Equivalent Models for Queueing Analysis of Deterministic Service Time Tree Networks

Transcription

1 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 10, PP , OCT Equvalent Models for Queueng Analyss of Determnstc Servce Tme Tree Networks Mchael J. Neely, Charles E. Rohrs, Eytan Modano Abstract In ths paper we analyze feedforward tree networks of queues servng fxed length packets. Usng sample path conservaton propertes and stochastc couplng technques, we analyze these systems wthout makng any assumptons about the nature of the underlyng nput processes. In the case when the server rate s the same for all queues, the exact packet occupancy dstrbuton n any queue of a mult-stage network s obtaned n terms of a reduced -stage equvalent model. Smple and exact expressons for occupancy mean and varance are derved from ths result, and the network s shown to exhbt a natural traffc smoothng property, where prelmnary stages act to smooth or mprove traffc for downstream nodes. In the case of heterogeneous server rates, a smlar type of smoothng s demonstrated, and upper bounds on the backlog dstrbuton are derved. These bounds hold for general nput streams and are tghter than currently known bounds for leaky bucket and stochastcally bounded bursty traffc. Index Terms stochastc couplng, network calculus I. INTRODUCTION Many modern data networks transmt nformaton usng fxed length packets. Often ths takes place at lower network protocol layers, where varable length packets from a source are segmented nto fxed length cells for transmsson over a sub-network. Such data segmentaton facltates network desgn and control and allows for many practcal advantages n terms of ppelnng gans, congeston control, and farness ssues. Fxed length packets are also advantageous from a queueng theory perspectve, as they mnmze queue backlog among all packet length dstrbutons wth the same mean [3]. It s thus mportant to develop methods for understandng and analyzng networks of determnstc servce tme queues. In ths paper, we consder feedforward tree networks wth arbtrary traffc streams exogenously enterng each node (Fg. 1). Packets from these streams flow through the multple stages of the tree toward a sngle output port at the head node. All packets have length L bts, and hence have determnstc servce tmes T = L/µ, where µ s the servce rate of packets n node (n unts of bts/second). In the frst half of ths paper, we assume processng rates µ are dentcal for all queues. Under ths assumpton, we use stochastc equvalence and stochastc nequalty relatonshps to show that the steady Manuscrpt receved March 1, 00; revsed February 18, 005. The materal n ths correspondence was presented at the 38th Annual Allerton Conference on Communcaton, Control, and Computng, Montcello, IL, October 000. M. J. Neely s wth the Department of Electrcal Engneerng, Unversty of Southern Calforna, Los Angeles, CA USA. C. E. Rohrs s wth the Dgtal Sgnal Processng Group, Research Laboratory of Electroncs, Massachusetts Insttute of Technology, Cambrdge, MA 0139 USA. E. Modano s wth the Laboratory for Informaton and Decson Systems, Massachusetts Insttute of Technology, Cambrdge, MA 0139 USA. Fg. 1. A 1 A μ 1 μ A 3 A 4 μ 3 μ 4 A 5 A mult-stage tree network wth multple exogenous sources. state occupancy dstrbuton at the head node of a multstage tree network s exactly preserved n a reduced -stage equvalent model. Exact expressons for the mean and varance of packet occupancy are derved from ths result, and the number of packets n the head node of the tree s shown to be stochastcally less than the number there would be f all prelmnary nodes were removed. We then consder networks wth heterogeneous server rates µ, and develop a smple upper bound on the packet occupancy dstrbuton at each of the nodes. Ths analyss contrbutes to a stochastc network calculus for analyzng general tree systems n terms of smpler one-queue or two-queue equvalent models. As queueng networks are non-lnear systems drven by stochastc events, exact analytcal results are largely lmted to systems wth the specal structure of tme reversblty [3] [4]. Exact analyss of non-reversble queueng systems s usually confned to small networks (see [4] [5] [6] for analyss of a sngle dscrete tme queue wth general nputs, and [7] for a moment generatng functon analyss of a two-queue tandem wth..d. arrvals every tmeslot). An approxmaton method s developed n [8] for modelng dscrete tme tandems wth arrvals and departures at each stage n the specal case when nputs have a specfed Markovan structure. Boundng technques for general networks are developed n [9] [10] [11] usng a determnstc calculus of network servce curves, and bounds usng stochastc calculus are developed n [13] [14]. An exact analyss of flud tandems wth a sngle nput havng exponentally dstrbuted ON and OFF perods s presented n [15] usng stochastc Itô calculus. An exact watng tme analyss s developed n [16] [17] for a tandem of determnstc servce tme queues wth Posson nputs. Ths analyss uses a smple nput-output nvarance property for tandems wth non-ncreasng servce rates. A smlar property s proven n [19] for dscrete tme trees wth slotted servce at each queue and wth ndependent nputs. Ths result s used n [18] to compute the average delay n each node of a dscrete tme tree when nputs are Posson. Our work uses a smlar nput-output nvarance property, but extends the technques to demonstrate equvalence of packet occupancy dstrbutons (rather than just averages) and to treat networks wth general stochastc nputs. μ 5

