^"i%i HD28 .M414 no. 35/9- ^3 ALFRED P. WORKING PAPER SLOAN SCHOOL OF MANAGEMENT Fluid and Diffusion Approximations of a Two-Station Mixed Queueing Network Vien Nguyen WP# 3519-93-MSA January 1993 MASSACHUSETTS INSTITUTE OF TECHNOLOGY 50 MEMORIAL DRIVE CAMBRIDGE, MASSACHUSETTS 02139 Fluid and Diffusion Approximations of a Two-Station Mixed Queueing Network Vien Nguyen WP# 3519-93-MSA January 1993 ^^-^^4 1993 Fluid and Diffusion Approximations of A Two-Station Mixed Queueing Network Vien Nguyen Sloan School of Management, M.I.T., Cambridge, MA 02139 Abstract The subject of this paper is a two-station mixed queueing network with two customer types: "Open" customers enter the network at station 1 and depart the system after receiving service. Meanwhile, a fixed number of "closed" customers circulate between stations 1 and 2 indefinitely. Such a mixed queueing network model can represent a single-stage production system that services both make-to-order for this network under the first-in-first-out service of the workload process at station This result is We present fluid and diffusion limits discipline. We find that the heavy traffic limit Brownian motion (RBM) on a finite interval. and make-to-stock customers. 1 is a reflected surprising in light of the behavior of the original mixed network model, in which the workload at station KEYWORDS: mixed 1 need not be bounded. queueing networks, make-to-order production, make-to-stock production, fusion approximation, reflected Brownian motion, performance analysis. Contents: 2. A A 3. The System Processes 4. The Limit Theorems 5. Proof of the Fluid Approximation 6. Proof of the Diffusion Approximation 7. Refining the Brownian Approximation 8. Numerical Examples 9. Appendix 1. Two-Station Mixed Queueing Network Make-to-order/Make-to-stock Production System - References January. 1993 dif- 'Open" customers c' X la m,. C, 'Closed" customers N Figure is A Two-Station Mixed Queueing Network A Two- Station Mixed Queueing Network 1 This paper 1: devoted to the analysis of the network pictured stations serving both "open" and "closed" customers. 1 and depart the network stations 1 and 2 departures, the by .V the number Open customers after being served. Closed customers, [3], We mi and m2 will on the other hand, circulate between to be the mean and 1 let One traffic intensities at stations 1 pi = Amo + P2 = I. expects that the is, approximations of We and /. denote be the Pi = Amo + P2 = Qr^- traffic intensity at station 2 Oy — 1 (1.3)-( 1.4) as A^ when In addition, let ('o(0 mean service time of 1 and is l/m2. Consequently, we repsectively, to be 2, mi m2 :i is given by a'^/m2. where traffic intensities are 1) oy is given by —^mi (13) m2 (1.-1) approaches 1 as the number of closed customers — oo and accordingly, equations (1.1)-(1.2) can N is ht^ taken as large. are interested in the process 1^1(0. defined to be the total at time We :i.2) than one. Moreover, the actual strictly less increases; that time mj < m2- From Chen and Mandelbaum For each finite N, the actual throughput rate of closed customers 1 mo one can verify that the relative throughput rate of closed customers number in service times of closed customers at stations aissume throughout this paper that can define the "relative" a enter the network at station of closed customers in the system. these customers. Set respectively. which consists of two of closed customers in the network remains constant Let A be the arrival rate of open customers at station 2, 1. Because there are no external arrivals of closed customers nor are there for service. number Figure in amount of work found at station and Ui(t) be that part of the workload corresponding 1 to open and closed customers, respectively. We show will in this paper that when station is 1 nearly saturated, namely Pj = Amo when the the following approximation holds is a reflected m2 »s 1, closed customer population « Here, l^i'() H is A'^ large: iW^{-),Uoi-)M:{-)). Brownian motion on the interval [0, 1712] 9- = N(\mo + — -l) CT- = Xml(cl + ct) with (1.5) drift and variance parameters (1.6) 2 + '^ic\ + c^^. (1.7) Moreover, the partial workloads corresponding to open and closed customers live in fixed propor- tions according to t^oC) These results seem counterintuitive = (^ - 1) in light where the workload process at station 1 Our if goal in this paper one is = AmotVrCO- of the behavior of the original mixed queueing network, On the other hand, neither (1.5) nor (1.8) should be familiar with certain properties of queueing networks in heavy traffic. to prove a heavy traffic limit is (1.8) need not be bounded above and the partial workloads are not subject to deterministic relationships. completely surprising ^i'(*) theorem that justifies the approximation stated in (1-5). For the reader to better understand the contributions of this paper, results within the context of diffusion Queueing networks are said it is helpful to cast our approximations of open and closed queueing network models. to be "multiclass" if the service time distribution as well as the routing of customers at each station can depend on the class designation of the customer. class" networks, "single- customers at each station are indistinguishable, meaning their service times are identically distributed and workload processes is all customers at each station follow the same routing mechanism. For Markovian routing, Reiman single-class networks with a reflected Brownian motion in [19] proved that the diffusion limit of the the positive orthant. a similar result for multiclass networks in which the routing the In same work, Peterson also showed that the is class specific by fixed proportions of the overall workload at that station. essentially requires that all Peterson [18] proved deterministic and feedforward. In workloads at each station are given The feedforward structure, which customers travel from lower numbered stations to higher numbered ones, turns out to be an important restriction. The generalization of Petersons work to include routing with feedback has proved to be quite difficult and the source of the difficulties contains deep and subtle theoretical issues. In the case of a multiclass single-queue network with feedback. Reiman [20] was able to prove a theorem approximation of the workload process to justify the by a one-dimensional reflected Brownian motion, and the proof due to Reiman was subsequently simplified by Dai (12, 13] insights drawn from these results. Harrison and Nguyen The Brownian system model proposed by Harrison and Nguyen Brownian motion on the nonnegative orthant, and reflected if With [7]. proposed a Brownian system model to approximate a general multiclciss queueing network with feedback. RBM and Kurtz was any queueing network. well defined for it is. essence, a in was generally thought that such an Nguyen Indeed, Dai and [8] have shown that the vector of wokload processes were to converge to any continuous limit, then that limit must be the Brownian system model described in [12, Wang However, an example by Dai and 13]. [9] conclusively verified that there exist queueing networks for which Harrison and .Nguyen's Brownian approximation do not irregularities [21], much Whitt exist. [23] provided another example that further illuminated the and nonconvergence of the workload process. Due progress has been made toward to the work of Taylor and Williams identifying sufficient and necessary conditions for the RBM's. However, the convergence of open multiclass queueing networks existence and uniqueness of with feedback remains a wide open question. Similar progress has been network models. the area of diffusion approximations for closed queueing workload process in 4] have proved fluid such a network is an RBM and diffusion [6] all