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STATISTICAL MULTIPLEXING 479

STATISTICAL MULTIPLEXING

INTRODUCTION

Conventional telecommunications networks, such as the pub-lic telephone network, were designed based on the synchro-nous transfer mode (STM) paradigm, which uses time-divi-sion multiplexing (TDM) for bandwidth allocation. In TDM,the link capacity is shared among contending connections us-ing TDM channels. A channel is uniquely identified by theposition of a time slot within a recurring synchronous struc-ture, known as a frame. For two end systems to communicate,a logical connection must be established between them,whereby one or more STM channels are reserved for that con-nection. When a channel is assigned to a connection, its band-width cannot be shared with other connections (see Fig. 1).The channel bandwidth is wasted when an established con-nection temporarily generates no traffic, as in the case of alistener in a phone conversation.

In contrast to conventional networks, the architecture ofBroadband-Integrated Services Digital Network (B-ISDN) isbased on a new paradigm, known as the asynchronous trans-fer mode (ATM). The adoption of ATM as the transfer technol-ogy for B-ISDN came in response to several considerations.B-ISDN will offer the means of communications to a widerange of applications, including conventional voice and dataapplications as well as new multimedia applications (e.g.,video-telephony, high-definition TV, and multimedia confer-encing). Integration of such diverse applications over a com-mon communication platform requires a simple, unifiedtransport technology, such as ATM, that is independent of thecharacteristics of the transported media. ATM is an attractiveswitching technology characterized by high-speed fiber trans-mission facilities and simple hardwired network protocols de-signed to match the huge transmission speeds of communica-tion links. Transported data in ATM are encapsulated intofixed-length packets known as cells. The size of an ATM cell is53 bytes; 5 bytes of which are used as a header. As a backboneswitching network, ATM is designed to minimize the over-head incurred in processing network protocols. Cell switchingin an ATM network is performed in hardware, unlike tradi-tional packet-switched networks in which packets are routedusing software processes.

BASIC OPERATION

One important difference between ATM and STM is that in-stead of TDM, ATM uses statistical [or asynchronous multi-plexing (SM)] as a means for resource sharing. In SM, cellsfrom various traffic streams share the link capacity on a needbasis. Bandwidth is dynamically allocated so that if a streamis temporarily idle, its bandwidth is given to active streams.SM results in a significant improvement in bandwidth use,particularly when traffic streams are characterized by alter-nating active and idle periods with the active periods being,on average, shorter than the idle periods (1). When statisti-cally multiplexed, ATM connections are no longer identifiedby the location of a time slow in a synchronous structure. In-stead, the header of each ATM cell contains connection identi-fiers that unambiguously identify the connection to which the

J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright # 1999 John Wiley & Sons, Inc.

480 STATISTICAL MULTIPLEXING

Figure 1. Time-division multiplexing (un-used bandwidth is wasted).

Stream #1

Stream #2

Stream #3

Timedivision

multiplexer

FrameFrameFrameFrame

Wasted bandwidth

cell belongs. This is illustrated in Fig. 2 where three streams Types of Networks Guaranteesare statistically multiplexed onto an output link.

In principle, QoS guarantees can be offered on a deterministicA two-level hierarchy of connection identifiers is used in

or a statistical basis (2–5). Deterministic guarantees are hardATM networks: virtual channel identifiers (VCI) and virtual

bounds on the transport performance (e.g., bounded packetpath identifiers (VPI). Each cell contains both identifiers in

transfer delay). Such bounds are relatively easy to support onits header. A VCI specifies a Virtual Channel Connection

an end-to-end basis. However, deterministic guarantees are(VCC). A VPI specifies a Virtual Path Connection (VPC),

often provided at the expense of a conservative use of networkwhich is a bundle of VCCs. VCIs and VPIs have local scope

resources because the bounds are obtained under worst-casethat is limited to a given switch. When a connection is estab-

traffic assumptions.lished, each switch along the path of a connection assigns VPI

While some applications require some form of transportand/or VCI values to each connection, independently of the

guarantees, others can tolerate infrequent violation of thesevalues assigned by other switches. When a cell arrives at a

guarantees. This is particularly true for ‘‘play-back’’ applica-switch, the VPI and/or VCI in the cell header are checked. If

tions in which the traffic consists largely of audio-visual datathe cell belongs to an admitted connection, the switch re-

units that are played back at the receiving end. Human insen-places the VPI and/or VCI in the header of the cell with the

sitivity to small variations in picture and sound qualityones assigned to the connection. The cell is then sent to one

allows for the loss of a small fraction of cells without any per-of the output ports. Switching can be performed at the VP

ceived impact on quality. For such applications, the networklevel with only VPIs being modified by a switch, or at the VC

can provide statistical QoS, which enables the network tolevel with both VPI and VCI being modified. Typically, VCCs

make use of statistical multiplexing to improve bandwidthare statistically multiplexing at the output port of a switch.

utilization. An example of such a guarantee is when the end-to-end cell transfer delay is ensured to be less than Dmax withprobability 1 � �, where 0 � � 1 (2). At steady-state, thisQUALITY OF SERVICE IN ATM NETWORKSmeans that (1 � �)% of cells should encounter a delay of nomore than Dmax; the remaining cells that arrive late can beMany multimedia applications require the underlying net-discarded without any perceived degradation in the signalworking infrastructure to provide a priori guarantees on thequality. Applications can compensate for some of the lossestransport of their packets. In the context of ATM, applicationsusing error concealment mechanisms (6,7).requirements are known as the quality of service (QoS), which

is measured by throughput, cell loss, and cell delay metrics(including cell delay variation). Providing guarantees on the Approaches to Providing QoS Guaranteesrequested QoS is particularly important for interactive and

Four approches have been identified for providing QoS guar-real-time applications, where the timely delivery of packets isantees (2). They represent different tradeoffs between sim-crucial to the coherent reception of the audio or video signalplicity and efficiency; a simple approach has practical appeal,at the destination. In the connection establishment phase, ap-but it often involves conservative allocation of resources.plications provide their QoS requirements to the network. A

connection is admitted only if the network can guarantee therequested QoS on an end-to-end basis without adversely af- Controlled Deterministic Approach. The controlled deter-

ministic approach provides deterministic QoS guarantees byfecting the QoS of already admitted connections.

Figure 2. Statistical multiplexing (band-width is allocated on demand).