2 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 10, PP , OCT. 005 Exact analyss s provded for networks wth homogeneous server rates, and boundng analyss s provded for systems wth heterogeneous server rates (wthout requrng the nonncreasng servce rate assumpton). Ths paper s structured as follows: In Secton II we develop an nput-output relatonshp for tree networks wth a sngle bottleneck. In Secton III, we consder networks wth homogeneous server rates and show that the steady state packet occupancy dstrbuton of any node n a multstage tree s dentcal to the correspondng dstrbuton n a smpler -stage equvalent model. Ths analyss combnes sample path queueng propertes wth stochastc equvalence relatons. In Sectons IV and V, exact expressons for the mean and varance of packet occupancy are derved, and a traffc smoothng result s developed to establsh upper bounds on the backlog dstrbuton at any node. In Secton VI ths boundng analyss s extended to systems wth heterogeneous servers. The resultng bounds explot the specal structure of determnstc servce tme queues as well as the probablstc nature of the underlyng nput processes, and are shown to be tghter than the network calculus bounds derved for specal classes of traffc wth varable length packets n [9]-[14]. II. EQUIVALENT MODELS Here we demonstrate an mportant nput-output sample path property for determnstc servce tme queues. We begn by presentng two smple lemmas. Consder a sngle queue wth a general nput stream consstng of fxed length packets arrvng at tmes τ 1, τ,..., wth arbtrarly dstrbuted and correlated nterarrval tmes. Packets are served accordng to any non-preemptve servce dscplne, such as the Frst-In- Frst-Out (FIFO) polcy. Let A represent the arrval process, formally wrtten as a sum of shfted mpulse functons: A = n=1 δ(t τ n). We assume that τ 1 > 0, so that all packets arrve after tme 0. Let T represent the servce tme of each packet n the server of the queue. Assume the system s ntally empty, and let D represent the correspondng process of queue departures durng the nterval [0, t]. Suppose the departures next pass through a delay lne of duraton d, for some value d 0. The delay lne accepts ncomng packets, stores them, and releases them after exactly d seconds, resultng n an output process D(t d). Lemma 1: (Delay Permutaton) The resultng output process D(t d) s unchanged f the delay lne s stuated before the queue, rather than after t. Ths elementary lemma follows because queues are tme nvarant systems, so that an nput of A(t d) leads to an output of D(t d). For the next lemma, consder the two-queue tandem of Fg.. Packets arrve to the frst queue accordng to an nput stream A 1, and these packets are then delvered to the second queue after passng through an ntermedate delay lne of duraton d 0. Addtonal packets arrve drectly to the second queue accordng to an nput stream A. Let D represent the resultng output process of the second queue. Let T 1 and T represent the servce tmes of packets n the frst queue and second queue, respectvely. A 1 1 d A D A 1 T 1 d A Fg.. A two-queue system wth determnstc servce tmes T 1, T (wth T 1 T ), and ts equvalent model wth queue 1 replaced by a tme delay. Lemma : (Queue Replacement) If both queues are ntally empty and f T 1 T, then D s unchanged f the frst queue s replaced by a pure delay lne of duraton T 1. The modfed system s smpler and forms an equvalent model of the orgnal system. Lemma was ndependently proven n [16] and [1] for the case when there s no ntermedate delay between the two queues. The case of an ntermedate delay lne of duraton d > 0 follows drectly from the result for no delay lne by usng Lemma 1 to swtch the order of the delay and the frst queue. Intutvely, the result of Lemma follows because the fnal node serves packets no faster than the frst node can delver them. Hence, the busy perod of the fnal node of the orgnal system cannot fnsh before the busy perod of the fnal node n the equvalent model. Note that although the D functon s unchanged, t s possble for the packet departure order to be dfferent n the equvalent model. A. Tree Reducton Prncple Consder a mult-stage tree network wth M queues and heterogeneous server rates µ, so that packets have servce tmes T = L/µ n each queue {1,..., M}. The nputs to each queue consst of an exogenous arrval process A together wth the endogenous departure process of upstream queues (Fg. 1). Let T M represent the servce tme of the fnal node (correspondng to servce rate µ M ), and let D represent the departure process of ths node. Theorem 1: (Output Invarance) If T M T for all prelmnary nodes, then D s unchanged f all other nodes M are replaced wth pure delay lnes of duraton T, as n Fg. 3. Proof: The proof follows by teratvely applyng Lemma. Specfcally, consder the two queue system composed of the fnal node together wth any other node stuated mmedately behnd t (possbly wth a delay lne n between). Let the prelmnary node represent queue 1 of Lemma, and let all of ts arrvals collectvely represent A 1 from the lemma. Lkewse, let all other streams nto the fnal node collectvely represent A. Then the departure process of the fnal node s unchanged f the prelmnary node s replaced by a delay lne of duraton equal to T. Recursvely repeatng ths procedure replaces all prelmnary nodes wth pure delay lnes wthout affectng the overall departure process of the system. Theorem 1 allows the departure functon of the fnal node n a mult-stage tree network to be modeled by the departures of a sngle queue wth a set of delayed nput streams, provded that the servce rate of the fnal node s less than or equal to all other servce rates. We note that a smlar reducton prncple for trees was developed n [19] for the specal case of dscrete D