theorems limit particular, the diffusion Extension of Chen difficulties as in the for a closed [21], open manufacturing job classes which are served at a station share a From Taylor and Williams RBM same propose a diffusion approximation system, but with the restriction that service time distribution. In on a simplex. to the multiclass networks involves the Dai and Harrison weak sense) a unique [3, queueing networks with .Markovian routing. and Mandelbaum's work counterpart. in Chen and Mandelbaum for single class closed limit of the made common one can verify that there exists (in a corresponding to the proposed approximation. However, there are no proofs to verify that the workload processes in fact converge to the said RBM. one may suspect that mixed queueing network models, as a combination In light of these results, of open and closed queueing networks, will exhibit similar properties under the diffusion limits. That is, the diffusion limit of the workload process can be cast in the form of a reflected Brownian motion, and on an in interval. particular, the workload due to closed customers can be expressed as an Moreover, one can conclude from the theory of multiclass queueing networks that the class-specific workload proceses at each station live processes due to open and closed customers at station in fixed I proportions, so that the workload is Because station substantially networks. more One may 1 is in then 1 must be an RBM on a finite essence a multiclass station, however, the proof of this network intricate than the corresponding proof of the single-class open and closed therefore view mixed queueing networks as an intermediate stepping stone between the well understood single models. It are deterministically related follows from these observations that the workload process at station interval. RBM class networks and the more challenginR multiclass queueing The remainder of the paper and proof, we discuss is Before we turn to problem formulation organized as follows. is next section a make-to-order/make-to-stock production system that in the naturally modelled by the mixed queueing network under study. Section 3 defines the processes that we use in our analysis. Our main results are stated theorems are then given in Sections 5 mappings, which we discuss These proofs 6. rely 4, and the proof of the we discuss in Section 7. limit on the properties of a certain pair of Appendix. Our approximation scheme in the of the Brownian limit, which and Section in is based on a refinement Finally, Section 8 contains the results of several numerical experiments. We end this section with product space of functions / some ; [0,oo) on (0,oo). The space D''[0,oo) — * D''[0,oo) £ is the r-dimensional ^^ that are right continuous on [0,oo) and have endowed with the Skorohod topology is of processes in D'"[0,oo) and .Y The space technical preliminaries. X"=^X we write D''[0,oo), [2]. mean to left limits For A"" a sequence A'" converges to A' in distribution. All vectors will be envisioned as vector whose components are On we occasion, column The dimension ones. all will also write e{t) to We vectors. mean of the identity e to denote a (column) should always be clear from context. e map confusion as to the appropriate interpretation of the letter For/: [0,oo)^$R, use the letter e{t) = t. Again, there should be no e. set = 11/11, sup |/(s)|, 0<s<( and for a vector- valued function = f {fi, f2, lt A = , fr)' ' [0, >2) — 3?'", we let (ll/illt,...,IIMI.)'. sequence of functions {/"} converges to a function / uniformly on compact sets each t > ||/" 0, — that /(x") D''[0,oo), > — f{x) f\\t u.o.c. we write 2 — A X^ —* as n — oo. > Finally, for a X u.o.c if We say that / is continuous at x if x" ^ A' if for x u.o.c. implies sequence of functions {-A"} on D'"[0,oo) and almost surely, X"' converges to (u.o.c.) A' a process in uniformly on compact sets. Make-to-order/Make-to-stock Production system Production systems are typically categorized as "make-to-order" or "make-to-stock," corresponding to the two scenarios for finished in which new jobs are triggered by customer orders or by replenishment orders goods inventory, respectively. In a make-to-order system, a system each time a customer places an order. A either in the over time. in form of orders waiting is released into the make-to-stock system, on the other hand, maintains a finished goods inventory from which customer inventory triggers a job release new job demands are filled. Each order fulfillment from the system; hence, the total number to be processed or as finished goods inventory, does not change of "jobs" in the system, ^^ o Make-to-order requests o Make-to-stock requests Inventory Replenishment orders Figure more often the It is 2: A Make-to-order/Make-tostock Production System case, however, that production systems employ a combination of both make- to-order and make-to-stock operations. Figure 2 shows such a system with a single processing stage. We make the cissumption that orders for make-to-stock products that cannot be of inventory are simply lost (no backlog). to model this system, One can employ the mixed queueing network we use station are serviced. A 1. Figure 1 1 naturally represents the production model the finished goods inventory from which make-to-stork orders 2 to service at station 2 signals that a make-to-stock request has been turn triggers a replenishment order for station to station in to lack where make-to-order products are represented by open customers and rlosed customers take the place of make- to-stock products. Station center and due filled In this case, a closed 1: that customer at is, .station filled, whirh iii a departure from station 2 then proceeds 1 represents an order for make-to-stork products, whereas a closed customer at station 2 takes the form of a finished good. We note that station 2 only approximates the In particular, consider a During The order to be it is times. [<i,<2] process for make-to-stock pro<luris during which the finished goods inventory this time period, make-to-stock requests continue arriving with first order, time interval demand — clear that t' That "excess filled is life" is, the first <2 the first one to arrive does not, in Denoting by same subject of this paper d. mterarrival is There is enipiv iiiii>'s the arrival time if ilus distribution as other interarri\.ii not significant in traffic limit (see Iglehart are, of course, several an "Order-up-to" policy in the sense that both systems and Whitt [16]). other policies that should be considered wiiirh a batcii of .V — ii in \<\ the behavior of such a system as the workstation operates umi-r example, make-to-order products (or similary, make-to-stock products) One can employ t' i service time of a busy period at station 2 should be characterized distribution. However, this difference first-in-first-out policy. t2- general, have the can be shown to converge to the same heavy The after i is may i hf I't receive higher pn-'niv make-to-stock recin-'-is -ir-' sent to the workstation whenever the inventory level n falls below interesting option is to process the In this example, the requested. are the rate for make-to-stock products P) be a probability space on which and identically distributed (i.i.d.) random variables {u{i),i squared coefficients of variation (SCV) of u(i) and designate open customers as customers of class and closed customers at station 2 as class 2. sequence of open customers to be {X~^u{i),i of the and open customer i at station 2 are defined as {mit;i(j),f one interprets A as the definitions, of open customers at station and 2 are = mi and m2, Setting u(0) = 0, 1. time let is server divided 1 1,2,...}, {vk{i),i 0, Vk{i), respectively. closed customers at station With = this categorization, 1,2, . . .