Stream #1

Stream #2

Stream #33 3

2 2

1 1 1 1

Statisticalmultiplexer

1 2 3 1 1 2 1 3


shaping the traffic to conform to a predefined traffic envelope the path. A drawback of this approach is that the bounds be-come loose as more nodes are traversed.(a time-invariant deterministic bound on the bit rate). An ex-

ample of a traffic envelope is the (�, �) model (8,9), which hasbeen extensively studied and has been used as the basis for Observation-Based Approach. In contrast to the previous

three approaches, the observation-based approach (11,12)the popular leaky-bucket traffic policing mechanism. The de-terministic approach, although easy to implement, has two provides no a priori quantitative commitments on perfor-

mance. Instead, it uses on-line measurements to determinemain disadvantages. First, traffic envelopes are inherentlyconservative, resulting in pessimistic performance predictions the current bandwidth demand and the admissibility of a new

connection under given QoS requirements. The guaranteesand poor bandwidth utilization. Second, because of the impactof statistical multiplexing on the characteristics of incoming are thus ‘‘predictive’’ based on the network status when the

connection was established. For this reason, advocates of thisstreams, enforcing a particular envelope requires shaping thetraffic after exiting each multiplexing node, which increases approach prefer the term assurances to indicate the qualita-

tive nature of the guarantees. Even though the observation-the hardware requirements of a switch.based approach is probably the simplest among the four ap-proaches, its qualitative nature precludes its use for nonadap-Approximate Stochastic Approach. This approach is appro-tive applications that require a priori, quantitative commit-priate for applications that contend with statistical guaran-ments on performance.tees. Here, traffic streams are characterized by stochastic

models that capture, to different degrees, the inherent ran-domness and fluctuations in the traffic. Statistical guarantees PRIORITY MECHANISMS ANDare provided by analyzing the performance of the multiplexer SCHEDULING AT A MULTIPLEXERas a queueing system. The multiplexer is modeled as aqueueing system of one or more finite-capacity queues that Traffic streams transported over B-ISDN are expected to haveare served by a common server. A queue in this context is a wide range of QoS requirements. Not only is this true forused as a surrogate to a memory buffer that accommodates a heterogeneous mix of traffic, but it is also true for certainarriving cells and queues them for switching onto the output individual traffic sources that generate cells with several losslink. The service rate is given by the transmission rate of the and/or delay requirements. For example, an MPEG (motionlink. An ATM cell that arrives at the multiplexer is served picture expert group) encoder uses layered coding to generateimmediately if the server is idle or is queued for service if a compressed-video stream that consists of a base layer andanother cell is being served. The problem of providing guaran- an enhancement layer. Information in the base layer is moretees can be formulated as follows: Given a number of traffic crucial to the reconstruction of the video signal. Ideally, thestreams that are statistically multiplexed and given a sto- network must guarantee the QoS for all connections while,chastic model that characterizes the individual streams or simultaneously, taking advantage of statistical multiplexing.their aggregate, find the bandwidth and buffer resources that To do that, the network may choose to provide indistinguish-must be allocated to these streams so that a given set (or sets) able transport service based on the most stringent QoS re-of statistical QoS are guaranteed. The statistical guarantees quirements. Such a strategy is too restrictive and signifi-are obtained from the buffer overflow probability and the cantly underutilizes network resources, particularly when theprobability distribution function for the delay in the queue. traffic streams with the most demanding requirements consti-The former gives an indication of the cell loss rate, whereas tute a small fraction of the total traffic. Alternatively, the net-the latter can be used to obtain various cell delay measures work can be designed to offer multiple bearer capabilities byat the node. Even though the end-to-end delay consists of assigning levels of ‘‘delivery’’ priority to incoming cells andpropagation, transmission, and queueing delays, only the offering differential service to these cells using priorityqueueing delay is variable and must thus be analyzed. One queueing mechanisms. Such priority mechanisms can be im-major problem with the approximate approach is that differ- plemented at various buffering stages in the network. The useent traffic models give rise to different queueing behaviors. of priority gives the network the flexibility to adjust dynami-Some sophisticated models are sufficiently accurate, but their cally to different traffic mixes, resulting in an increase in thequeueing analysis is not analytically tractable. total admissible load as compared to nonprioritization (13).

Priority mechanisms are also useful in other areas of trafficcontrol such as traffic policing.Bounding Stochastic Approach. Instead of employing de-

tailed stochastic models as in the approximate stochastic ap-Types of Priority Mechanismsproach, the bounding stochastic approach contends with sto-

chastic bounds on the number of arrivals in any interval of In general, the design of a priority mechanism involves twotime of length T, possibly for several values for T (10). This aspects: a service (or scheduling) discipline, which determinesapproach is the stochastic counterpart of the deterministic ap- the order in which cells in the buffer are served, and a bufferproach, where the traffic envelope here is specified in probabi- access discipline, which deals with admitting cells to the buff-listic terms. Aside from the bounds, no assumptions are made ers (14). Explicit or implicit priority rules may be applied toon the actual arrival pattern. The end-to-end guarantees are either or both disciplines. Accordingly, two types of priorityobtained by first obtaining stochastic bounds on the traffic at queueing mechanisms can be identified, based on where thethe edge of the network, which are then used to bound the priority rule is enforced: delay and loss priority mechanisms.departure traffic at that node. In turn, the bounded departuretraffic of one node is used to bound the arrival traffic at the Delay Priority Mechanisms. In a delay priority mechanism,

the priority rule takes place at the output of the buffer. It isnext node, and the procedure is repeated for all nodes along


in essence a scheduling algorithm, with higher priority cells BURSTINESS AND TRAFFIC CORRELATIONSreceiving preferential service over lower priority cells in the

Burstiness is an important characteristic of ATM traffic thatscheduling order. Delay priority mechanisms are quite usefulhas a profound impact on the multiplexing performance. Infor time-critical traffic, such as alarms and real-time controlsimple terms, burstiness indicates the presence of nonnegligi-messages in manufacturing environments. Examples of delayble positive correlations between cell interarrival times. Itpriority mechanisms are Head-Of-the-Line (HOL), Earliest-arises naturally as a consequence of segmenting variable-Deadline-First (EDF), Queue-Length Threshold (QLT), Mini-length packets at the sender into fixed-length cells that aremum-Laxity Threshold (MLT), and HOL with Priority Jumpsinjected into the network. After segmentation, a traffic stream(HOL-PJ) (3,15–17). A delay priority scheme can be static orlooks like a sequence of alternating active (ON) and idle (OFF)dynamic. In the former type, the priority rule does not adaptperiods where each ON period consists of a ‘‘train’’ of cells fol-to changes in the traffic mix or load conditions. In contrast,lowing each other. Therefore, even if the interarrival times ofpriority levels in a dynamic priority scheme are adjusted dy-packets are uncorrelated, cell interarrival times are stronglynamically to cope with traffic conditions. Both QLT and MLTcorrelated. The discovery of the bursty nature of ATM trafficare of this type.was a turning point in the study of network traffic. Tradi-tional renewal models, including the Poisson model, are noLoss Priority Mechanisms. The priority rule in this case islonger adequate as they tend to severely underestimate theapplied at the input to the buffer. Cells of higher classes havequeueing behavior at the multiplexer (25). And even if thepriority over cells of lower classes in terms of accessing thetraffic of a single source is approximated by a renewal pro-buffer. A loss priority mechanism is, therefore, a selectivecess, the traffic resulting from the superposition of a finitecell-discarding scheme, where a cell of a given class isnumber of sources is a complex nonrenewal process that isdropped (rather than delayed) to accommodate a higher prior-modulated (i.e., controlled) by the number of active sourcesity cell. Loss priority mechanisms were first introduced toat each instant (26). Correlations between arrivals have beencontrol congestion in ATM nodes (18). They are needed to pro-found to cause considerable degradation in network perfor-tect an ATM node from the stochastic fluctuations in the traf-mance (as measured by cell loss rate and delay jitter), whichfic, which may temporarily deplete network resources andcannot be predicted by a simple renewal model. The termcause congestion to develop. These schemes are also used toburstiness has been traditionally used to indicate a correlatedguarantee different cell loss rates for various classes of traffic.process in which the variance of the interarrival times isThe use of a loss priority scheme results in an increase in thegreater than the variance of the interarrival times of a Pois-total admissible load compared to no priority. Examples ofson process (27). However, burstiness is better quantified by