3 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 10, PP , OCT tme systems wth slotted servce and ndependent nputs. In the next secton we use ths theorem to analyze tree networks wth homogeneous servers. As an asde, we note that for heterogeneous networks, Theorem 1 mples that routng accordng to a shortest-path tree s optmal when sendng fxed length packets from multple sources to a sngle bottleneck node M, where µ M µ for all nodes. Specfcally, consder a network of queueng nodes defned by an arbtrary graph (wth no tree structure yet determned), and defne the path length of a route between any two nodes as the sum of servce tmes T = L/µ over all nodes of the route. It s clear that under any routng polcy, every exogenously arrvng packet s delayed from enterng the fnal node by at least the sum of servce tmes over ts shortest path. Each packet s delayed by exactly ths amount n the shortest path tree model where all prelmnary nodes are replaced by pure tme delays (Fg. 3). Hence, at every nstant of tme the number of departures n ths reduced model s greater than or equal to the number of departures under any routng scheme n the actual network. However, Theorem 1 mples that the departures of ths reduced model are dentcal to the actual departures n the network under shortest path tree routng, and hence shortest path tree routng mnmzes the total number of packets n the system at every nstant of tme. III. TREES WITH HOMOGENEOUS SERVER RATES Here we consder tree networks wth homogeneous server rates, so that packet servce tmes satsfy T = T for all. Theorem : If T = T for all, then at every nstant of tme the number of packets n the fnal node of the tree s the same as the number of packets n the fnal node of a reduced -stage tree, where all nodes more than one stage beyond the fnal node are replaced wth delay lnes of duraton T. Proof: The endogenous nputs to the fnal node are the departure processes from the prevous stages. By Theorem 1, these departure processes are unchanged when ther prelmnary queues are replaced wth tme delays. Because the endogenous and exogenous nputs to the fnal node reman the same n the reduced system, the number of packets n the fnal node s unchanged. Theorem allows the occupancy dynamcs n the fnal node of a tree to be modeled by a reduced -stage system wth prelmnary tme delays. Intutvely, t s clear that f the nputs A are statonary and ndependent of each other, then the steady-state occupancy dstrbuton n the fnal node of the reduced system s preserved when the delay lnes are removed. A 1 A T 1 T A 4 A 3 T 3 T 4 A 5 μ 5 D Fg. 3. Replacng all prelmnary nodes of the network n Fg. 1 wth delay lnes. The output process D s unchanged f µ µ 5 for all {1,..., 5}. Here we use stochastc couplng to formally prove ths result, and show that steady state behavor exsts n the equvalent model f and only f t exsts n the orgnal system. We frst present the basc concepts of stochastc couplng theory. A. Stochastc Domnance and Equvalence Defnton 1: A random varable N 1 s stochastcally greater than another random varable N f there exsts a thrd couplng varable Ñ1 such that N 1 Ñ1, and Ñ1 has the same dstrbuton as N. In ths case, we wrte N 1 N. An equvalent defnton can be stated by couplng wth respect to N, so that there s an external varable Ñ wth the same dstrbuton as N 1 and such that Ñ N. It s well known that N 1 N f and only f P r[n 1 α] P r[n α] for all real numbers α (see []). Ths fact mmedately mples that stochastc nequalty relatons are transtve: If N 1 N and N N 3, then N 1 N 3. If N 1 and N have the same dstrbuton, we wrte N 1 = N, and say that the random varables are stochastcally equvalent. It s also essental to have a noton of stochastc equvalence for random processes: Defnton : A random process A 1 s stochastcally equvalent to another process A f E {h(a 1 )} = E {h(a )} for all measurable operators h( ) that map a process A to a sngle real number. It s clear that f stochastcally equvalent nput processes A 1 and A are appled to dentcal queues at tme 0, they produce stochastcally equvalent packet occupancy processes N 1 and N, and that at any partcular tme nstant τ, the random varables N 1 (τ) and N (τ) satsfy N 1 (τ) = N (τ). Indeed, ths can be seen from the defnton by defnng the operator h( ) that maps an arrval process to the number of packets N(τ) at tme τ va the queueng law. B. Removng Delays Va Stochastc Couplng Consder now the -node tandem wth nodes 1 and and nput streams A, B, and C delverng packets destned for node, as shown n Fg. 4. Ths system represents any two sequental nodes of a mult-stage tree, where the general nputs consst of the combned exogenous streams and endogenous streams from other nodes. As before, we assume all arrvals occur after tme 0, so that A = B = C = 0 for all t 0. A B d 1 C Fg. 4. A canoncal -queue system wth nput X delayed by tme d. Note that the A process s explctly shown wth a tme delay of duraton d. We show that f A, B, and C are ndependent and statonary, the delay can be removed wthout affectng the steady state dstrbuton of packets n node 1

4 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 10, PP , OCT or node. We frst defne the notons of steady state and statonarty. Defnton 3: Let the stochastc process N represent the number of packets n a queue as a functon of tme. The steady state dstrbuton F [n] for the queue s defned: F [n] = 1 lm t t t τ=0 P r[n(τ) n]dτ, n {0, 1,,...