} among = 0, 1,2, amount we = and c^. arrival rate of — the 1 customers, set the interarrival time 1,2,...}, respectively. open customers and mo as the mean mean — convenient to and we denote by moi'o(0 the 1,2,...} and {m2V2{i),i Similarly, the time servici^ 1 With these time servict^ service time of closed customers at stations u{0) : + + u(i) < 1 1 as Xt}. (3 1) t. At station the two customer types, and we denote by To(t) and 2i(<) the = amount To{i) + 1 . this nf time T\{l) ) be the counting process associated with class k service times. = ma\{i > 1 and Ai{t) ib c^ as class 1 of time that server j has spent working up to time arrival processes for class I4(t), 1,2,...}, k respectively. Skit) Denote by = We denote by We will find it has devoted to open and closed customers, respectively. (Clearly, Bi{t) Let Sk{t), k The = define the arrival process for o'pen customers at station Bj{t) be the mi < m2- Service time sequences for closed customers at stations 1. AqU) = max{j > Next, given by l/m2. are defined four independent sequences of independent where these random variables are positive and have unit mean. 0, 1,2, is The System Processes 3 !F, Another which make-to-order products are at natural, in the context of this example, to consider only cases in which Let (Q, . processing times for make-to-order and make-to- demand stock products, respectively. Finally, the It is mean critical level n' cyclical basis. parameter A corresponds to the rate mo and mi Similarly, two product types on some 0, 1,2, = 2 : mfci>fc(0) + + rrikVkii) < t). (3 2) customers, respectively, are then given by S2{B2{t)), the partial A2it) sums process Vkit} = ir.i) for class k service times, = ^7nkVkii}. :=1 Si{T,{t)). '31) Setting Mk(t) it = = mkVk{l) + VkiAkit)) follows from (3-l)-(3.5) that Mif(t) is amount the have arrived to the associated station by time Li{t) then L,{t) We with all for station those customers at station i at time = (3.6) customer classes at station ail = M2(t), = Q\{0) i. = N; and (32(0) that W,{0) = I. is, the Letting defined to be the remaining service time associated i, t. who define L2(t) \fi(i), process for iii|)ut (35) closed customers at station 2 and no open customers in station denote the workload process all we now assume throughout the paper that Qo(0) system starts with W^,(<) + Mo(i) the immediate workload is will = mkVk(Ak(t)), of immediate work from riass k customers If t. + either 0, queued or receiving service, we have Wm = V2{Nl (3.7) + (3.8) and W,(t) Defining I,(t) = / — = W,(0) L,(t)-B,(t). B,{t) to be the cumulative idleness process at station X,(t) = - L,(t) and i (3.9) t to be the workload netflow process, write (3.8) as W,{t) We = W,{0) + + X,{t) (3.10) I,(t). require the idleness processes satisfy the following properties: The first statement is continuous and nondecreasing with /, is /, increases only at times t when is = = (3-11) (3-12) U. a simple consequence of the properties of an idleness process, and the second statement holds for any work-conserving system. That when there IV',(/) /,(0) is, it states that the server remains idle only no work to be processed. The vector processes A', IV, I, U.Q are then dpfined in the obvious manner. It remains to characterize the "allocation" processes T{t) arrival time of the customer currently With FIFO service discipline, in service at station 1 if = (To{t),T\(t))' W\{t) > and . set Let denote the r](t) ri(t) = t otherwise. we must have Tk(t) = Mk(riU)) + eik(f), (313) where fifc(f ) is eik(t) = amount the otherwise. of service the current customer has received The amount are denoted by Uq and Ui of work at station is of class k and associated with open and closed customers 1 and are given by respectively, , that task if Uo{t) = Mo{t) - Toit) = Mo{t) - - eio(0 (3.14) Uiit) = Miit) - Tr{t) = M,(t) - Mi(q{t)) - en(t) (3.15) Moiriit)) Next, define Qk(t) to be the queue length process associated with class k customers (including any customer who may be in service). It follows from the previous definitions that Qo{t) = Aoit) - Ao{r]{t)) (3.16) Qi(t) = Ai{t) - Arirjit)) (3.17) Q2(<) = N + A2{t) As we expect, Qi(f) + Q2(0 = Qii0) + Q2{0) does not fluctuate = A2irj{t)). (3.18) ^, so the number of closed customers Furthermore, Q2(0 in time. - in the network completely determined by Qi(t). Finally, observe is that r^(t)^t-\\\{ii{t)) where 62(0 'f '^ W'lO = and otherwise currently occupying station The limit is (3.19) equal to the remaining service time of the customer theorems proved here apply to systems that we satisfy conditions of and service times terms of the basic sequences of unitized random variables {u{i) = To construct a sequence 0, 1, 2. > {A("',n 1}, {mj^ ,n > 1}, and service times are taken the n system, A'"' is fc = of networks nii^ to be u'"'(f) sequence is and m^. set to be n, that The convention superscript "(")" The • here is is, traffic," and : ; > network are defined 1}, {i'A:(') : ' > 1}, of positive constants 0,1,2. In the n"' system of the sequence, the interarrival times = u(!)/A'"' and v]^ (i) the arrival rate of open customers and m)^ in place of A for the we further require sequences = m]^ is various customer designations. Define the relative traffic intensities pj and "heavy require the construction of a "sequence of systems" to be indexed by n. Recall that the interarrival times fc e2{t), 1. in order to rigorously state these conditions, in + Finally, the closed we define AT*"' to denote a For example, Aq = the i'/ic(')' mean respectively. For service time of the as in (1.1)-(1.2) using A*"' customer population of each network in the n. parameter or a process associated with the n'" system by the refers to the external arrival process of open customers in the results in this paper apply to processes that have been "scaled." Let A'*"' denote a "generic" process associated with the n"" system. The "fluid scaled version" and "diff"usion scaled n system. version" of the process A"*"', denoted as A"(<) = -A(")(n') n A" and A", respectively, are defined via and X"(/) = -A<")("^0n We also define = \x^"Hri^t) = .V"(0 --V"(<) n^ assumed that the following conditions hold It IS the arrival rates and i = mean — oo < < 9 — pj . = Amo + mi/m2 Theorem Theorem Theorem and 3.1 .4s n = where .4^(0 Xt, mfc, (3.20) 1 number of closed must become approximately — oo, Vo{i) = 1. "sufficiently fast." is following two theorems are direct consequences of the functional strong law of large bers, Donsker's num- the functional central limit theorem for renewal processes: W''2'(0) mot. V; — = mo almost m,t. = l'/ and surely m^t. = ^t. S'o S', = ^t . S'^ = ^t. 3.2 iA^,V^,V{',V.p)=>{A^,Vo ,Vi' ,V{) where A^ Js(0,Ac^) Brownian motion and V^ (0,m^c^) Brownian motion, k — The — assumed that there requires that as the the It large, the relative traffic intensity at station Moreover, the rate of convergence is is it and m^" —.^asn^oo. the heavy traffic condttton. (3. 20) is called customers becomes The Furthermore, network. First, oo such that n(p',"'-l) Condition for the input processes of the service times converge to finite constants, A*"' -^ A This implies that p," 0, 1,2. exists n 0.l.'2. following result, which establishes that "remainder" terms converge to zero under scalmg, will be needed in our proofs. For j = 1,2, let fu(0=-4:;'(n'0, e-^(i) = -e[';\nt), n and ^2(0 Lemma 3.3 For lkl()l|. -0 Proof. It j = 1,2 = -4"'(n'0. r,{t) = -e["\nt). n and each n t > 0, \\('^j()\\t — 0, \\^ji-)\\t ~ 0, asn-^oo. follows from the definitions of fij(() an<l f^(t) ihat < < ~ fi,(0 e2(/) < ~ < max max max »noio(')+ miii(i) I<i<52(<) l<i<.4o(() rM|ii(;) 1<I<.