loss priority mechanisms are Push-Out (PO) (19) and Partial other measures, including the ones described next.Buffer Sharing (PBS) (13,20–22). In PO, cells enter oneshared buffer up to the maximum buffer size. If a high-prior- Indices of Dispersionity cell arrives at a saturated buffer that contains low-priority

The index of dispersion for intervals (IDI) and the index ofcells, a low-priority cell is dropped and its place is given todispersion for counts (IDC) are two popular measures ofthe high-priority cell. Despite its efficiency, PO requires aburstiness (25,27–29). Let �Xk, k � 1� be a sequence of inter-complicated buffer management to preserve the sequencing ofarrival times of a stationary arrival process. Let Sk � X1 �cells. PBS achieves loss priority by means of threshold-basedX2 � . . . � Xk, for all k. The IDI is defined by the sequencecell discarding. We now describe a generalized form of it,�c2

k, k � 1� (30), whereknown as Nested-Threshold Cell Discarding (NTCD) (21–24).Under NTCD (see example in Fig. 3) the buffer is partitionedby n thresholds, T1, . . ., Tn, that correspond to n � 1 priorityclasses. Cells of priority class i enter the buffer up to thresh-old level Ti. When the buffer level is above Ti, arriving cellsof class i are dropped. Note that only new arrivals are

c2k = kVar(Sk)

[E(Sk)]2 = Var(Sk)

k[E(X1)]2

= c21 +

∑ki, j=1,i �= j cov(Xi, Xj )

k[E(X1)]2 , k ≥ 1

(1)

dropped; class-i cells that are already in the buffer are neverdropped and are eventually served. NTCD results in a and cov(Xi, Xj) is the covariance of Xi and Xj. For k � 1, c2

k

slightly less total admissible load compared to PO (13), but it measures the cumulative covariance (normalized by theis less complex to implement in hardware. square mean) among k consecutive interarrival times of a

source. The significance of the IDI measure is related to thefact that the multiplexing performance is influenced by thecumulative effect of covariances, rather than the individualcovariances (25).

Let N(t) be the counting process associated with �Xk�:

N(t) �∞∑

k=1

1[Sk ≤ t] (2)

where 1[.] is the indicator function. The IDC is defined by thefunction

Source 1

Source 2

Source m Tn T2 T1

Shared buffer

. . . . .

Server

Incomingtraffic .

. .

. .

Figure 3. A loss priority mechanism at a multiplexer: NTCD withn thresholds.

I(t) = Var[N(t)]E[N(t)]

, t > 0 (3)


For a Poisson process, I(t) � c2k � 1 for all t and k. For a mentary service distribution:

renewal process, c2k � c2

1 for all k. Accordingly, we can test theappropriateness of the renewal assumption in a modelingproblem by examining the IDI of the empirical data.

ρ(t) �∫ ∞

0Gc(x)Gc(t + x) dx (7)

Typically, we are interested in evaluating the multiplexingperformance for a finite number of sources. In this case, it is where Gc(x) � 1 � G(x). Then peakedness can be written as

(32)more appropriate to consider the IDI and IDC for the aggre-gate traffic that is obtained from the superposition of severalsources. Let c2

k(n) and I(t, n) be the IDI and IDC for the super-position of n processes, respectively. Then, for mutually inde-

PG = 1 + µ

λ

∫ ∞

−∞[k(x) − λδ(x)]ρ(x) dx (8)

pendent and identical renewal processes, we have (30)where �(x) is the Dirac delta function and k(x) is the covari-ance density of the arrival process, which is defined bylim

t→∞I(t, n) = lim

k→∞c2

k(n) = c21(1) (4)

for any fixed n. The right equality says that for a fixed num-∂2cov[N(u),N(v)]

∂u∂v= k(u − v) = k(v − u), where u, v > 0 (9)

ber of streams, as more interarrival times are considered, theIDI tends to the coefficient of variation (ratio of variance to This definition of peakedness applies only to point processes.mean) of a single stream. This interesting result points Mark et al. (33) extended the definition to fluid processes andclearly to the inadequacy of Poisson modeling because, in batch arrivals. The new measure, known as modified peaked-practice, c2

1(1) � 1. Note, however, if n � � and the mean ness, relies on the concept of a rate process �R(t) : t � ��,arrival rate of each stream is �/n (i.e., individual processes where R(t)dt represents the amount of work (i.e., volume ofbecome ‘‘sparse’’ as more streams are added), then the super- arrivals) in the interval [t, t � dt). The counting process canposition process tends to a Poison process [the Palm Theorem now be expressed as N(t) � t

0 R(�)d�, for t � 0. The work is(31)]. In practice, n is finite, and the mean arrival rate is al- offered to a fictitious service system that is characterized bymost constant, independent of n. a service process �U(t), t � 0�, where U(t) can be interpreted

as the time the work R(t)dt spends in the system (the equiva-Peakedness lent of the service time). The continuous-time process �U(t),

t � 0� is stationary with marginal distribution G. For any pos-Peakedness is another statistical measure of burstiness thatitive t1 and t2 with t1 � t2, U(t1) and U(t2) are i.i.d. randomwas first used by teletraffic engineers to estimate callvariables. The busy-server process is �B(t), t � 0�, whereblocking probability at trunk groups (32). Consider a station-

ary point process with rate �. Each point corresponds to thearrival of a customer (a cell in the context of ATM). Let�N(t), t � 0� be the counting process associated with the ar-

B(t) �∫ t

01[U (x)>t−x](x)R(x) dx (10)

rival process. Arrivals are offered to a group of infinite serversModified peakedness is then defined bywith i.i.d. (independent and identically distributed) service

times and common service distribution G, which is also inde-pendent of the arrival process. Each customer is handled byits own server. Let B(t) be the number of busy servers at time

PG � limt→∞

Var[B(t)]

E[B(t)](11)

t. The peakedness of the arrival process with respect to a ser-vice distribution G is defined as Specifying the arrival process in terms of a rate process

makes it possible to define the modified peakedness measurefor processes that do not have the property of point arrivals,such as fluid processess and processes with batch arrivals.