} (1) whenever the lmt exsts. We now defne the noton of statonarty for processes that only have arrvals after tme 0. For any arrval process A and any postve delay d, we defne the partally deleted process à d as follows: { à d = 0 f t d A f t > d Thus, à d can be vewed as a verson of the A data stream n whch packets durng the frst d seconds are thrown away. Defnton 4: An arrval process A s statonary f for any postve delay d, the delayed process A(t d) s stochastcally equvalent to Ãd. Note that for two statonary arrval processes A and B that are also ndependent, the superposton A(t d) + B s stochastcally equvalent to the superposton Ãd + B, and hence ether superposton appled to a queue yelds the same steady state packet occupancy dstrbuton, provded that the dstrbuton exsts. We further note that the total number of packets n a determnstc servce tme queue cannot ncrease f some packets from the nput stream are deleted []. Theorem 3: For any general nputs A, B, C that are ndependent and statonary, the steady state occupancy dstrbuton n node 1 of Fg. 4 exsts f and only f the steady state occupancy dstrbuton exsts when the tme delay on the A nput stream s removed. If the dstrbutons exst, they are dentcal. Smlarly, the steady state dstrbuton n node s the same wth or wthout the tme delay on the A stream, provded the dstrbuton exsts. Proof: It s useful to defne N [A] (τ) as the number of packets at tme τ n a queue that s ntally empty wth a general arrval process A appled at tme 0, where A could represent a superposton of processes. Note that N [A] (τ) s always greater than or equal to the number of packets n a queue at tme τ wth the same nput process but wth some of the arrvng packets deleted. Hence, the followng nequaltes hold determnstcally for all tme nstants τ: N [ à d + B d ] (τ) N [Ãd+B] (τ) N [A+B](τ) () The random varable N [A+B] (τ) on the rght of the above nequalty represents the number of packets n node 1 of Fg. 4 at tme τ n the case when the A and B streams are appled drectly wth no tme delay, whle the mddle term of the above nequalty represents the correspondng number of packets when arrvals durng the frst d seconds are deleted from the A stream. Lkewse, the leftmost term consders the case when packets from both the A and B streams are deleted durng the frst d seconds. However, by statonarty, the arrval process Ãd + B d s stochastcally equvalent to the process A(t d)+b(t d), whch represents a delayed verson of A + B. Lkewse, by ndependence, the arrval process Ãd + B s stochastcally equvalent to the process A(t d) +B. We thus have the followng stochastc equaltes for all tme nstants τ: N [ à d + B d ] (τ) N [ à d +B] (τ) = = N [A+B] (τ d) N [A(t d)+b] (τ) Usng these stochastc nequaltes n () yelds: N [A+B] (τ d) N [A(t d)+b] (τ) N [A+B] (τ) The upper and lower bounds n the above nequalty are tme delayed versons of the same process, namely, the process of packets n node 1 of Fg. 4 when A and B are appled drectly. It follows that ther tme average dstrbutons (defned n (1)) are equal, and converge f and only f the mddle term converges. The mddle term represents the process of packets n node 1 when the A stream frst passes through the d second delay. Thus, the steady state dstrbuton n node 1 s unchanged f the tme delay s removed. The proof of the correspondng property for node s smlar, and follows from the fact that, because both nodes have the same servce tme T, the number of packets n node s greater than or equal to the resultng number of packets f some arrvals from A, B, or C are deleted before enterng the system (see Appendx A). It follows that for all nstants τ: N 1 (τ) N (τ) N 3 (τ) where N 3 (τ) represents the packet occupancy n the second node of Fg. 4 when the streams A, B, and C are appled wth no tme delays, N represents the correspondng occupancy n the second node n the case when all packets arrvng from A durng the frst d seconds are deleted, and N 1 represents the occupancy n the case when arrvals from all streams are deleted for the frst d seconds. The result follows by notng that N 1 (τ) = N 3 (τ d), and that N (τ) s stochastcally equal to the number of packets n the fnal node of Fg. 4 (wth A delayed by d seconds). C. The -Stage Reducton Theorem We can now present the man reducton theorem for homogeneous tree networks wth general statonary and ndependent arrval processes. Consder a mult-stage tree network wth exogenous arrval processes A at each node, and defne ts -stage equvalent model as the system wth the same nputs, but wth all queues more than 1 stage behnd the fnal node removed. Theorem 4: (-Stage Equvalent Models) If exogenous nputs A are statonary and ndependent of each other, a steady-state occupancy dstrbuton exsts n the fnal node of the orgnal network f and only f a steady state occupancy dstrbuton exsts n the -stage equvalent model. Furthermore, f the dstrbutons exst, they are exactly the same. Proof: Reduce the mult-stage tree network to a -stage network n tandem wth delay lnes, as descrbed n Theorem

5 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 10, PP , OCT A 1 A A G W 1 W A 1 A A G A 0 Z N 1 N N G A 1 A k-1 A k+1 A G A 0 Y (a) W 1 W k-1 W k+1 W G A 0 Z k Defne Q T (A A j ) to be the expected occupancy n a sngle queue wth determnstc servce tme T and wth a superposton of the arrval streams A A j enterng as nputs. Thus, the mean occupancy n the frst stage queues of Fg. 5a can be wrtten as E {N } = Q T (A ) for {1,..., G}. Now consder the system of Fg. 5b, where the frst stage queues are replaced by delay lnes of duraton T. Let W = W W G represent the sum number of packets n the delay lnes (where W represents the number of packets n the delay lne from stream A ). Let Z represent the number n the fnal node. We have: W G (b) (c) A k N k E {N N G + Y } = Q T (A 1 )+...