^-J(() An application of Lemma 3.3 from Iglehart ami \\ liiti [16] proves the lemma. Ikji)!!' - ^' ""(^ The Limit Theorems 4 Theorem 4.1 (The Fluid Approximation) heavy If the condition (3-20) holds, then traffic (W^",/",Q",r')^(iy*,r,Q',f') u.o.c. where = W,'{t) (4.1) W;{t)^m2 Qoii) /,•(<) f*it) Theorem 4.2 = 0, (4.2) = Q'lit) = i;(t) = = Xmot, (4.3) 1 0, (4.4) T^{t) (The Diffusion Approximation) = —t. Ff the =^ (IV",r,C/",Q") Qm = 0, (4.5) heavy traffic condition (3.20) holds, then {W'J',U',Q'} where W:it)=Cx{t} + r{t)-^i;(t) ^j is a {6,a'^) py;(0 = m2 - Browntan motion (4.7) ^^^(O (4.8) = XmoW^it), C/o-(0 (4.6) U'{t) = ^P^i'(i), C/2(0 = W2(t) (4.9) 771.2 QUO = /* is — ' k f^fc'(0, = 0.1/2 (4.10) continuous and nondecreasing with /*(0) /' increases only at times t with W*{t) = = (4.11) (4.12) 0. Equations (4.6)-(4.8),(4.11), and (4.12) characterize IV{ as a one-dimensional reflected Brovvnian motion on the interval [0,m2] with and (1.7), respectively. drift 9 and variance a^ , where 9 and a^ are given by Properties (4.9) and (4.10) e.^press the deterministic relationships between queue lengths, partial workloads, and overall workloads that are characteristic of Brownian of queueing networks (for example, see [12, In Section 1, equation (4.9), states that in of the (limiting) traffic nii.xing principle." the heavy limits 13]). we noted that the boundedness be viewed as a consequence of the "heavy in (3.20) traffic limit, workload process at station This principle, which can born out the class specific workloads at each station are deterministically proportional to the overall workload at that station. We rationale for explaining the boundedness of the workload process at station 10 is 1 1, can develop another which perhap.s may Open customers be more intuitive, by considering the arrival process to this station. station While station at rate A 1 "2 not empty, the arrivals of closed customers to station is When resembles a renewal process with rate l/mj. all closed customers are at station the arrival process for closed customers are temporarily "turned the rate at which work arrives of time Let which station 2 in falls below the empty, station is heavy critical Theorem [0.6]. and — Tr{dz) _ our = b m2. = = ^2 '"" 1' during the period If 9 = then 3i 0, = 32 = and (T^/2b tt is 29/(7^, ^-—E, _ g-«6' (4.13) 1 p{:)d: where (''') = -pfzi pi^) The is, t Otherwise, setting k p«6 That period of time, [10]. 4.3 (Proposition 5.5.5. [10]) Set uniform distribution on the traffic level. for that lim \i;{t) t—'OO have the following result from Harrison and however, and set , = /?, We off," 1. 1 displays "non-heavy traffic behavior." 1 be the steady state distribution of \V^ IT arrive to following deterministic time change theorem, due to Whilt [22], will be helpful in proving results. Theorem sequences 4.4 (Deterministic inn where Cn is Time Change Theorem) nondecreastng with Cn{0) = 0- Let {/n,n If {fn,Cn) > 1} ind {c„,n > converges u.o.c. to a 1} he continuous pair (f,c), then /n(cn(0) converges u.o.c. to f{c{t)). Proof of 5 Lemma Proof. 5.1 for each The lemma t > is 0, ||tVi"(-)||( tlic — Fluid Ai)proximation a.s n — proved via a bounding argument sequence of open queues with two customer types Aq and S2 and wp let from type 1 (3.1) •x,. and (3.2). We denote by and .4o in 1. which W^ bounded above by a Recall the definitions of the processes the arrival process of type customers arrive according to the renewal process ^j 11 is The sequence customers of service . times for class k customers sums associated partial given by {m)^ Vk(i),i is X^-)it) = l/o'"'(4"*(0) Zi^)(t) = ^(")(i) y(")(<) = queue provides an upper bound < Applying the fluid 0}, and as in (34), we denote by the V^. process. Defining + + vf>(55"'(0)-< (5.1) y(")(<) (5.2) sup {x^"n«)}~ * (5.3) -' 0<s<t this > for station 1 in the sense that Wl''\t) < Z(")(<) for all t > 0. (5.4) scaUng to x^"^ equations (5.1)-(5.3) become nt) = Ki<it)) + KiS^{t))-t z"(<) = y"(t) = x^{t) sup 0<i<t It follows from Theorem almost surely for each ll^"(-)ll( -^ 0. and y"{t) {xT- > as n — oo. (5.7) — 4.4 that ||x"(.)||f Because the mappings (5.6)-(5.7) are continuous, follows from (5.4) that for each it (5.6) and the Deterministic Time Change Theorem 3.1 < + (5.5) t > 0, ||W^r()i|( ^ almost surely as n — oo. I Proofof Theorem 4.1 Because |B,"(0-flr(s)l < l*-s| and |77(i)-T;"(s)| we can conclude from the Arzela-Ascoli theorem that there which is Tq'' , < |<-s| almost surely, subsequence nt, exists a fc = 1, 2, . . . on T"*, B"*, and Sj* converges almost surely to continuous limits and the convergence uniform on compact Observe that I^{t) Applying the From equation = sets. - t Denote the corresponding , = B^{t), hence /f* -^ /• u.o.c. where /*(<) fluid scaling to (5.12), limits by Tq T*, 5', and t - flji respectively. B'{t). equations (3.8) and (3.16)-(3.19), we have W^(t) = W^{0) + V^^{SnB^it)))-B^{t) (5.8) QS(f) = i"(0-i"(r)"(0) (5.9) Q'iit) = S^iB^{t))-S^{B^iri''{t))) (5.10) Q^{t) = l-Q'^it) (5.11) ;j-(t) = t-Wi'{fj^it)) + ^{t). (5.12) we have limsup||.-rj"(.)||, < limsup||P^i"(77"(-))||( < limsup||^^i"(.)||i n—'oo < + limsup||e-^(.)||, n—-oo n—*oo n—oo + limsup||6-^(.)||i n— oo (5.13) 0, 12 where the result of inequality follows from the observation that first Lfmma 3 3 Lemma and 51. Scaling (313) = T?it) Applying Theorem follows that T^ — and the t the fluid convention, l/,"(5?(fl2"(7j"(0))) (3.1), the Deterministic r* in < Tj(t) + last inequality is a we obtain f7i(0- Time Change Theorem Lemma 4.4, 33, and (5.13), it where u.o.c. = \mot TZit) and f,-(f) = ^flj'CO. 1712 Again applying Theorem (31) and Theorem we 4.4 to (5.8)-(5.1 1), get (W2''.Q'"') — (iVj'Q*) where u.o.c. W^{i) = ^2 + —f^{t) - fljC) mi = ^2 (5.14) and Q5(O = QI(O = 0, Finally, (5.14) implies that there exists a finite n' all < > and n/t > = for all > T7(0 = ^<, Si'(0 = that /2"*(0 that Because n*. may I^ t, and and positive increase only at times and n^ > n*. < Q\{t)=\. 0, /,'(0 = Hence. /^(O l^it) = = 0, for all i such that IV'^''(0 e for t (5.15) which tV2'(0 = ^ ^ 0, it converges to the same limit Theorem 4.1, > Because each subsequence 0. t, from which we may conclude that (IV, /",Q", it Txit) - /ij,"'(0 = .4i"'(0 »7<")(0 = r?(")(<)_<. (3. 13) and (3.19) = in this V;'"*(0 A(")< = i;<"'(/)-m<.'''f way, we obtain V,(S2{B2{T](t)))) + m,S2{B2(n(t))) - —hinii)) + 77)2 rri = follows in turn T"") itself I . begin with the definition of "centered" processes Centering follows = Proof of the Diffusion Approximation 6 We in for S.J(0 of (PV", /",Q", T") contains a convergent subsequence and each of these subsequences converges to the limit described ^s- Vi(S2{B2{rj{t)m+ miS2{B2{r](t))) - -—Wi(n(t)) + —(2{t) + mj mj (u(t) 13 1 —hinV)) U}2 —m + '"2 ^u(f) = + V,iS2{B2{n{t)))) - m,52(52(r,(<))) — X,(r,(0) (712 -^/2(r?(0) + m2 — m2 e2(0 + - —hivit)) m2 fn(<)- (6.1) Next, set 6(0 = VoiMt)) + moAo{t) + + miS2iB2{t))+(xino + ViiS2{B2it))) V 6(0 = t^2(0) + V'2(5i(Ti(O)) +m2S2{B2{ri{t))) One can + — +— mi + m25,(Ti(<)) - Unit)) + £2(0 — -\)t m2 (6.2) J V'i(52(fl2(r?(0))) m.\ eii(0- (6.3) invoke (6.1)-(6.3) together with the observation that /i(f) = /i(rj(0) to express the netput processes (3.9) as Xi{t) = 6(0-— ^2(0 m2 (6.4) ^2(0 = )/2('7(s)). 6(0-A(0-(l-— m2/ (6.5) V Applying the diffusion scale to (3.10)-(3.12), we have the following expressions for the scaled work- load process: (n) w'r(o = -er(o - ^/2"(o + /r(o (6.6) 7712 wm = ait) /" is + hit) f 1 - -J^j - i^i^it) /2"(r?"(0)) continuous and nondecreasing with /"(O) /" increases only at times when Wp{t) = t + -j;^i2(t) = (6.7) (6.8) 0. (6.9) where 6"(0 = Ki^oit)) + r"|,"'/iS(0 ^^ + AHm<"' = 6"(0 = W^2"(0) + + + Vi"(52"(fl2"(0)) ml"'52"(SJ(0) + nf 1 j (6.10) (")= = = V'2"(5r(Ti"(<))) = = + m<"'5r(7r(0) + ^Vr(5J(B2('?"(O))) TTlj = = ,,""(<) = t + -W^i^-it)) + n By in - ^r('?"