PG = limt→∞

Var[B(t)]E[B(t)]

(5)

The peakedness of an arrival process is sometimes definedPERFORMANCE ANALYSIS OF A STATISTICAL MULTIPLEXER

with respect to a family of service distributions that have thesame form but differ in the value of one parameter, typically

Providing QoS guarantees necessitates analyzing the multi-the mean. In this case, peakedness is indicated as a function

plexing performance at various locations in the network andof that parameter. For example, Pexp(�) indicates the peaked-

computing the resources (bandwidth and buffer) needed to at-ness of an arrival process with respect to an exponential ser-

tain a certain level of QoS. Ideally, the performance shouldvice distribution with mean �. The peakedness has an appar-

be evaluated using measurements taken from an operationalent similarity to the IDC [see Eq. (3)]. In fact, the limit of the

ATM network. At this point, ATM is still evolving and it hasIDC function can be expressed in terms of the peakedness

not yet been deployed at a wide scale. In the absence of anwith respect to an exponential distribution (33):

Internet-like ATM network, studying the performance ofATM multiplexers is typically done by means of analysis or,when analysis is intractable, by simulations. In either case,lim

t→∞I(t) = 2Pexp(0+) − 1 (6)

the multiplexer is modeled as a queueing system. Its inputtraffic is characterized by some stochastic process or by a setIt is possible to express peakedness in terms of the second-

order statistics of the arrival process and the service distribu- of ‘‘real’’ traces that are captured from an experimental ATMtestbed. When real traces are used, the queueing performancetion. Let �(t) be the autocorrelation function for the comple-


is studied by means of discrete-event simulations, and the ap- Then, �St� is a Markov process ifproach is known as trace-driven simulations. When a stochas-tic model is assumed, the queueing performance is often ob-tained analytically. However, some stochastic models do not

Pr{St1≤ x1/St2

= x2, St3= x3, . . ., Stn = xn}

= Pr{St1≤ x1/St2

= x2} (12)lend themselves to tractable queueing analysis. Such modelscan still be used to generate synthetic traces (realizations of for any t1 � t2 � � � � � tn and any x1, . . ., xn in �. If thethe underlying model), and the multiplexing performance is state space is discrete, the process is called a Markov chainthen evaluated by means of trace-driven simulations. The and is associated with a transition probability matrix P thatsimulation-based approach has the disadvantage that it does describes the probabilities of going from one state to another.not provide on-line results that can be used in connection ad- The time spent in a given state is exponentially distributedmission control. Nonetheless, it can be used, for example, to with a parameter that depends on the current state. Typi-dimension network resources off-line to guarantee a predeter- cally, the transition probabilities are stationary, and P is ex-mined level of QoS. Moreover, a simulation-based approach pressed as P � [pij], where pij � Pr�Sn � xj/Sn�1 � xi�; xi andseparates the issues of traffic modeling and queueing evalua- xj are two states of the chain (discrete states are often takention, allowing highly accurate models to be employed (even as integers). When representing a traffic stream, transitionsif these models cannot be studied in an analytical queueing between states could represent cell arrivals.framework). It should also be mentioned that the queueingperformance under traffic models, although analytically trac- Markov-Renewal Models. A Markov-renewal model consiststable, is not always given in closed form. Thus, numerical of two processes: a Markov chain �Sn : n � 0, 1, . . .� and ancomputations must be performed to determine the measures associated transition-times process �Tn : n � 0, 1, . . .� (35). Atof interest, such as the cell loss rate. These computations can time n, the pair (Sn�1, Tn�1) of the next state depends only onbe quite expensive, precluding their use in on-line admission the current state Sn. A transition from one state to anothercontrol. could indicate a cell arrival. This model has the advantage of

allowing arbitrary interarrival times to be used, whereas onlyexponentially distributed interarrival times are possible in

TRAFFIC MODELS IN ATM NETWORKS the basic Markov model.

Markovian Arrival Process. Markovian arrival processesWe now discuss some of the traffic models that have been(MAP) are a subclass of Markov-renewal processes. MAPsused in studying the performance of an ATM multiplexer. Thehave recently attracted much attention because of their ver-vast majority of traffic models are stochastic in nature, sosatility and analytical tractability (36). As in phase-type re-they can be used to provide statistical guarantees only. Deter-newal processes, interarrival times in a MAP are obtainedministic traffic models, which are not discussed here, includefrom the time to reach absorption in a k-state Markov chainthe (�, �) model (8,9), the D-BIND model (34), and other enve-with one aborbing state and k � 1 transient states. However,lope-based models.in contrast to a phase-type renewal process, the distributionthat is used to restart the chain depends on the last transientRenewal Modelsstate from which the most recent absorption took place. This

Historically, queueing systems have been analyzed under re- way interarrivals are correlated in a Markovian fashion. Onenewal traffic models in which the interarrival times are i.i.d. important property of MAPs is that they obey a ‘‘superposi-A well-known example of renewal models is the Poison pro- tion rule’’: The superposition of two independent MAPs is acess, in which the interarrival times are exponentially distrib- MAP with an extended sample space. This property is quiteuted. Renewal models include the so-called phase-type re- useful in evaluating the performance of a statistical multi-newal processes (35) in which the interarrival times are plexer. For example, if one traffic stream is modeled as aderived from a continuous-time Markov process with discrete MAP, then the multiplexing performance for n such streamsstate space �0, 1, . . ., M�. State 0 is absorbing, whereas all is given by the queueing performance under a single MAPother states are transient. The Markov chain is initiated with with a larger state space. Extensions of MAPs includesome probability distribution. The first interarrival time is the batch MAP (BMAP) (37,38) and the discrete-time BMAPtaken as the time to reach absorption. Subsequent interar- (D-BMAP) (39).rival times are obtained similarly by restarting the chain withthe same initial distribution. Markov-Modulated Models. A modulated process is a dou-

bly stochastic process whose parameters are modulated (i.e.,Interest in renewal models stemmed from their simplicitycontrolled) by another stochastic process. Modulated pro-and analytical tractability. However, given the burstiness andcesses play an important role in traffic modeling. Their versa-the inherent correlations in ATM traffic, renewal models sig-tility enables them to capture traffic randomness at multiplenificantly underestimate the queueing performance, which istime scales. The simplest type of modulated processes uses agreatly affected by traffic correlations.Markov process for modulation. Here, the probability law ofthe modulated process depends on the state of a modulatingMarkov and Markov-Modulated ModelsMarkov chain. Each state gives rise to a different probability

Preliminaries. To account for traffic correlations, Marko- law. Typically, one parameter (e.g., the mean) is modulated.vian stochastic processes, which exhibit correlated interar- Popular Markov-modulated processes include the Markov-rival times, have been extensively studied. Let �St : t � �� be a modulated Poisson process and Markov-modulated fluid

models.continuous-time stochastic process with a sample space �.