+Q T (A G )+E {Y } Fg. 5. (a) The canoncal -stage equvalent model of a homogeneous tree network, and (b), (c) reduced systems wth the same total packets.. Note that ths does not change the packet occupancy process N n the fnal node. By the delay removal theorem (Theorem 3), we can teratvely remove each of the delay lnes wthout changng the steady state dstrbuton n the fnal node. The resultng system has no tme delays and s exactly the - stage equvalent model. Note that any node of a tree can be vewed as the fnal node of the smaller network consstng only of nodes wth arrval streams that pass through node. Hence, the steady state occupancy dstrbuton of any node of a tree network can be exactly analyzed accordng to a -stage equvalent model. IV. MEAN AND VARIANCE ANALYSIS Here we use the -stage reducton theorem to develop smple expressons for the mean and varance of packet occupancy n terms of the correspondng moments n systems wth only one queue and two queues, respectvely. Consder a mult-stage tree wth homogeneous servce tmes T n each node, and wth statonary and ndependent exogenous arrval processes A. By Theorem 4, the steady state occupancy n any such network can be analyzed by a -stage equvalent model. The canoncal -stage model s shown n Fg. 5a, and has G frst stage queues. In ths model, the nputs A 1,... A G represent superpostons of the exogenous nputs of the orgnal mult-stage network, and A 0 s the exogenous nput to the fnal node. Assume ths system exhbts steady state behavor, and let N 1,..., N G represent the steady state number of packets n the frst stage queues. Let Y represent the steady state occupancy n the fnal node. That s, the collecton {N 1,..., N G, Y } can be vewed as random varables wth jont dstrbuton gven by the steady state system occupancy dstrbuton. Alternatvely, these random varables can be vewed as samples of queue occupancy at a tme when the system s n steady state. A. Mean Occupancy Here we compute E {Y }, the mean occupancy n the fnal node of the canoncal -stage tree of Fg. 5a. Assume nputs A 0, A 1,..., A G are rate ergodc wth arrval rates λ 0, λ 1,..., λ G, and defne the sum rate λ = λ λ G. E {W + Z} = E {W } + Q T (A 0 + A A G ) = (λ λ 0 )T + Q T (A 0 + A A G ) By Theorem 1, we know that the departures of both the system of Fg. 5a and Fg. 5b are the same for all tme, and hence the total number of packets n the two systems s always the same. The equaltes above can thus be equated, yeldng the followng exact expresson for E {Y }: E {Y } = (λ λ 0 )T +Q T (A 0 +A A G ) Q T (A ) =1 The above equaton expresses the expected occupancy n any node of a homogeneous mult-stage tree network n terms of the expected occupancy n sngle-queue systems wth a superposton of the orgnal exogenous nputs, and can be evaluated whenever the average occupancy n such sngle queue systems can be computed. We note that expected occupancy for tree networks wth Posson nputs was derved prevously n [18]. We can obtan the result of [18] from the above formula by usng the Q T (A) functon for the specal case of Posson nputs. In ths case, Q T (A) can be wrtten as a pure functon of the arrval rate λ, and s gven by the standard Pollaczek-Khnchn formula for expected occupancy n an M/D/1 queue: B. Occupancy Varance Q T (λ) = λt + (λt ) (1 λt ) To compute the varance V ar(y ) for packet occupancy n the fnal node, consder the followng alternatve modfcaton of the canoncal -stage system n Fg. 5a: Replace all prelmnary nodes {1,..., G} {k} wth tme delays of duraton T, but keep the prelmnary node k unchanged. By Theorem 1, ths modfcaton also contans the same aggregate number of packets as the orgnal system. Let Z k represent the correspondng number of packets n the fnal node of ths modfed system. Lemma 3: The varables Y, Z, {Z k }, {W k }, and {N k } satsfy: Y = (W N )(W j N j ) (3) k=1 Z k (G 1)Z + j

6 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 10, PP , OCT Proof: Usng the fact that all three systems n Fg. 5 have the same total number of packets wthn them, we have: N N G + Y = W W G + Z = N k + Z k + {1,...,G} {k} The above equaltes hold for all k {1,..., G}, and hence: W W k N k = Z k Z for all k {1,..., G} (4) Y = Z + [W k N k ] = Z + [Z k Z] (5) k=1 k=1 Squarng both sdes of (5) (workng only wth the Z k and Z varables) and then usng (4) establshes the result. Theorem 5: (Varance) If A 0, A 1,..., A G are statonary and ndependent of each other, then: V ar(y ) = V ar(z k ) (G 1)V ar(z) (6) k=1 Proof: Note that eq. (3) of the precedng lemma s smply an algebrac statement about any varables Y, Z, {Z k }, {W k }, and {N k } that satsfy (4) and (5). Any random varables satsfyng these two lnear equatons wll also satsfy these equatons n ther expected values. Hence, we can replace Y, Z, {Z k }, {W k }, and {N k } n (4) and (5) wth ther expectatons E {Y }, E {Z}, {E {Z k }}, {E {W k }}, and {E {N k }} to fnd that the lemma also mples: E {Y } = E {Z k } (G 1)E {Z} k=1 + j E {W N } E {W j N j } (7) Takng expectatons over (3) and subtractng (7) yelds: V ar(y ) = G k=1 V ar(z k) (G 1)V ar(z) + j E {(W N )(W j N j )} j E {W N } E {W j N j } Because the nput processes are statonary and ndependent, (W N ) s ndependent of (W j N j ) whenever j. The last two terms of the above equaton thus cancel, provng the theorem. Expressons for the varance V ar(z k ) for a tandem of queues wth Posson nputs and for a dscrete tme tandem wth..d. arrvals and general arrval dstrbutons are presented n [] usng a moment generatng technque developed n [7]. These expressons can be used wth Theorem 5 to yeld exact varance expressons for any node of a mult-stage tree wth such nputs. V. TRAFFIC SMOOTHING Here we show that tree networks of homogeneous, determnstc servce tme queues naturally act to smooth traffc, makng the patterns better for downstream nodes to receve. All nputs are agan assumed to be ndependent and statonary. Theorem 6: (Smoothng) The steady state packet occupancy n the fnal node of a homogeneous tree s stochastcally less than ts resultng occupancy when all prelmnary nodes are removed and the exogenous nputs are appled drectly to the fnal stage. Proof: By Theorem 4, we can frst reduce the multstage tree to ts canoncal -stage equvalent model (as n Fg. 5a), where the nputs to the equvalent model represent superpostons of the nputs to the tree. The total number of packets n ths -stage system s the same as the total number of packets n a modfed system where all nodes at the frst stage are replaced by tme delays of duraton T, as n Fg. 5b. However, t s clear that the number of packets