(0) + f5(0 -e^(t). n the Skorohod representation theorem, we Theorem (") = +m^2^S^iB^{ri'^(t))) + ^^11(0, (6.11) (6.12) may and 3.2 holds u.o.c. 14 will assume henceforth that the convergence — Lemma 6.1 Proof. Let Z'"' be the process defined is For each > t - 0,||r7"() ||( — as n in Lemma oo. 51, and note that Z" -~ Z' uo.c. where Z' a one-dimensional reflected Brownian motion with hmsup n— CO hmsup < -n''{-)\\t • II n—"oo < = Proof. The convergence as n u.o.c. — > . Because ir|" (<) < H n—'OO n — oo -||Z,"()||, Ti 0. = '"j— m"(') oc where ^' is{9.(t^) Broitnian moftoi} nnrf^^iO of ^" follows from from (3.20), Theorem Theorem 3.2, 4.4, and Li nima Define 6.1. = (,7(0 V2"(5T(:^(0)) + (")= = C2"(0 An cr^ + Hmsup -||f?()||, 1 Hmsup n ^ ^' and variance limsup-!-||PVr(-)lk < 6.2 ^" /i -UPV^H'T"!))!!. n—'oo Lemma drift application of Theorem 3.2, Theorem Theorem 4.1, (6.13)-(6.14) that ||C()II( ^2"(o = ((113) = = ^Vr(52"(fl?(r/"(0))) + m^"'Sr(77(/)). Iglehart in 1 "'2'''^2"(B?(r7""(/))) and Whitt [1.5], shows that S" (6 11) — —m,~^^^V' *• u.o c Fmm and the Deterministic Time Change Theorem, we can conclude from — as n 1^7(0) — :c for each + cr(n + C2"(o - / > and s^r(rr(o) = i 1.2. + f^o + Writing ,n 2 -jto '"i^'^' ml '1 the theorem follows as a result of Lemma 62 proof, that the that Theorem 31, Lemma 33, and Lemma 61. implies that e'^"(<) > m2/2 we may therefore assume that ^" £ for all n sufficiently large. D^ ,2 '^o'" ^" " - ^ From (6 I For the purpose .5). we may also <,{ .>iir ciiiicln.l.- 4J(0- /"(O - (1 - "«*r'/"i2"V2'( '?"{«)) has no jumps downward, from which it follow. 1I1..1 assumption of Lemma 9.7 is satisfied because /" is a nondecreasing process. Finallv. iictmn 17" £ A, we conclude from Theorem 91 that there satisfying (6.6)-(6.9), and that (WJ") is given by the 15 exists a unique pair of processes (II mapping (IT",/") = (*. *)(4"- »?") " /'' I Proof of Theorem 4.2 From Lemmas Brownian motion, continuous, W) (/*, ^2J{t) = m2 — Theorem follows from it = where {I',W') u.o.c. ^i{t), I' ($, 6.1-6.2, (^".^J.rj") -^ (^t.^^rj*) u.o.c. and q'{t) 9.1 that *)(r, = Wi is and variance is continuous at v')- Specifically, {^',t]'), we have from nondecreasing and continuous with /*(0) is /* increases only at times implying Because Brownian motion t. ($,^) with W*{t) t where ^j = is is {9,cr^) almost surely hence {I^,W^) —* Lemma 9.2, = 0, a one-dimensional reflected Brownian motion on the interval [0,m2] with drift 9 cr^. Next, observe from (3. 19) and (5. 13) that ;)" — t)' » where u.o.c. 77*(f) = -W'(i). Centering we obtain (3.14)-(3.15), USit) = Mo"(t)-Mo"('?'"(0)-A(")m("'^"«) U^{t) = M,"(0-Mr(/?"(f))--J;^'?"(0, 7712 with = Mo(<) Kr(^o(0) + = Because Brownian motion and Theorem and (/" — [/j* is 4.4 that ||A/,"(-) u.o.c. = continuous, - m<,"'.4S(0 (") = it follows MP(jfi-))\\t -^ from Lemma for a.s. each t 61, Theorem > and i = 3.1, 1,2. Theorem Thus, U^ where and Uoit)^\moWi{t) = —Wi{t). u;{t) 1712 Similarly, noting that Qlit) = iS(f)-.4S(r,"(0)-A(")77"(0 QUO = .4^(<)-.4!,'(^"(<))-^^"(<) Qlit) = A^it) 1 - A^rj-it)) - -^nt) where A-,{t) = S^{B^{t))--L^I^{t) and 16 .45(0 = 5r(T,"(0)--|;^rr(0, — 3 2. TJ it follows that Q" — Q' with u.o.c. Qo(0 = g-(0 = —u:(t) xw;(t) = "'0 — -Ltv7(o = c/,-(o r7i2 and the proof of Theorem 4.2 is complete. Refining the Brownian Approximation 7 We now turn to the following important question: Given a mixed network with parameters A. c^, m,, and c^, = i 0, 1,2, and a finite number customers, ;V of closed estimates for the network using the theory developed in how do we obtain performance the previous sections? the following approximation for the workload process at station Theorem 4.2 suggests 1: j^(Wr{N'')^\V'{) where (1.7), is IV'j* an RBM and variance a^ are given by (1.6) = .V (xmo + V a^ = Xml{cl + — m2 1 m? cl) + ^{cl + m2 cl). reversing the scaling in equation (7.1), one obtains the approximation W{)^\V{) where W is an RBM on the interval [0,m2N], whose H and whose variance of and namely, 9' By drift 6' on the interval [0,m2] whose (7.1) W is again cr^. We = Xmo + (7.2) drift — ^ - /i is given by (''S) 1. in 2 then apply and other performance measures of Theorem interest. In 4.3 to obtain the steady-state distribution particular, writing W''('^5o) to random variable associated with the stationary distribution of the process {n'(f)./ setting 6 = rriiN, k = 2^/ct^ , mean > 0} the and we have EW{oo) = =1, { ,' ,L ** 17 --,. , - otherwise. (71) Morover, the throughput rate obtained from the Brownian approximation a = I Equations 12, 13]), - ^ \T^ much ajm^ where (7.5) otherw.se. form one possible approximation traffic to developing our refinement is for a two-station approximations however, one typically needs to "refine" the Brownian limit that, as given by ^ previous works on heavy in performance estimates. The approach method 1 (7.4)-(7.5) in particular, As was noted mixed network. [6, and (7.2), V2M "^; J is in (for example, order to obtain good to arrive at an approximation as possible, yields estimates that agree with the exact solutions in those known. Henceforth, our special czises for which the exact solutions are results will be benchmarked against the following special case. From the theory of quasi-reversible queues product form solutions [17] (see also [1]), the mixed network in Figure 1 has if mo = mi = m, Ca = Co = c? = c^ and (7-6) = (7.7) 1- In this case, observe that = 2m(l (? Denoting by at station 1, -K/j). P(fc, /) the steady-state probability oik open customers at station and N — / closed customers at station P(fc,/) 2, = Gf ^1' NoSi and G is - Am, closed customers we have where rjo 1, / /yi = (7.8) m , the normalizing constant G=(i-^)f:(-^) -'70/ TT'c.y^ /c=0 and Let us write Qo(oo) {Qo(0'^ ^ 0}i ^nd customers at station to mean the steady-state similarly, let us use 1. From (7.8), random variable associated with the process Qi(oo) to mean the steady-state headcount of closed we obtain the following statistics: . and = a' l-^P(n,.V) .V (Recall that a'/mj is the throughput rate of closed customers.) In particular, if p = r;o + ^i = 1. equations (7.9)-(7.11) simplify to Eg, Finally, when p < 1, = we have the following (7.13) J limit as the number of closed customers in the system mcreases: Refinement We 1: Replace b by lim NTco EQo lim Ntoc EQ, = 1 - '/O I - ^0 = (7.15) - ^1 - 71 '^-^ (7.16) m2N/a. justify this modification via the following argument. increases whenever that station "functional" Little's interpreting l/m2 Law which is The empty, or equivalently, at times follows from equations (4.9)-(4.10), idleness process at station 2 t when Q\(t) = we have Q\[t) = By the .V. (l/m2)\'V{t), as the throughput rate of closed customers. This refinement essentially replaces the naive throughput rate by the approximated throughput rate be verified from (7.5) and (7.14) that a^ = 2m and In the case p = U. it can the refinement gives \-2bj in = a/m^. (7.17) 1-f from which we obtain a Note that in this case, Refinement 2: Set .18) V + 1 the Brownian appro.ximation agrees with the exact solution (7.14) Q,(/) = [& / m2)W (t) / p 19 This modification of equations (4.9) and (4.10) consists of two parts. rate used in the so called "functional" Little's a weighting factor of l/p processes. As noted in is taken to be d/m2 the throughput rather than l/m2. Second, then applied to the relationship between queue length and workload is Section 6 of may be obtained with such Law First, [13], empirical experience suggests that a better approximation a refinement. Writing Qi(oo) to > with the stationary distribution of the process {Qx{i)J we have 0}, random the variable associated = 0, the approximation is the special case of in /i ^EPV(oo) = EQi(oo) mean m2p a { 7712 p V m2N 2Qf N_ ' 2 which agrees with (7.13). asymptotically exact as A^ Moreover, it follows from (7.4) that = Qo{t) This refinement is - %- = J?! lim EQi(oo). l^- oc XWit)/p. essentially identical to the previous one, in that a factor of l/p can also be shown that with this modification, the approximation N —> oo whenever < is is introduced. asymptotically exact as 0, lim EQo(oo) = -\-fIp p iV-»oo \ no 1 - - '70 = = EQo(oo) lim EQo(co). '^^<: ^l However, note that we do not obtain the exact solution for finite -ElV(:c.) P A /m2,V p \ '2a 'to 1 where the 0, rn2p It p. < 1 = 1 3: Set // — oo: lim EQi(oo) N—'oo Refinement when last equality follows because rjo + '/i - = 20 I 'to / .V + V 2 1 N , even when p = 0, since To summarize, our approximation procedure RBM on the interval [O^m^N/a] with drift The queue /i and variance a^ given by (73) and Qo{t) = -rV(/) P (7.19) Qi(0 = -^W{t). (7.20) m2p < d < = Theorem Proof. 