Markov-Modulated Poisson Process. The Markov-modulated tion of cell-level and burst-level time scales, which is the un-derlying theme in the fluid approximation. Third, unlike thePoisson process (MMPP) is a Poisson process whose arrival

rate is a random variable that is modulated by the state of a point process approach, the computational complexity of fluidanalysis is independent of the buffer size, making the fluidcontinuous-time Markov chain. It is a correlated process that

enjoys a tractable queueing analysis. One of its interesting approach particularly useful for systems with large buffers.Fluid models were originally developed for data and voicefeatures is that, similar to a MAP, the MMPP obeys a super-

position rule: The superposition of two MMPPs is another sources (44,45). When transmitted over a constant-rate line,bursty data and packetized voice streams (with silence detec-MMPP with expanded state space (40). An MMPP can be

characterized by a generator matrix for the Markov chain and tors) exhibit the ON/OFF behavior. In the fluid model, the du-rations of the ON and OFF periods are random. During ON peri-an associated arrival rate matrix.

Various MMPP-based traffic models have been proposed in ods, the fluid arrives at a (constant) peak rate. The ON periods(as well as OFF periods) are i.i.d., often with exponentially dis-the literature. One model uses a two-state MMPP to charac-

terize a single stream that alternates between active (ON) tributed durations (the distribution for the ON periods is dif-ferent from that of the OFF periods). Other scenarios have alsoperiods and idle (OFF) periods. In this case, the arrival rate

during OFF periods is zero, and the MMPP reduces to an inter- been studied in the literature. The attractiveness of the expo-nential distribution is that it gives rise to a superposition rulerupted Poisson process (IPP). More commonly, an n-state

MMPP (with n � 2) is used to characterize the aggregate of that, in fact, applies to all Markov-modulated models (an im-portant property of the exponential distribution is that thevoice sources, each exhibiting an ON/OFF behavior (26) (the ON

periods in a voice source correspond to talkspurts, whereas minimum of several independent and exponentially distrib-uted random variables (rv) is another exponentially distrib-the OFF periods correspond to silence). Various results related

to the queueing performance under an MMPP arrival process uted rv). Consider the multiplexing of n homogeneous ON/OFF

fluid sources; ON and OFF periods are exponentially distrib-are summarized in Ref. 40 (see also Refs. 24 and 41–43).uted with means ��1 and r�1, repectively. Let � be the arrivalrate from one source during ON periods. The arrival process

FLUID MODELS characterizing a single stream is a two-state Markov-modu-lated fluid flow (MMFF) process, which is parameterized by a

The models discussed so far are all based on point processes, 2 � 2 infinitesimal generator matrix Q and an arrival-ratewhere the arrival of a cell is represented by a point on the vector � � (�0 �1) � (0 �), where �i is the arrival rate in statetime axis. A different approach to traffic modeling based on i, i � �0, 1�. The superposition of the n streams is an (n � 1)-the fluid approximation (44–46). Here, a traffic source is state MMFF with generator matrix Q(n) given byviewed as a stream of fluid that is characterized by a flowrate. The notion of discrete arrivals is lost as packets are as- Q(n) = Q ⊕ Q · · · ⊕ Q (n times) (13)sumed to be infinitesimally small (see Fig. 4). The fluid ap-proach has been found particularly appropriate to model the where � is the Kronecker sum. A state i in the Markov chaintraffic in ATM networks for a number of reasons (47). First, of the expanded MMFF means that i sources are simultane-this approach captures the bursty nature of ATM traffic. Sec- ously active (ON) and the remaining n � i sources are idle.ond, the traffic granularity, caused by small-size cells that are During state i, i � �0, 1, . . ., n�, the total arrival rate is �i.transmitted at very high speeds, makes the impact of individ- Fluid models enjoy tractable queueing analysis. In general,ual cells insignificant. This gives a justification for the separa- the queueing performance is obtained by formulating a set of

first-order linear differential equations that describe thebuffer occupancy at equilibrium in terms of the traffic param-eters and the service rate. This set is then solved as a general-ized eigenvalue/eigenvector problem (see References 44, 46,and 48 for details). Analytical results are also available forqueues with priority scheduling: Elwalid and Mitra (47) ana-lyzed a queue with multiple loss priorities and NTCD sched-uling; Zhang (49) analyzed a two-buffer system in which oneof the buffers has a complete preemptive priority (i.e., themultiplexer dedicates up to its full capacity to the high-prior-ity buffer, with the low-priority buffer being served only whenthe high-priority buffer is empty).

Even though the fluid approach is mathematically tracta-ble, the queueing results are often obtained numerically (ex-cept for a few cases in which closed-form solutions are avail-able). Unfortunately, the numerical procedure suffers frominherent numerical instability caused by the need to invertbadly scaled matrices. Significant computations are needed tocondition such matrices. The problem pertains to queues of

Cell Silence Burst

Time(a)

(b)

(c) finite capacity or queues that are partitioned by thresholds.Tucker (50) found a way to overcome the numerical problem,Figure 4. Point process and fluid representations of an ON/OFF trafficbut his solution works only for a finite-capacity queue with nosource: (a) actual stream, (b) point-process representation, and (c)

fluid representation. thresholds. Historically, the fluid approach used to suffer


techniques have been developed for video. These techniquesvary in their compression efficiency, which usually comes atthe expense of increased complexity in the encoder and de-coder design and, therefore, higher encoding/decoding delay.Delay can be a deciding factor in the selection of a compres-sion scheme. For example, interpolative motion compensa-tion, which is part of the MPEG compression technique, isa very efficient compression scheme. However, its complexityprevents it from being used in real-time video conferencing.

Many regression models have been proposed for various

Bit rate

MA

2AA

Time

ModelActual

types of video under different compression techniques. EarlierFigure 5. Quantization of the aggregate bit rate of multiplexed models are based on autoregressive (AR) and autoregressivevideo-phone sources. moving average (ARMA) processes (51,53–56). An AR process

of order p, AR(p), is a random process �Xn : n � 1, 2, . . .� thatis described by

from a ‘‘state explosion’’ problem resulting from a large statespace, but this problem was overcome by decomposition of thestate space followed by a functional inversion (48).

