V, π (V, π (V 3,... The row vector π is calle a fixe point of the infiniteimensional system of ifferential equations ( to(7 if π lim t (u, v, whereπ (W lim t u for π (V l lim t v l for l.itiswellnownthatifπ is a fixe point of the limiting expecte fraction vector (u, v, then lim t t u, lim t t v. (55
36 For two sequences of positive numbers {x : } {y l : l } with x y, we introuce two function notations for L (x ; y, y C m (y y m (x y m, m L (x, x ; y, y C m (x x y y m (x y m. m Taing t in both sies of the system of ifferential equations ( to(7, we obtain π (W π (W θπ (V, (56 ( λ π (V π (V L π (W ; π (V, π (V θπ (V, (57 for, ( λ π (W π (W L π (W, π (W ; π (V, π (V π (W π (W θπ (V (58 ( λ π (V π (V L π (W, π (W ; π (V, π (V θπ (V, (59 with the bounary conition π (V π (W. (6 To solve the system of nonlinear equations, it is a ey to first compute the bounary probabilities π (V π (W in the fixe point. The following theorem provie an effective metho to etermine the bounary probabilities. Theorem 4 If λ <,thenthebounaryprobabilitiesπ (V π (W are given by Proof For,wehave ( π (V π (V L π (W ; π (V, π (V π (V λ π (W λ. m π (V π (W m m π (V π (V π (V π (W π (V π (W,
37 ( π (V π (V L π (W, π (W ; π (V, π (V π (V π (V m π (V π (V π (V π (W π (W π (V π (V π (V π (V { π (V π (W π (W π (W π (V m π (W m } π (W ( π (W π (W L π (W, π (W ; π (V, π (V π (W π (W π (V π (V π (W π (W { π (V π (W π (V } π (W, this gives ( π (V π (V L π (W ; π (V, π (V ( π (V π (V L π (W, π (W ; π (V, π (V ( π (W π (W L π (W, π (W ; π (V, π (V π (V π (W π (V π (W π (V π (W, { π (V π (W π (V Together with π (V π (W, it follows from Eqs. 56 to 6 that { ( π (W λ π (W π (W L π (W, π (W ; π (V, π (V π (V π (V L ( Thus, using π (V π (W we obtain π (W ; π (V, π (V } π (W ( } π (V π (V L π (W, π (W ; π (V, π (V λ. (6 π (V λ. This completes the proof.
38 Remar 8 Note that ( π (W π (W L π (W, π (W ; π (V, π (V ( π (V π (V L π (W, π (W ; π (V, π (V, ( π (V π (V L π (W ; π (V, π (V the sequence { ( ( π (W π (W L π (W, π (W ; π (V, π (V ; π (V π (V L π (W ; π (V, π (V, ( } π (V π (V L π (W, π (W ; π (V, π (V : is a probability vector. Now, we provie an iterative algorithm for computing the fixe point of the infiniteimensional system of ifferential equations (56to(6. We efine F (x θ x λ (ξ x L (δ, δ ; ξ, x,. (6 We assume that ξ λ δ λ. UsingEq.6, wetaethatπ (V ξ δ. We enote by ξ asolutionin(, ξ to the nonlinear equation π (W set F (x θ x λ (ξ x L (δ ; ξ, x, (63 Let ξ be a solution in (, ξ to the nonlinear equation set δ δ θξ. (64 F (x θ x λ (ξ x L (δ, δ ; ξ, x, (65 δ 3 δ θξ λ (δ δ L (δ, δ ; ξ, ξ. (66 We assume that the pairs (ξ, δ, (ξ, δ,...,(ξ, δ have been obtaine iteratively. Then we enote by ξ asolutionin(, ξ to the nonlinear equation F (x, set δ δ θξ λ (δ δ L (δ, δ ; ξ, ξ. (67 It is clear that < ξ < ξ < < ξ < ξ λ < δ < δ < < δ < δ λ. The following theorem expresses the fixe point of the infiniteimensional system of ifferential equations (56to(6by means of the iterative algorithm.note that it is a ey in the proof that we nee to inicate the uniqueness of the sequences {(ξ, δ : }.
39 Theorem 5 If λ <, then the fixe point π (π (W, π (W, π (W 3,...; π (V, π (V, π (V,...is uniquely given by π (V l ξ l, l, π (W δ,. Proof It is obvious that π (V ξ λ π (W δ λ. It follows from Eq. 57 that ( λ π (V π (V L π (W ; π (V, π (V θπ (V, that is Let λ m m ( m λ π (V π (V λ θπ (V. F (x θ x λ C m λ xm (x λ m m θ x λ λ x λ. Then F ( λ λ <, F (ξ θξ > for x (, ξ x F (x θ λ x λ >. Note that F (x is a continuous function for x (, ξ, thus there exists a unique positive solution x ξ to the nonlinear equation F (x for x (, ξ. Hence, π (V ξ.itfollowsfromeq.56 that π (W π (W θπ (V δ θξ δ. It follows from Eq. 59 for that ( λ π (V π (V L π (W, π (W ; π (V, π (V θπ (V, that is λ ξ π (V L ( δ, δ ; ξ, π (V θπ (V. Let F (x θ x λ ξ x L (δ, δ ; ξ, x ξ x θ x λ (ξ δ (x δ. ξ x δ δ
40 Then F ( λξ L (δ, δ ; ξ, <, F (ξ θξ > for x (, ξ ( ( x F (x θ λ (ξ x(ξ δ λ (ξ x(x δ x ξ x δ δ x ξ x δ δ θ λ (ξ δ (ξ x δ δ (ξ x δ δ λ (ξ x(ξ δ (ξ x δ δ λ (x δ (ξ x δ δ (ξ x δ δ λ (ξ x(x δ (ξ x δ δ λ (ξ x(x δ (ξ x δ δ (ξ x δ δ >. Since F (x is a continuous function for x (, ξ, there exists a unique positive solution x ξ to the nonlinear equation F (x for x (, ξ.hence,π (V ξ. It follows from Eqs that ( π (W 3 π (W θπ (V λ π (W π (W L π (W, π (W ; π (V, π (V δ θξ λ δ δ L (δ, δ ; ξ, ξ δ 3. We assume that for l, π (V ξ π (W δ, where < ξ < ξ < < ξ < ξ λ < δ < δ < < δ < δ λ. Then for l, itfollows from Eq. 59 that ( λ π (V π (V L π (W, π (W ; π (V, π (V θπ (V, that is λ ξ π (V L ( δ, δ ; ξ, π (V θπ (V. Let F (x θ x λ ξ x L (δ, δ ; ξ, x ξ x θ x λ (ξ δ (x δ. ξ x δ δ Then F ( λξ L (δ, δ ; ξ, <, F (ξ θξ > for x (, ξ ( ( x F (x θ λ (ξ x(ξ δ λ (ξ x(x δ x ξ x δ δ x ξ x δ δ θ λ (ξ δ (ξ x δ δ (ξ x δ δ λ (ξ x(ξ δ (ξ x δ δ λ (x δ (ξ x δ δ (ξ x δ δ λ (ξ x(x δ (ξ x δ δ λ (ξ x(x δ (ξ x δ δ (ξ x δ δ >.
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