n the set of frst stage queues s always greater than or equal to the number n the correspondng delay lnes. It follows that the number of packets Y n the fnal node of the -stage system s less than or equal to the number of packets Z n the fnal node of the modfed system. Thus, Y Z. However, from Theorem 3, we know that Z = Z, where Z represents the steady state occupancy of the modfed system when the prelmnary delay lnes are removed. Hence, Y Z, provng the theorem. Ths result provdes a smple bound on the occupancy dstrbuton of the fnal node of a homogeneous tree n terms of the occupancy n a sngle queue wth the same exogenous nputs. VI. TREES WITH HETEROGENEOUS SERVICE RATES Note that the results of the prevous secton are derved from the output nvarance propertes of tree networks wth homogeneous servce rates, as characterzed by Theorem 1. The theorem holds whenever the servce rate of a gven node of the tree s less than or equal to the rates of all of ts prelmnary nodes. Hence, all results for homogeneous trees derved n the prevous secton apply equally to trees wth non-ncreasng servce rates on every path to the destnaton. However, t s common for servce rates to ncrease at one or more stages, so that downstream nodes have the ablty to support the sum traffc load from earler queues. In ths secton, we consder the general case of trees wth arbtrary servce rates µ at each node. Specfcally, we consder a tree wth M nodes wth labels {1,..., M}, where the fnal node s labeled as node M (see Fg. 6a for the case M = 5). Let A represent a general arrval process of packets exogenously enterng the network at node, and let λ represent the arrval rate. All packets have fxed lengths L wth servce tmes L/µ n each node. Let N M represent the steady state packet occupancy n the fnal node. For each {1,..., M} let N (γ) represent the steady state occupancy n a vrtual queue wth servce rate γ and nput process A enterng t alone (we assume throughout that a steady state exsts). Theorem 7: (Stochastc Occupancy Bound) For any vrtual servce rates {γ } such that γ γ M = µ M, we have: (a) If exogenous nputs {A } are statonary and ndependent, then varables {N (γ) } are ndependent, and: N M N (γ1) 1 + N (γ) N (γ M ) M (8) (b) If exogenous nputs are not statonary and ndependent,

7 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 10, PP , OCT then there exst varables { N M ˆN (γ) ˆN (γ1) 1 + (γ) ˆN } such that: (γ) ˆN ˆN (γ M ) M (9) = N (γ) for all {1,..., M} (10) The stochastc bounds descrbed above can be easly vsualzed n terms of the parallel queue system of Fg. 6d, where each of the M exogenous nputs of the tree s gven a separate vrtual queue. Notce that the rates γ of the vrtual queues must sum to the rate of the fnal node, but are otherwse left unspecfed. Hence, the rates can be chosen for convenence, or can be optmzed to acheve the tghtest upper bound. Before provng the theorem, we demonstrate ts mplcatons. Example 1 (Averages): Let Q L/γ (A ) represent the average occupancy n a sngle queue wth servce rate γ, packet length L, and nput stream A. From Theorem 7, we have the followng upper bound for the average occupancy n the fnal node of a mult-stage tree: E {N M } mn P M =1 γ=µ M M Q L/γ (A ) The Q L/γ (A ) functon can be shown to be a convex functon of γ [0], and hence the mnmzaton above represents a convex optmzaton. A smpler bound s obtaned by the λ assgnment γ = µ M λ (where λ = λ λ M s the sum nput rate). Ths choce of the γ values ensures all vrtual queues are stable whenever the orgnal network s stable. Example (Leaky Bucket Inputs): Suppose the exogenous nputs {A } are leaky bucket constraned wth rate and burst parameters (λ, σ ) [9], so that wth probablty 1 a queue wth nput A and server rate γ wll never have more than σ packets, provded that γ λ L. If the sum arrval rate λ from all nputs to the tree satsfes λl µ M, then usng the proportonal rate allocaton γ = µ M λ /λ guarantees each vrtual queue receves a servce rate γ λ L. Theorem 7 thus verfes the well known result that the number of packets n the head node s always less than or equal to M =1 σ [10] [1]. Furthermore, the probablty of achevng ths worst case backlog can be bounded f more detaled statstcal nformaton about the arrval processes s avalable. Example 3 (Moment Generatng Functons): If nputs {A } are statonary and ndependent, then we have from part (a) of Theorem 7 that for any rates {γ } that sum to µ M : E { e rn M } =1 =1 M } E {e rn (γ ) for any value r 0. Hence, the moment generatng functon for the number of packets n the fnal node of a tree s less than or equal to the product of moment generatng functons for queues wth sngle nputs A and server rates γ. If nputs are not necessarly statonary or ndependent, then (γ) part (b) of Theorem 7 mples there exst varables { ˆN } such that the occupancy N M n the fnal node of the tree (γ1) satsfes N M ˆN ˆN (γ M ) (γ) M, where ˆN = N (γ) for each {1,..., M}. Let {p 1, p,..., p M } be any collecton of non-negatve numbers that sum to 1. Usng the fact that M =1 x have: E { e rn M } E max {1,...M} [x /p ] for any values {x }, we for any r 0. Because functon satsfes: {e r max[ ˆN (γ ) /p ] } ˆN (γ) E { e rn M } M =1 } E {e r ˆN (γ ) /p = N (γ), the moment generatng M =1 } E {e rn (γ ) /p Example 4 (Complementary Occupancy Dstrbuton): A bound on the complementary occupancy dstrbuton can smlarly be derved for the general case when nputs are not necessarly statonary or ndependent. Agan let {p 1, p,..., p M } be a collecton of non-negatve numbers that sum to 1. Usng the same technque as [14], we observe the followng ncluson (γ1) of events: { ˆN ˆN (γ M ) (γ1) M > k} { ˆN 1 > p 1 k}... { ˆN (γ M ) (γ) M > p M k}. Hence, because P r[ ˆN > x] = P r[n (γ) > x] for any x, we have from the unon bound: [ P r[n M > k] P r[n (γ1) mn P