Xn = a0 +p∑

r=1

arXn−r + εn, n > 0 (14)

In addition to ON/OFF sources, the fluid approximation wasalso used to model variable bit rate (VBR) sources generated where in the context of video, Xn is the size of the nth frameby video-phones (51). Let Nvd be the number of video sources or, less commonly, a smaller unit than a frame, such as a slicearriving at a statistical multiplexer. In this model, the aggre- (a horizontal strip in a frame). The sequence ��n� consists ofgate bit rate of the Nvd sources is quantized into a number of i.i.d. random variables, known as the residuals, that give thediscrete levels, as shown in Fig. 5. The quantized aggregate AR model its stochastic nature. The residuals are often nor-bit rate constitutes a Markov chain. Transitions between mally distributed, with mean zero, which implies that Xn isstates follow the birth-death transition diagram of Fig. 6. At also normally distributed but with different mean and vari-state i, the total arrival rate is �vd(i) � Ai, i � �0, 1, . . ., M�, ance. In general, the autocorrelation function (ACF) of anwhere A is the quantization step (difference between two suc- AR(p) model is a mixture of damped exponentials and har-cessive levels) and M is the number of quantization levels. monics. It can be written as a difference equation (52):Transitions occur only between adjacent states (i.e., arrivalrate increases gradually). A transition from state i to statei � 1 occurs at an average rate of (M � 1)rvd. Likewise, the ρk =

p∑r=1

arρk−r (15)

average transition rate from state i to state i � 1 is i�vd. Theprocess has the tendency to go to a lower level at high rates

where �k is the ACF at lag k (autocovariance at lag k dividedand to a higher level at low rates. The values for rvd and �vd by the variance). An AR(1) model was used to characterize theare found by matching the mean, standard deviation, and au-

frame-size sequence of a video-phone under the conditionaltocorrelation function of the model and the empirical data.

replenishment compression algorithm (51). For other types ofvideo, higher-order AR models have been suggested, including

Regression ModelsAR(2) (53), and composite AR/Markovian models (54). ARMAmodels have also been applied to the modeling of videoRegression models have been extensively used in fitting em-

pirical time series arising in various domains, including fi- streams (57). For full-motion video, several regression modelsthat explicitly incorporate scene dynamics have been investi-nance, biological sciences, and engineering [cf. (52)]. In tele-

traffic studies, regression models are found particularly gated (58–61). In simple terms, a scene is a segment of amovie with no abrupt changes but possibly with some pan-suitable for characterizing compressed video streams. To

maintain constant-quality video, a video encoder generates ning and zooming (62). Frame sizes within a scene tend to bestrongly correlated. To model scene dynamics, a discretevariable-size compressed frames at a contant frame rate (e.g.,

30 frames per second in the NTSC standard), so that the out- AR(1) [DAR(1)] process has been suggested (53,63), in whichframe sizes are generated according to a finite-state Markovput stream has a variable bit rate. Characterizing the VBR

stream is equivalent to modeling the frame-size sequence. chain. After the chain enters a state, it stays there for a geo-metrically distributed random time, which corresponds to aFrame sizes are significantly affected by the scene dynamics.

More dynamics means less temporal redundancy in the video, scene length. The frame size stays constant during a scenebut varies from one scene to another according to a negativeand thus larger encoded frames. The size of a frame is also

influenced by the type of compression. Various compression binomial distribution (53) or a lognormal distribution (58,64).

Figure 6. State transition diagram in thevideo-phone fluid model.

0 1 2 M – 1

M vd

(M – 1) rvd 2 rvd rvdMrvd

M

µ(M – 1) vd µ2 vd µ vd µ


To provide better predictions of the queueing performance, Both types of background sequences can be shown to be Mar-kovian with a uniform marginal distribution, irrespective ofFrater et al. (59) enhanced the DAR(1) model by using a dif-

ferent distribution for scene durations. As in the original FV. However, the transition probabilities for �U�n � are time-

dependent (i.e., �U�n � is a nonhomogeneous Markov process).DAR(1) model, bit-rate variations within scenes were not in-

corporated. An elaborate scene-based model for MPEG-coded The modulo 1 operation results in sample paths that lookmore discontinuous at the origin than at other points. Tomovies was introduced (61). It incorporates the three types of

MPEG frames (I, P, and B). The model is composed of three make the sample path of a TES look more ‘‘homogeneous,’’ asmoothing operation is applied prior to the transformationsubmodels, one for each frame type, which are intermixed in

a deterministic, periodic manner. The sequence of I frames F �1. This smoothing operation is called a stitching transfor-mation and is given by the functionare modeled by the sum of two individually correlated pro-

cesses: one process captures the variations between scenes,whose durations are assumed to be exponentially distributed,whereas the other is an AR(2) process that captures the in-trascene variations. Without the AR(2) component, the I-

Sξ (x) ={

x/ξ, if 0 ≤ x < ξ

(1 − x)/(1 − ξ ), if ξ ≤ x < 1(19)

frame submodel is simply the DAR(1) model with lognormalframe-size distribution. Two renewal processes were used to where � is called a stitching factor. The stitching transforma-model the P and B sequences, with frame sizes having lognor- tion preserves the uniformity of the marginal distribution ofmal marginal distributions. the background sequence. The foreground sequence is now ob-

tained using Xn � F �1[S�(Un)], where Un is either U�n or U�

n .Note that the transformation FB is not needed because theTES Modelsbackground sequence consists of uniform variates.

Correlated random sequences can also be generated using the The TES approach was used to model various types ofTransform-Expand-Sample (TES) technique (65–67), which is ATM traffic, including H.261-encoded video (68), JPEG-en-a form of nonlinear regression. The TES approach attempts coded motion picture (69), and MPEG-encoded frame-levelto simultaneously capture the marginal distribution and the video (70). In addition, a Markov-modulated TES process thatACF of an empirical record. The marginal distribution can be accounts for scene dynamics was used to model JPEG-en-exactly matched, while the ACF is approximated. coded motion picture (71). The general approach to TES mod-

A TES model consists of two processes: a background pro- eling proceeds by searching for appropriate (�, FV) that resultscess �Un : n � 0, 1, . . .� and a foreground process �Xn : n � 0, in a sequence �S�(Un) : n � 0, 1 . . .� with an ACF that fits the1, . . .�. The background process defines a random walk on ACF of the empirical data. After this pair is determined, thethe unit circle (using modulo-1 arithmetic). It consists of a transformation F �1 is applied, resulting in a correlated se-sequence of correlated and identically distributed random quence with a desired distribution F. Because �S�(Un)� has avariables with a common distribution FB. The foreground pro- uniform marginal distribution, the transformation F �1 pre-cess is obtained by transforming the background process such serves the correlation structure of �S�(Un)�. This way, the twothat the resulting sequence has some target marginal distri- aspects of fitting the ACF and the marginal distribution arebution. In the general TES method, two transformations are decoupled.needed to generate a foreground sequence with a target mar- One drawback of the TES approach is that the ACF of aginal distribution F: TES process cannot be given analytically for lags beyond one.