1 that satisfies equation (7.5) for o// ranges of the parameter set A, 7.1 /i Let b = 0, = m2N/a. Then it is easily verified there exists a unique from (7. .5) a = /i c^, m,, 0,1,2. For Consider must be itself approximation to be consistent, we must now show that there In order for the exists a unique i an length processes are then approximated via the mappings approximated. c^, . (1.7), respectively. Note that the formulation of the approximation includes the quantity d, which and W consists of replacing the workload process by < 0. We < d < 1 that satisifes (7.5). that the unique solution is given by 1 ) to the transcendental N + o-2/2mi need to show that there exists a unique solution r S (0. equation /(x)= (l-e-"/")(l-r)--^/i = where First, (7.21), a.?^ observe that /(I) = — (m2/mi)/i > [ dx Because a < solution X G 0, it follows that df(x)/dx (0, 1) to (7.21) (see Figure 8 7. 0. (7.22) >d as j Next, f(x) ], 0. Finally, differentiating we obtain df{i) The (7.21) subject of this section is + > 3). e-"/^ [[ - ax-'^{[ < J < for The proof for 1 - x)) (7.23) . and consequently there ^ > exists a unique I proceeds similarly. Numerical Examples the performance of the refined approximation described in Section Using product-form networks (namely, those whose parameters satisfy conditions (7.6)-(7.7), we compare estimates obtained from the Brownian approximation against exact theory predicts that the approximations are asymptotically exact, that 21 is, as the solutions. number Our of closed ""2 „- . Figure In 6. cases, the all approximation 1% within is N of the exact solution for expect, the quality of the approximation increases as p\ becomes closer to > System One can 2. The accuracy is see that the approximations are of these estimates are clarified the performance measure of interest 10% with N of exact solutions for > As Figure 10. of closed customers. Figure 11 bears out our expectation that N of but are required for good estimates. > in all cases, ^V Some X £ > 0. (iii) = So or a{t) X = < = si a, (xi,j;2) < < for all t • • for • t G Dj, > t = 0; < s,v £ [s,_i,s,). a < D € 1, larger values solutions. in the proofs of Sections 5-6. Chen and Mandelbaum [5]. that satisfy the following conditions: > e'i{s) > j(0) £ e for all s (i) j:(0) > 0; Define C^ to be the continuous [0, t]. 0, > e'r(s) ( for ail / each finite < and constants t, < oq there ai < a finite is < a2 particular, observe that e(<) In < c > o} that have the following properties: for (iii) A, and £ : D Denote by A the set of functions a € a{t) closer to namely, , C^ = {j e C^ < is queue length Appendix Let D^[0,<] be the set of x D^ length of open customers 10 shows, the estimates are pi devoted to characterizing some mappings that are used no downward jumps; and hcis functions in (ii) 10% from exact 15 gives estimates that are of the results here are adapted from the work of Fix (ii) is N Here, the estimates perform poorly for small values of A^, 9 This section when respectively, 1, Finally, Figures 11-12 displays estimates for 5. also intprmediate values of for The queue Figures 9-12. in Figures 9 and in good even As we 1 Figures 7 and 8 display the queue lengths of open and closed customers at station for 15. < 1, let u' = mapping (w,y) = {^.^)(x,a) defined by the following wi{t) = <l>i{x, a){t) = xi(t) + W2{t) = ^2{i.a)(t) = X2{t) - yi(0 + yiit) yi are nondecreasing with y,(0) y, increases only at limes t - (u'i,[r2), y • = — (i) a is number of • such that f is nondecreasing; subintervals either a(1) an element of A i For {y\.y2) be a solution of the properties: cy2{t) ( — 1 - (9.1) c){y2(i) - y2{a(t))) + cy^it) — (9.2) (9.3) where w,(t) = (9.4) Observe from (9.1)-(9.2) that e'w{t) due to the monotonicity of 2/2. = e'x{f) + The main (1 - r){y2{t) - .V2(n(0)) result of this section 23 is > e'x{t) the proof of the following: (9.5) Theorem 9.1 For each x satisifes (9-l)-(9.4)- x € Cj and a{t) process The if — X2 following Lemma {I is £ D^ and is A, there exists a unique pair of processes {uj,y) that In other words, the mapping ($,^) well defined on is — t, then ($, 'J') is continuous at {x,a). — c)(y2 ° a) has no jumps downward. a special case of Theorem 2.5 of Chen and 9.2 Suppose that x G and {w,y) G a D^ and = a{t) Madelbaum Then {w,y) t. — Finally, y D^ x A. '^(i,a) is Moreover, a if continuous [5]. are uniquely defined by (9.1)-(9.4), uniquely given by wi{t) = xiit) + yi{t) - cy2it) W2it) = X2{t) - yi{t) + cy2{t) yi{t) = sup {xi{s) - cy2{s)}~ 0<3</ - sup {x2(s) - yi(s)}~ = y2it) C Moreover, fixing a(t) Lemma = 9.3 Given x G t, "^ D is 0<s<( continuous on C^. and a G A, define fi{z){t) = - sup {x{s) {l - c)z{a(s)}}- (9.6) 0<s<t f2{z){t) = - sup {x(s) + (l-c)(z(s)-z(a(s)))}-. C There exists Proof. a unique solution The key to the proof to (9.6) is and is a contraction the solution uniquely satisfies (9.7). the observation that a{t) \\h{z){-)-f,{z'){-)\\i Hence, /i mapping (9.7) 0<s<t in < t, from which we obtain < {l-c)\\z{a{-))-z'{a{-))\\t < (l-c)||z(.)---'(.)||t. x and there exists a unique solution to (9.6). the fix point solution of (9.6) and observe that z{t) = sup {x{s) — (1 — c)z{a{s))}~ + (1 - c)(z(5) - z{ais))) - (1 - c)z(s)r + (1 - c)izis) - z{a{s)))}- + (1 0<s<t = sup {x{s) Q<a<t < sup {x{s) = C/2('Z) - c) sup {---(s)}" 0<s<t 0<s<t + (1-C)Z(0, 24 Denote by where the first inequality follows because monotonicity of r. < Therefore, z{t) f2{-){t) < < s <. On last equality is a result of the the other hand, noting that + xis)-{l-c)z{a(s))>0 z(t) for all nonnegative and the z is we have = f2(~)(t) + z(s)- - sup {x(s)-ll- c)z{a{s)) C > cz(s))- 0<j<( - sup { — c;(s)}~ c 0<s<.t = and Now a solution of (9.7). 2 is = fi(z'){t) ^(0. let z' be another solution of (9 7) We have {x(s)-(l-c)z'(a(s))}- sup 0<3<t = sup {x{s) + {l-c){z'is)-z'{a{s)))-{l-c)z'{s)}- 0<s<t < sup {x{s) + {l-c){z'{s}-z'{a{s))) + cz'{s)}- + sup {---'(s)}" 0<s<t 0<3<t = At) and = Mz'){t) sup {xis) + {I- c)(z'(s)-z'{ais)))-i\-c)z'(s)} 0<s<( > sup {-;'(«)}" 0<3<( = because i(s) to /i , but Remark: /i + (1 h;i-s - r)(z'{s) -''(0 - + cz'(s) > z'{a(s))) a unique solution so Suppose that a{t) = for all s > we can conclude that for all t > z' and x{0) > z{t)= sup {j(a)}- = - sup {x(s) + C 0<a<t Lemma 9.4 Given x 6 the interval [0,r). Thm D? and a e there exists — have shown that :' is a >;ohiiion I z. It then follows from (l -c)r(s)}" Lemma 'J :} lii.ii . o<j<( A, suppose that (ir.y) a We 0. is the unique extension of(u'.y) 25 unique solution of (9.1)-('J to the interval [O.r]. i ) "" Proof. To extend "^('")iy(^) - the definition of {w, y) to the endpoint r, observe from (9.2)-(9.4) that (u(r) — y('"~)) satisfy wi{t) = (wi{t-) + xi{t) - xi(r-)) + (y^ir) u;2(r) = {w2{r~) + X2(r) - 12(1"")) - (1 y2(a(r))) - + (1 - c) - (y2(r) - - yi(r-)) c){y2{T~) - (yi(r) - - c(y2(r) y2ia{T~ - y2(r-)) (9.8) ))) yi(r-)) (9.9) +c(y2(r)-y2(r-)) (9.10) y.(r)-y.(^")>0 (9.11) Mr)'(y(r)-y(r-))=0. (912) Because e'iwir-) x{t) - i(r-)) - = e'.c(r) + (y2(r)-y2(a(r))) > e'i(r) > 0, where the [5] + first inequality is (1 a result of - - (y2(r-) c) y2(afr-))) — that (9.8)-(9.12) produces a unique solution for y{r) For X £ T) Theorem follows from (9. 