γ=µ M, P p =1 1 > p 1 k] +... ] +P r[n (γ M ) M > p M k] (11) The above bound s smpler and tghter than the bounds derved for varable length packet systems n [13] [14] wth nputs that are characterzed by a class of exponentally bounded bursty arrvals (EBB) or stochastcally bounded bursty arrvals (SBB). Indeed, n the specal case when nputs are EBB or SBB, then neglectng the optmzaton over γ and smply choosng the proportonal rate assgnment γ = µ M λ /λ, the bound of (11) becomes dentcal to the bound offered n [13] [14] for the occupancy dstrbuton of a sngle queue wth a superposton of EBB or SBB sources. When ths queue s stuated wthn a mult-stage network, the boundng technques of [13] [14] can be used recursvely to compute updated bounds by consderng the effects of each stage. However, these updated bounds requre more computaton and have the dsadvantage of gettng progressvely larger and larger at each stage suggestng that congeston may ncrease wth the sze of the network n systems wth varable length packets. The bound n (11) holds equally well for any node at any stage of the network. Hence, t s always tghter and s mmune to any pejoratve effects when the sze of the network s scaled. Ths ndependence to scalng s due to the ntrnsc traffc smoothng propertes of determnstc servce tme queues. Furthermore, the bound of (11) holds for any general nput processes, and ncorporates the partcular statstcal propertes of each process by relatng performance to the resultng backlog n a system of sngle queues wth each of these nputs appled ndvdually. A. Dervaton of Theorem 7 To prove Theorem 7, we use two prelmnary lemmas that hold for one queue and two queue systems wth fxed length packets. Consder a queue wth server rate µ and wth a superposton of M nput processes A A M. Let N represent the number of packets n ths queue as

8 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 10, PP , OCT a functon of tme (assumng the system s ntally empty). For a gven rate γ µ, let N represent the correspondng number of packets n a queue wth server rate γ and wth nput stream A alone. Lemma 4: (Multplexng Inequalty) For arbtrary nput streams {A } and for any rates {γ } such that γ γ M = µ, we have at every nstant of tme: N N N M The lemma s an mmedate consequence of the multplexng results proven n [1]. Intutvely, Lemma 4 follows by observng that the sngle queue system s ether empty or s processng data at a rate that s greater than or equal to the sum processng rate of the combned system of M queues. Next consder two tandem queues wth a sngle nput. Packets of length L bts arrve to the frst queue accordng to an arrval process A wth rate λ, and these packets enter the second queue after servce at the fr Servce rates of the frst and second queues are µ 1 and µ, respectvely. Assume the system exhbts a steady state, and let random varable N represent the steady state number of packets n queue. Let Ñ represent the correspondng steady state occupancy n the case when the frst queue s removed and all packets drectly enter queue. Lemma 5: (Smoothng n Sngle-Input Tandems) For arbtrary servce rates µ 1 and µ, we have: N Ñ. That s, removng the frst queue creates a stochastcally greater packet occupancy at the second queue. Proof: If µ 1 µ, then the proof s the same as the proof of Theorem 6, as the total number of packets n the tandem s unchanged f the frst queue s replaced by a pure delay lne. In the case µ 1 < µ, there s never more than 1 packet n the second queue. Thus, P r[n > k] = 0 P r[ñ > k] for all ntegers k 1. Furthermore, by Lttle s Theorem we have that P r[n > 0] = P r[ñ > 0] = λl/µ. Thus, N s stochastcally less than Ñ. These two lemmas can be used teratvely to bound packet occupancy n any node of a heterogeneous tree. Consder the fnal node n the network of Fg. 6a, whch has rate µ 5. Let N 5 represent the random number of packets n ths fnal node when the system s n steady state. We see from the fgure that there are three separate streams flowng nto ths node. Thus, n the frst teraton of our reducton technque we splt ths node nto three vrtual sub-nodes wth rates γ 5, γ 3, and γ 4 that ndvdually servce the three streams (where γ 5 + γ 3 + γ 4 = µ 5, see Fg. 6b). From Lemma 4, the resultng number of packets N 5, N 3, N 4 n the sub-nodes satsfy N 5 N 5 +N 3 +N 4. (Here, the t=1 superscrpt desgnates values obtaned on the frst teraton of the reducton, and the subscrpt ndex represents the hghest-numbered exogenous nput correspondng to the partcular sub-node). Now notce that several -queue tandem stuatons have been created (Fg. 6b). Consder for nstance the queue wth rate µ 3 n tandem wth the N 3 queue. From Lemma 5, removng ths front queue creates a stochastcally greater occupancy Ñ 3. Lkewse, the queue wth rate µ 4 can be A A 1 3 μ A 1 5 A 3 A A μ 1 5 μ 3 1 μ A N 5 A μ μ 5 μ 3 μ A 4 4 A μ 4 4 (a) (b) A 1 A A A 3 5 μ 1 μ A 4 N 5 N 3 N 4 (c) γ 5 γ 3 γ 4 A 1 A A M (d) N 1 (γ1) N (γ) N M (γm) N 5 N 3 N 4 γ 1 γ γ M γ 5 γ 3 γ 4 Fg. 6. An llustraton of the teratve reducton technque to stochastcally bound the occupancy n the fnal node. Note that P γ = µ 0. removed to generate a new varable Ñ 4 that s stochastcally greater than N 4, creatng the smplfed system n Fg. 6c. The number of stages n ths smplfed system s one less than the orgnal. For a second teraton, the same procedure can be appled to splt queue Ñ 3 nto queues wth rates γ [t=] 1, γ [t=], γ [t=] 3 such that γ [t=] 1 + γ [t=] + γ [t=] 3 = γ 3. Proceedng ths way, we remove nodes and splt nodes untl we are left wth a parallel collecton of 5 queues of rates γ 1,..., γ 5 such that γ γ 5 = µ 5, and each new node has ts own exogenous nput stream A. At each step of the teraton the component varables N are stochastcally ncreasng. Thus, we are left wth varables N (γ1) 1,..., N (γ5) 5, where each N (γ) s dstrbuted