Thus, the pair (�, FV) is obtained by systematically searchingXn = F−1{FB[(Un)]} (16)in the parameter space of � and FV. There is no guaranteethat the search will result in a good match to a given target

where F �1 (the inverse function) is known as the distortion ACF. Nonetheless, the TES approach is still one of the bestfunction. The target distribution F is often expressed in the available methods for generating correlated identically dis-form of a histogram. Typically, FB is a uniform distribution, tributed random variates.so only one transformation is needed to generate the fore-ground sequence: Xn � F �1(Un).

Long-Range Dependent ModelsTES processes can be classified into TES� and TES�, whichdiffer in the sign of the ACF at lag 1. Let x� indicate the A common aspect of all the models presented so far is thatmodulo 1 of x (i.e., the fractional part). The background se- the interarrival times are either uncorrelated or are corre-quence in TES� is defined by lated with an exponentially decaying ACF (e.g, Markovian

models). Such models give rise to a summable ACF (i.e.,��

k�0�k �). Recently, a number of studies supported by ex-tensive measurements indicated the presence of persistent

U+n =

{U0, if n = 0〈U+

n−1 + Vn〉, if n > 0(17)

correlations in various types of network traffic, includinglocal-area network (LAN) (72,73), wide-area network (WAN)where U0 is uniformly distributed on [0, 1) and �Vn� is an ‘‘in-(74), and VBR video traffic (75,76). This phenomenon, which isnovation sequence’’ of i.i.d. random variables with marginalknown as long-range dependence (LRD), has long been knowndistribution FV. The innovation sequence is independent ofin other domains of science, such as hydrolics and economicsU0. In TES� processes, the background sequence is given by[see (75) and the references therein]. It has been argued thatthe correlations persistence in network traffic cannot beadequately captured by Markov-like models. Instead, newmodels that exhibit the LRD behavior should be used to char-

U−n =

{U+

n , if n is even1 − U+

n , if n is odd(18)


acterize network traffic and capture its correlations at multi- means that the statistical properties of a stochastic processare invariant to the time scale. From Eq. (23), it can be shownple time scales.

The indication of the LRD phenomenon in network traffic thatspurred an ongoing debate on whether LRD models should beused in network dimensioning and resource allocation. The lim

m→∞ρ(m)

k = (1/2)δ2(k2−β) ≈ ck−β (for large k) (24)ramifications of LRD modeling are quite significant. For ex-ample, in contrast to Markovian models in which the bursti- where �(m)

k is the autocorrelation in �X(m)k � at lag k, �2(.) is the

ness is significantly tempered by statistical multiplexing, the central second difference operator, and c is a constant. A pro-multiplexing of LRD traffic streams can even increase traffic cess that satisfies Eq. (24) is said to exhibit asymptotic sec-burstiness. Although the persistence of traffic correlations is ond-order self-similarity (80). In contrast, a process exhibitswidely acknowledged, some researchers believe that captur- exact second-order self-similarity if �k � (��)�2(k2��) for all k,ing these correlations at all time scales is not needed to engi- which implies that �(m)

k � �k for all nonnegative integers k andneer the network. More specifically, they argue that because m. Note that for SRD processes �(m)

k � 0 as m � �, for all knetwork buffers are finite in size, correlations beyond a cer- (i.e., the process tends to white noise). In general, a processtain critical lag have no impact on the queueing performance �Xt� is said to be self-similar with parameter H, which isat a multiplexer (77–79). known as the Hurst parameter, if �Xat : t � 0� and �aHXt : t � 0�

have identical finite-dimensional distributions for all a � 0.LRD and Self-Similarity. Consider a second-order stationary In other words, a self-similar process exhibits the same statis-

process �Xn : n � 1, 2, . . .� with mean X and variance v. Let tical properties (scaled by aH) at all time scales. Interest isCk �

� cov(Xn, Xn�k) � E[(Xn � X)(Xn�k � X)]. The ACF is given often limited to second-order self-similarity.by �k � Ck/v, for k � 0, 1, . . .. An equivalent representation Of the several LRD models known in the literature, we willto the ACF is given by its power spectral density: examine two important ones: the fractional autoregressive in-

tegrated moving-average (F-ARIMA) process and the frac-tional Gaussian noise (FGN) process.g(ω) � (1/2π)

∞∑k=−∞

ρke−ikω (20)

Fractional ARIMA Model. Long-range dependence is dis-For m � 1, 2, . . ., let �X(m)

n � be the time series obtained by played by the F-ARIMA process, which is an extension of theaveraging the original series �Xn� over nonoverlapping blocks conventional ARIMA processes (52). An ARIMA(p, d, q) pro-of length m, that is, cess �Xn� is defined by

φ(B)∇dXn = θ(B)εn (25)X (m)n = 1

m(Xnm−m+1 + · · · + Xnm), for n = 1,2, . . . (21)

where �(B) and �(B) are polynomials of orders p and q, re-The variance of the new time series is given byspectively, in the delay operator B and � is the differencingoperator. In the F-ARIMA model, d is a fraction between 0and ��. The F-ARIMA(0, d, 0) model (i.e., p � q � 0) has beenvm � var(X (m)

n ) = vm

+ 2m2

m−1∑p=1

p∑q=1

Cq (22)used to characterize VBR video streams (76,81). Letting�(B) � �(B) � 1, F-ARIMA(0, d, 0) can be written as

The process �Xn� is said to be LRD if it satisfies any of thefollowing (virtually equivalent) conditions (80): ∇dXn = εn (26)

1. ��k�0�k � �. The fractional differencing can be expanded as follows:

2. g(w) � � as w � 0.3. mvm � � as m � �. ∇d = (1 − B)d =

∞∑k=0

(dk

)(−1)kBk (27)

If, on the other hand, ��k�0�k is finite, g(0) is finite, or

limj��mvm � constant, then �Xn� is said to exhibit short-rangedependence (SRD). Accordingly, all Markov-like models ex-hibit SRD. To exhibit LRD, the ACF of a model must drop off

(dk

)= �(d + 1)

�(k + 1)�(d − k + 1)(28)

slowly, so that the autocorrelations have an infinite sum.where �(x) �

� �

0 tx�1e�tdt is the gamma function. When dNote that LRD is determined by the asymptotic behavior ofis a positive integer, �(d � 1) � d!. The ACF of thethe ACF (the sum of correlations up to a finite lag, no matterF-ARIMA(0, d, 0) model behaves asymptotically ashow large, does not determine whether or not the model is