5), it + (1 - c) 4.3 of - (y2(r) y2(a(r))) Chen and Mandelbaiim y{T~). I and a G A, define the mappings u(0 = yi(x,a)(t) = sup {x(s) + {I - c){u{s) - u(a{s)}})- ('J 13) 0<j<t v{t) For a sequence Tk, k = = g2{x,a)(t) x{t) + il-c){u{t)-uia(t)) + cuit). (9 11) 1,2,..., let us define the "shifted" processes (uo It is = x'^it) = v{n) + u^'it) = u{t «)*=(<) = u{a{t v''{t) = v(t x{t + Tk) - x{n) + Tk)-u(Tk) + + Tk))-u{aiTk)) Tk). straightforward to verify from (9.13)-(9.14) that u''(t) = yf(x^a)(0 = - sup {x^ is) + il-c){u'' is) C v^{t) = g^^{x\a){i) = 0<s<t x^{t) -{no a)' (s))}~ (D !.->) ' ^ + {\-c){u\t)-{uoa)^(t)) + 26 cuHt). i9l(3> Remark: If we define the mappings = /i,(z,a)(0 - sup {x(s)}- (9.17) C 0<j<« = /.2(x,a)(0 it is clear that (/ii./jj) + x(t) cfn{x,a)(t), a special case of (31,92) with a equal to is (918) where e e(t) = < is the identity map. Lcmnia 9.5 For each / G Dj and (here exists a pair of functions G A, a (w,y) that sattstfes (9.1)-(9.4). = we Proof. Noting that We may cissume without loss of generality that (9.5) that ifi(O) > it suffices to set f It is = (<I>, prove the first > it'2(0) lemma for i 1/2 (otherwise, it G C\. < Fi.x follows from i £ i5 < 1/2. C\ and 1/2 and we proceed similarly). For an increasing sequence of times {iv,y) 1. vl')(x, a), it will k Tjt, be necessary to = and a pair of functions (u\y) satisfying 1,2 refer to the following "shifted" processes: x*(0 = w,{Tk) y*(<) = y,{t {y2oa)'(t) = y2(a(t u;*(0 = wit + x,{t + Tk)-x,{Tk) (9.19) + n)-y,(Tk) (9.20) + n))-y2{a{Tk)) (9.21) + n). (9.22) straightforward to verify that the mappings (9.I)-(9.2) yield u{-(0 = xt(0 + y{''(0-cy2'(0 w^it) = j:^(0-yf(0 + (l-c)(y^(0-(y2oa)'--(0) + (9-23) (9.24) cy^'(<); moreover, e'w''{t) = e'x^t) + (1 - c)(y^(0 - (y2 o a)'(t)) (9.2.5) and e'l^O + (l-c)(yl^(t)-(y2oa)''(t)) (1 = - e'x{t c)(y2{t + + n) + Tk) Set To 0, y^(t) = 0, e'w(Tk) - y2{n)) - (l-c)(y2(t where the equality follows directly from equation = = (1 - + e'x(t c)(y2(a(< + n)-y2(a{t + + n)-e'x(n) + = /ii(r?,a), - y2{a{n))) (9.26) Tk))), (9.5) and ob.serve that by monotonicity, y2('i(0) by the mapping (9.17), namely, y? T,)) and 27 let = for all / > U. Define y^ w° be given by equations (9.23)-(9.24) with fc fc = = Observe that 0. and hence w^ are uniquely defined by Theorem j/j = If f 1 = > inf{t = hand, a if the identity is = < oo. - ti + . . + . tk where the - (1 c)(y2(0 = + e'x°{s) {w'^,y^) — for < < = (y2 ° a)'^(0 on (2/2 is nondecreasing. Hence, we - c){y^{s) - o a)''(<)) y^ia{s))) - for k = Wiitk+i) > e'x''itk+i) = e'x{Tk+i) > 1-6 > = monotonicity of shift time to 7\ via the y2 be defined by (9.15), namely, y2 > = on the other hand, = a{t) t, [0, s; definition of (u;, Using the special structure of tk+x (1 = inf{t the definition of Ti, — c)(y2(0) that e'x*^(<) conclude ^2 Define for ifc [0,<i], £ t — + > = and we Define [0,<i]. 0, + + - c){y^(tk+i)- {y2oaf{tk+i)) - (1 - (1 - c)(y2(Tfc+i) - y2(a(Tk+,))) 6 6 (9.28) 8, mappings = (9.19)-(9.2I) setting 3j(x2,a). — then y2(0 — (y2 ° Ti) whether a{t) y) can then be The = I: Set y\(t) I. 2/2(^(0))) (1 0. - — and functions w\, yl are then defined using a)HO = for = t, in which < < s/ case — Ti = for Thus . y2 where a a, is > we have from ~ c)(y2(f) "^2(0) — — (9.28) that w\(0) = "^{(O) = For - + (y2 o a)*^(0) In addition, w\{s) - >0. c)(y2'(0 e'x^it) : S( — either a constant or the identity all t 1 ( > ^- ^^^ — ^t(0 "^^.n A: = 1 has no - wi{Ti) > w^^it) 6 is Ti using map si — Ti. uniquely 2.2.3 of Lemma 9.3 Lemma 9.4. over intervals of < (9.29) ^}- and consquently e'x^(O) same argument as before to + show jumps downward, from which we may + {l-c)(y](s)-{y2oa)^{s))- e'x^{s) < let y'2ia{t))) use the t we invoke Theorem extended to the endpoint time, one can thus uniquely define {y^,w^) for By x € or a{t) takes on a constant value, in which case the remark following [10], The conclude that Let [s;_i,s/) denote the interval corresponding to the function a that I. defined over the interval applies. In either may 6 for contains Ti. If a{t) takes a constant value over this interval, then (y2 o a)^{t) Harrison [0,si). has no negative jumps on u;f (<) w^{s) the other y2. (9.23)-(9.24) with k If, — On si. a solution of (9.1)-(9.4) over the time period is and observe that - (1 = a)''{t) (1 associated with the function first interval ~ y^ + equality follows from (9.26) and the last inequality follows from (9.5) and the first We now + (9.27) 6}. note from (9.25) that e'x°(0) First, Let [0,si) be the 8. - w^t) < (yj o a)'=(t)) over this interval, then y2{t) e'x''{t) = Moreover, w^{s) 0. wi{Tk+i) let > c){y^{t) <i no downward jumps and have shown that {w,y) Tk - (1 us cissume let w^{0) map we may conclude that Si) because x has h > + e'x*(0 a takes a constant value over this interval, then (y2 o If case, : oo then we are done so c)(y2(0)-y2(a(0)))-t«i(0) [0, For [10]. 0, set tk+i a. of Harrison 2.2. 3 wl{s) > 6 for s G [O.f^]- 1 ' y(0 = <i 0<i<Tk y{t) ^, _-^~ y{n) + y^{t-n) t>Tk. '_, . ,,, 28 (9-30) . The pair {w,y) thus constructed observe that for = it and In particular, = y* let a solution of (9 l)-(9.4) over the time period if > e'i*(<fc+, = e'xin+i) + > 1-6 > + ) ( - 1 ) - (y2 o a)*((t+, = In either case, w'' is process (w,y) on the interval (9. 22). Our [0,T)t] is some finite (9.5) and the t for which Tn construction for i < > for all " t )) Finally, - 6 6 pair {w,y) that satisfies (9.1)-(9.4) on the interval — ^"'^ ^^^ y\ ''i(^f.'j)- If ''^ odd, we set y^(t) 's > G Cj To do t. If 1 = defined according to (9.23)-(9.24). Similarly, we use is thus complete is if map given we can show that in (9.30) for each us suppose to the contrary that there so. let t The even or odd, respectively. is then constructed via the concatenation there exists finite n' with Tn« t, r2] (9.31) either (9. 27) or (9.29) to define tk+\ depending on whether k fixed [0, 6. we can construct a is c)(y2*(/fc+, - c)(y2(n+i) - y2(a(n+i))) - {I even, set j/2(0 fc g'l{x2,a). and property g t 1, Iterating in this way, [O.Tfc]. is is even, we have the following inequality due to definition of Tk a-,(7fc+,) - > 1*^1(7;) \-6 + e'x{Tk+x)- e' x(Tk) + {\ - c){y2{Tk+,) - y2(a(Tt+, (1 - c)(y2fT;.+ - y2{a(Tk+i))) ))) -(\ - c)(y2{n) - y2(a(Tk))y, odd, we have for k u'2(7it+,) - > u;2(7;.) Because a G A, there are a identity there map ^ - 6 -(I - finite + e'x(n + i)- r'r{n) + - c)(y2(Tk) number of intervals partitioning [0,<] sucli that a interval [s/_i,S() such that Tk (1 - C) [(y2(T^+,) ) xj2(a(Tk))). From the or a constant value over each subinterval must be an , G - y2(«(n + , - U2{Tk) - either the finiteness of these subintervals. [^/-i.s/) for all k ))) is > k' . For such k. either y2(a(Tk))] = or (1 - c)[(y2(71. + ,) - y2(a(n+,))) - i'MT,) - depending on whether a In either case, is the indentity we may conclude that max sup J Tk<t<t<Tk+i {wj{t) map for all k '/i(.i(r,))] = (I - c)(y2(r,+, ) - y2(Tk)) > 0, or a constant value on this subinterval, respectively. > - wj{s))> - - k' f' ->i\\Ax 2 J 29 sup r»<3<<<Tt+i \xj(f) - Xj(s)\. (932) It is Mandelbaum straightforward to extend identity 2.8.G. of Ciien and [5] to this setting, which states that - (w(t) sup "S'^'^" Substituting (9.33) max E for all s,t > for all k It [0,t] with — |s < <| t). > k* imply that T^ tj - \xj{t) The so there exists [0,t], Xj{s)\ < f 3 of jumps over each > for all k k' x(s)|. (9.33) 77 2 > ~ (9,34) such that V < t However, this contradicts the finiteness oft. . Lemma 94 interval [0,<]. unique extension at each jump point of Lemma - inequality (9. 