as a packet occupancy n a sngle queue wth nput stream A and processng rate γ (Fg. 6d). In ths way, we can bound packet occupancy n the head node of a mult-stage heterogeneous tree wth arbtrary exogenous nputs A 1,..., A M by smply usng the parallel queue pcture of Fg. 6d. However, there s a subtlety n the above teraton procedure: Whle t s true that N 5 N 5 + N 3 + N 4 when the network s changed from Fg. 6a to Fg. 6b, and t s true that N Ñ 5 5, Ñ 4 when the network s N 3 Ñ 3, N 4 changed from Fg. 6b to Fg. 6c, t s not necessarly the case that N 5 Ñ 5 + Ñ 3 + Ñ 4. Ths ssue s formally treated by usng stochastc couplng to ntroduce the auxlary ˆN (γ) varables of Theorem 7. We frst requre two lemmas. Lemma 6: If N, A, and B are random varables such that N A + B, then there exst varables  and ˆB such that N Â+ ˆB, and  = A, ˆB = B. Furthermore, f A and B are ndependent, then  and ˆB can be chosen to be ndependent. Proof: By defnton of stochastc nequalty, there exsts a random varable Z such that N Z and Z = A + B. Gven ths Z varable, form the par of random varables (Â, ˆB) by choosng them accordng to the jont condtonal dstrbuton p A,B Z (a, b A + B = Z). Thus, the vector (Â, ˆB) s coupled

9 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 10, PP , OCT to the Z varable through the jont dstrbuton functon for the par (A, B). It follows that  + ˆB = Z N, and that  = A, ˆB = B. If A and B are ndependent, then  and ˆB wll also be ndependent. Lemma 7: If N, A, and B are random varables such that N A+B, and f there exst varables Ã, B such that A à and B B, then there exst varables Â, ˆB such that N  + ˆB, and  = Ã, ˆB = B. Proof: By defnton of stochastc nequalty, because A à and B B, there must exst varables A and B such that A A, B B, and A = Ã, B = B. It follows that N A +B. By the prevous lemma, there must be varables  and ˆB such that N  + ˆB and  = A, ˆB = B. By transtvty, we have  = à and ˆB = B, provng the lemma. We can now prove Theorem 7. Proof: (Theorem 7) We proceed by nducton on the teratve procedure outlned above. At the begnnng of teraton k, assume we have a set of sub-nodes I k wth steady state occupances Ñ [t=k] ( I k ) and correspondng varables and such that N M I k ˆN [t=k] [t=k] ˆN = Ñ [t=k] ˆN [t=k] all I k. After splttng each node wth occupancy Ñ [t=k] nto a set of S() parallel sub-nodes wth new occupances N [t=k+1] j t follows that: (j S()) such that Ñ [t=k] ˆN [t=k] j S() for j S() N [t=k+1] j, N [t=k+1] j, I k (1) Next, the nputs to the parallel sub-nodes are un-smoothed by removng any prelmnary queues, and new varables are formed that are stochastcally greater than Ñ [t=k+1] j N [t=k+1] j, that s, for each I k and each j S(): N [t=k+1] j Ñ [t=k+1] j (13) Applyng Lemma 7 to (1) and (13), for all we can fnd [t=k+1] auxlary varables ˆN j for each j S() such that ˆN [t=k] [t=k+1] [t=k+1] j S() ˆN j, and ˆN j = Ñ [t=k+1] j for all j S(). Defnng the set I k+1= Ik S() establshes the nducton hypothess for the next teraton, provng the theorem. VII. CONCLUSIONS We have used sample path observatons and stochastc couplng technques to analyze determnstc servce tme tree networks wth arbtrary nput streams. The analyss yelds quanttatve probablstc expressons for network congeston n terms of smpler systems. For homogeneous tree networks wth multple stages, a reduced -stage equvalent model was developed and shown to exactly preserve the steady state occupancy dstrbuton. Exact expressons for occupancy mean and varance n any node were obtaned usng ths analyss, and a smoothng result was proven, showng that packet occupancy at any node s stochastcally ncreased f ts prelmnary nodes are removed. For tree networks wth heterogeneous server rates {µ }, a smple upper bound on the packet occupancy of any node was developed n terms of the occupancy n a set of vrtual queues that ndvdually serve the exogenous nput streams A. The bound s ndependent of the locaton of the node wthn the tree, and s tghter than prevous bounds derved for varable length packet systems wth traffc envelope constrants. Ths work contrbutes to a theory of stochastc network calculus, and provdes powerful technques for analyzng complex queueng systems. APPENDIX A Here we show that the number of packets n the second node of the -queue tandem n Fg. 4 s greater than or equal to the number there would be f some arrvals from any of the nputs are deleted before enterng the system. To see ths, frst note by Theorem 1 that the total number of packets n the tandem s unchanged f the frst node s replaced by a pure delay lne of duraton T. The total number of packets n ths equvalent model (delay lne plus queue) cannot ncrease f some arrvals are deleted, and hence the total number n the -queue tandem cannot ncrease wth deleted arrvals. Thus, f the frst node of the -queue tandem s empty, then the total number of packets n node s greater than or equal to the correspondng amount f some arrvals were deleted. It s not dffcult to show that ths property s preserved durng tmes when the frst stage node s busy. Indeed, durng such busy perods the frst node delvers at the maxmum rate of 1 packet per T seconds, and the number of packets n the second node cannot decrease below the correspondng number n the deleted system. REFERENCES [1] M. J. Neely and C. E. Rohrs. Equvalent models and analyss for multstage tree networks of determnstc servce tme queues. Proceedngs of the 38th Annual Allerton Conference on Communcaton, Control, and Computng, pp , Oct [] M. J. Neely. Queue occupancy n sngle server, determnstc servce tme tree networks. Master s thess, Massachusetts Insttute of Technology, Laboratory for Informaton and Decson Systems (LIDS), March [3] P. Humblet. Determnsm mnmzes watng tme. MIT LIDS Techncal Report P-107, May 198. 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