LRD). One particular form of LRD which received much at-tention is when ρk = �(1 − d)

�(d)k2d−1(29)

ρk ∼ ck−β as k → ∞ (23)Thus, for 0 d 0.5, the model exhibits LRD.

where 0 � 1 and c is a constant. The slow decline of thepower function results in a nonsummable ACF. Fractional Gaussian Noise Model. FGN is an exactly second-

order self-similar process that is obtained from the stationaryRelated to the LRD phenomenon is another interestingconcept known as self-similarity, which in general terms increments of a fractional Brownian motion (FBM). In itself,


FBM is a self-similar Gaussian process with Hurst parameter EFFECTIVE BANDWIDTHH � (0, 1). For the discrete-time case, the ACF of the (dis-crete) FGN is given by Statistical multiplexing improves network utilization by

allowing bursty sources to share bandwidth on demand, sothat the allocated bandwidth per source is less than theρk = 1

2

(|k + 1|2H − 2|k|2H + |k − 1|2H)

(30)source peak rate. For the network to take advantage of statis-tical multiplexing, it should be able to determine the approxi-For H � (0.5, 1) �k � H(2H � 1)k2H�2 as k � �, and the FGNmate minimum required bandwidth per source as a functionprocess is LRD.of the QoS, the buffer size at the multiplexer, and the trafficparameters. This bandwidth is commonly known as the effec-M/G/� Input Processestive bandwidth (or equivalent capacity). After the bandwidth

M/G/� processes constitute a versatile class of models that is is computed, the effective bandwidth can be used as the basiscapable of displaying many forms of correlations, including for call admission control (CAC). Intuitively, the effectiveshort-range and long-range dependence. They arise naturally bandwidth of a stream lies between the peak rate and theas the limiting case for the superposition of many ON/OFF mean rate of that stream.sources with ON periods having a ‘‘heavy-tailed’’ distribution As a surrogate to a statistical multiplexer, consider an in-such as the Pareto distribution (82). Recently, M/G/� pro- finite-capacity queueing system with a single server and first-cesses have been proposed as a viable modeling approach for in-first-out (FIFO) service discipline. Input traffic consists ofvarious types of network traffic (83,84). So far they have been n independent, possibly heterogeneous, Markovian sourcesapplied in the modeling of VBR intracoded video streams (85). (e.g, fluid sources, MMPPs, MAPs). Let W be the waiting time

An M/G/� process can be defined as follows: consider a of an arbitrary cell in the queue before it gets served. Un-discrete-time M/G/� queue in which customers arrive in i.i.d. der very general assumptions, the asymptotic behavior ofPoisson batches of mean �. Let �n�1 be the size of the (n � the complementary distribution G(x) for the waiting time is1)th batch (i.e., the number of arrivals during time slot [n, given byn � 1)). Upon arriving at the system, customers are presentedto an infinite group of servers. Arrivals during time slot [n, G(x) � P[W > x] ∼ αeηx, as x → ∞ (33)n � 1) are serviced at the beginning of slot [n � 1, n � 2). Let�n�1,1 . . ., �n�1,�n�1 be the integer-valued service times for cus- where and � are called the asymptotic decay rate and asymp-tomers 1, 2, . . ., �n�1 of the (n � 1)th batch, respectively. It totic constant, respectively.is assumed that service times are i.i.d. with a common distri- The literature provides several approximations for the ef-bution G. We use � to indicate a generic rv for the service fective bandwidth. The simplest of these is based on approxi-time of a customer. Initially, there are b0� customers in the mating Eq. (33) bysystem with corresponding (residual) service times �0,1, . . .,�0,b0

�, which are mutually independent. Let bn be the number P[W > x] ≈ eηx (34)of busy servers (i.e., remaining customers) at time n�, n � 0,1, . . . (after counting arrivals and departures at the start of (i.e., � is set to one) and using as the basis for computingslot [n, n � 1)). The process �bn : n � 0, 1, . . .� is known as the the effective bandwidth (87). This one-parameter approxima-M/G/� input process. tion, which we will refer to as the asymptotic-rate-of-decay

It has been shown that �bn� can display various forms of (ARD) approximation, is quite appealing from a practicalpositive autocorrelations, the extent of which is controlled by standpoint because it allows CAC to be designed solely basedthe tail behavior of G (83). In fact, it can even exhibit LRD on basic characteristics of the input streams. Interestingly,when G is a Pareto distribution (80). In general, �bn� is not according to the ARD approximation, the effective bandwidthstationary, but it admits a stationary and ergodic version of a source is independent of the characteristics of the other�b*n� (86), which is typically used as the basis for modeling. sources at the multiplexer. For general Markovian processes,The ACF for the process �b*n � is given by the ARD approximation of the effective bandwidth of a source

is determined based on the maximal real eigenvalue of a ma-trix that is derived from the source parameters, network re-ρk = e−uk , for k = 0, 1, . . . (31)sources, and service requirements. Let p be the target over-

where uk �� ln P[� � k] and � is the forward recurrence flow probability to be guaranteed by the network. For a buffer

associated with the service time �: of size B, the QoS is satisfied if G(B) � p, where G( � ) wasdefined in Eq. (33).

One special type of Markovian processes for which theARD approximation was computed is given by Markov-modu-

P[σ = i] = P[σ ≥ i]E[σ ]

, i = 1, 2, . . . (32)

lated fluid flow processes. First, consider a multiplexer withonly one input source, which is characterized by an MMFFEquation (31) relates the monotonic behavior of the ACF to

the service distribution G. By varying G, we can obtain vari- process with generator matrix M (the infinitesimal generatormatrix of the Markov chain) and arrival rates � � (�1 � � �ous correlation structures. For example, a Weibull-like G was

chosen in characterizing VBR video streams (85), so that the �S), where �i is the fluid-flow rate during state i, i � �1, . . .,S�. Then, as p � 0 and B � � with log p/B � � � [��, 0],resulting ACF has the form �k � e��k, which provided a good

fit to the empirical ACF. Even though this ACF is summable the ARD approximation of the effective bandwidth is given bythe maximal real eigenvalue of the matrix � � (1/�)M, where(i.e., the model is SRD), it does not exhibit a Markovian

structure. � � diag(�). Now consider N multiplexed MMFF sources that


are characterized by (M(j), �(j)), j � 1, 2, . . ., N. The effective traffic descriptors for VBR video is a challenging research is-sue that awaits further work.bandwidth for the superposition of these sources is given by

the sum of their individual effective bandwidths, which aregiven by the maximal real eigenvalue of the matrices

BIBLIOGRAPHY�(j) � (1/�)M(j), j � 1, 2, . . ., N. The ARD approximation isalso available for MMPP sources, where the effective band-

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