34) together with the assumption that T^ remains to extend the construction to r G Dj. This number |j;(<) \xj{t)-Xjis)\>l(l--8] >0. sup uniformly continuous on is sup u<s<t<v we have in (9.32), max However, x < w{s)) is is done by noting that x has only a then applied to show that finite tliere exists a I x. 9.6 Lei x and a satisfy (he conditions Theorem Then 9.1. (9.1)-(9.Ji) have unique a solution. The proof proceeds Proof. {w,y) be the process constructed satisfying (9.1)-(9.4). 1. y and 2. if y(r) = 3. if y{t) = y'(7') at y'{t) in the proof of Lemma and 9.5, let [.5]. Let (w',y') be another process Suppose we can show that coincide on y' Chen and Mandelbaum as in the proof of Proposition 2.4 of on [0, S] some t e < [0, for > some 0, 5 > 0; then the also two coincide on r) then y{T) = [r, r + 6] some for positive 6; y'(r). Defining it follows from (1) that r T = > 6. sup{< > Suppose y(s) : < r = y'{s) for all Then oc. < (3) holds This contradicts the definition so we can conclude that r = 00. s < /}, hence y and y' The proof now coincide beyond rests r. on establishing (l)-(3). The proof of (1) follows from the construction y" then follows from (1) by applying a time application of Lemma shift as 111 in Lemma 9.5. The proof of (2) Lemma 95. The proof of (3) is an the proof of the proof of I 9.4 30 Lemma 9.7 then y continuous. is We Proof. (9.8)-(9. 12) of If X2{) — - c)(y2oa)() has no downward jumps where (1 = only need to show that y(t) Lemma 94. As argued in y{t~) for all the proof of this > t Lemma, (u),y) satisfies (91)-(94), Consider the problem posed there exists a unique solution for y(t) — y(t~). Because X has no downward jumps, u;i(r~) + ii(r) — ri(r~) > iii(r~)>0. because we assume that X2{-) — (1 — c)(j/2 o a)() has no negative jumps, - - - W2it-) + > w2{t-) + il-c)(y2(t)-y2(t-)) > U!2{t~)>0. Hence, y{t) — y{t~ ) J-2(0 = X2ii' ) ( 1 c) - y2(a(r ))) + (y2(a(0) ( 1 - c) (yjCO Moreover, - y2(r )) I the unique solution. is in Let us define the modulus of continuity u„Ax)= |x(r)-r(s)|. sup (9.35) 0<r.3<(.|r-l|<r) Lemma 9.8 Let £ x. x' D and a, e - {u,v) £ A where e{t) {9i,g2){x,a), — Define t. (i/, v') = (yi, yj)!-^', e), and {u'^v') > and = Fix t We have the following inequalities: = (u'.v''') (yf,y.t)(x*.a), {glg'^){x'' ,e). set ll"()-"(a())||, < -u;,,,(x), (9.36) c ||u*(.)-(uoa)*(-)||, < (9.37) -a;,,+r.(-r), c Proof. ||u*(.)-»''()lk < l\\x''()-x''()\\, + 2(^^y,,^TAr). (9.38) l^^(-)-v'^)||, < 2\\x''{)-x'''{)\U + 4(^^^^r,.,+n(x). (9.39) From Lemma 9.3, we have the equivalent representation u{t)=gi{x,a}{t)= sup {j:(s) 0<3<t 31 - ( 1 - c)u(a(s))}- . (9.40) With we have (9.40), < - u{t) u{ait)) = sup - {ix{s) x{a{t))) - - {I c)(u{a{s)) - u(a(a{t))))}- a(t)<s<.t < sup \x{s) - x{a(t))\ + {I - sup c) a(t)<s<t \u{a{s))- u{a{a{t)))\. a{t)<s<t Thus ||u(.)-ti(a(.))||t and equation (9.36) \\u^{-) - is a)'=(.)||t - u,r,A^) + {I < u;,,,(x) + (l-c)|lu(.)-u(a(.))||t proved. Equation (9.37) (u o - < + c)\\u{a{-)) u{a{a{-))}\\t proved similarly by observing that is + < ||"(- < 2||u(.)-u(a(.))||,+ r, < -w^,i+Tfc(^) Tk) -u{a{- n))\\t + |u(Tfc) - u(a(T^))| 2 where the Next, last inequality we obtain from is an application of (9.15) (9. 36). and (9.37) < ll"'(-)-"''(-)ll^ -\\x>'i-)-x''(-)\\t + ^\u'(-}-{uoa)\-)\\t c < c -J\x'{-)-x''i)\\t + 2(^'^y,,+TA^). Finally, observe that li'^'(-) - v''{-)\\t Proof. + Wx'i-) < 2\\x'{-)-x''i-)\\, and the proof of the lemma Lemma - < is x''{-)\\, t > 0. We will construct the processes {w,y) - + u''{-)\U +(1 - c)\W'{-) - (u oa)'{-)\ + 4(^^y,,+T,{^) finished. 9.9 Suppose that x E C^ and a(t) Fix c\\u'(-) make — = t. Then ^ is continuous at (x,a) use of the procedure described in the proof of ($, ^)(a;,a) in the interval [0,<]. Lemma Let 8 be the postive constant used in the procedure, Tk the sequence of (increasing) times obtained from the construction Tk — t. We may T)t-i), and let n* the (finite) assume that Tn« = t. iteration (starting with iteration 0). number of (tk — iterations required to construct {w,y) up to time Denote by {w ,y We 9.5 to ) write (u*^,u^) to 32 the shifted processes defined on the k mean the processes obtained by applying ) same mappings used to (z',a') the procodure + and w(t Fix \\z' - = Tk) w^(t) for We want to n and [W - a||, 0. < x\\t nhtaining (u'*, y*). Wp then define {v,u) by the concatenation in (9.30): > ( in "0 = < < < < n{t) i ~ q, <Tk ~ 9-41) /*+i. show that there < t , { exists > rj such that for any £ C^ and x' a' g A with then <f. ||vl',(i',a')-*,(x,a)||, Note that - ||*,(i',a') From Lemma < *.(x,a)||, ||*.(x'.a') we can conclude that there 9.2, - *.(x'.«)||, exists > rji any \\x' — j;||( < r/i If . we can show thai there exists - ||*.(x',a) (9.42) vJ/.fi-, a)||,. such that <'- ||v]/.(x',a)-*.(i,a)||, for + r/2 > such that \\a' - a\\t < 772 implies \\^,{x',ci')-^.{r\a)\U<'-, lemma then the is proved by setting rj = min(r/i . r;2). Because c^ \ f i' is continuous, there exists > »72 such that f / A-l and "...(-')<* where /I We = + 1 first 2/c. We [8 (i^)(i^)..(if^)]"', assume henceforth that .show that (iv,rj) and (I'.ii), \\a' — a\\t < (M4, r?2- constructed as described previously, satisfy the following condition: max||y,(.)-ti,(.)||T, From (9.45) and max||y,(-)-u,()||, lemma is proved First, we prove (9. 45). |' ^^ ( / (9.45) we can conclude (9.43), so the <2(i-^) i!_l L^^ if <2(^—^j ( -^—^j- j a-^ ,, ( we can establish tho processes we constructed = w^(k)(n) + 0\t) = v,(k){n)^A(t^Tk)-x\[n) 33 )< ^, satisfy {v. u) Define a*(0 x' r\(t+Tk)-x\{n) = (<I>, *)(j' . a ). where = j(ib) ^m{Tk)- 1 if ifc is odd = and_7(ib) 2 even or zero. In either case, observe that a is A: (t) — l3 (t) = i>j(k)iTk) so l|a'(-)-/?*(-)llt We if + < l|yi(-)-"i(-)llT. < 2max||y,(.)-u,(-)||T,. ||yi(-)-"i(-)llT. (9.46) proceed by induction and the observation that sup From (9.38) max and (9.46), \y,{t) - |y,(Tfc) - + Uj{Tk)\ ||y,^(-) - u'^{-)\\t. we can conclude \yj(t)-Uj{t)\< (l sup < u,{t)\ + -)ma.x\\yji-)-Uj{-)\\T,+-2(^-^]u>r,^j{x'). (9.47) straightforward to verify that It is max sup \yj{t) - Uj{t)\ - = \\yji-) < 2(—2-) w^2,ri(-r')- Uj{-)\\t, Equation (9.45) thus follows with an inductive argument. It remains to show that {v,u) as constructed satisfy {v,u) enough to show that odd. It suffices Because e'x'{t) \\e'x'{- Here, the v'-,f,Jt) > < for to do so for k even, for the = e'w{t), < ^fc+i, where j{k) argument = 2 (<t>, if for the case of k t ^)(j:', a'). is To do even and j{k) = .so, I if it is A: is being odd proceeds similarly. we have + n)-vU-)-uj'2i-)\W,, first < = = l|u^f(-)-t'i'(-)lk,. = \\92i<^'.e)C)-g',{p',a')(-)\\t,^^ < 2\\a'i-) < 4max||y,(-)-Uj(-)||T, < 6. - l3'{-)\\tk+, + 4 e'x'it + Tk) (9.44), respectively. Hence, - vi{t + n) > 34 W2{t ^,„T., (^') +4(^^)u;,„r,^(x') inequality follows from (9.39); the second inequality two inequalities follow from (9.45) and (^i^) is a result of (9.46); and the last by the definition of + Tk)-6>0 7^., for < < < ^+1. But V2{t + Tk) = e'z'{t > e'r'{t+n)-vx(t + n) + Tk) + {l-c){u:i{t + n)-U2(a{t + n)))-Vi{t + Tk) > and the theorem is proved. Acknowledgements: I am I grateful to during the course of this research. discussed in Section I Mike Harrison and Larry Wein would like to for their helpful comments thank Marty Reiman for sharing the interpretation 4. 35 - ez 03 5J en i2 0) S o O (A u CT at « «5 I d a. 01 r .SP o T 001 06 08 2 -- £ OZ o u CI, CI, nj « - - 09 OS (U o «0 3 3 V (A G, o X00 3 O |4 ft* J3 Ot' a; C a> o Ui CI, 0£ > a* Cti OZ (30 01 o o c«i o o 00 J0JJ3 ^uaDJaj o o o o o o T S3 I (0 r o *^ 8 I I < I o IT) o o o in CO r4 o o o in jojjg )ua3i3 J o o o If) o o o to C/5 (A £ o c a o o "-•-I c o X o u Cu to c 3 tN 3 GO o s o o 00 o o IN o o ^ o o in o o O sjduio]sro jO jaqiuHN O o o o o o o T 001 r4 Si en 2 Qj 2 B o en T3 V (A o § .s (d 00 (U IX, SJduio)sro JO Jdqumf^j T 001 06 08 OZ L r—i s o s T3 CO O Q 1 T— 2 St (A ££ 52 B o (A o e3 -- § SI 2 m References [1] Baskett, Chandy, K. 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