Stochastic Models in Telecommunications for Optimal Design, Control and Performance Evaluation
N. Gautam Department of Industrial and Manufacturing Engineering, The Pennsylvania State University, 310 Leonhard Building, University Park, PA 16802 [email protected]
Abstract
In this paper stochastic models for high-speed networks viz. traffic models, buffer content process models, LAN models, etc are considered. The stochastic models are used to obtain end-to-end Quality-of-Service (QoS) measures that the network must guarantee users in the future. The QoS measures or performance measures can be used in optimal design problems and admission control problems. Other aspects such as TCP, routing, leaky buckets, wireless networks, etc are also considered.
1 1 Introduction
One of the greatest success stories for stochastic models in engineering is in the field of telecommunications. Discrete and fluid queueing models have played a major role in the development of computer and communication networks. There are several branches of telecommunications that use stochastic models, however, in this paper the main focus is in networking systems and other stochastic systems that aide high-performance networking. A few other applications are briefly mentioned later in this introduction. There are several interesting scenarios in the Internet and the emerging next generation networks (such as Internet2, NGI, etc) where stochastic modeling is applicable. The future networks will carry a wide variety of traffic (Data, Voice, Video, etc) and the users will demand very high-quality from the networks. Therefore it is very important to consider certain performance issues known as Quality-of-Service (QoS). There are four well-known end-to-end QoS measures, viz., loss probability, delay, delay-jitter and bandwidth. They are briefly described as follows: When messages flow from a source to a destination (end-to-end) through a network, parts of a message or the whole message may be dropped due to unavailable resources (buffer capacity) to store the messages. The probability of delivering a message with some data loss is termed as loss probability. The time between the source sending a message and the destination receiving it is called latency or delay. Typically real-time or multimedia traffic (such as live video conference) can tolerate some loss but have very stringent delay requirements. However data traffic such as emails, fax, file transfers, etc can tolerate some delay but almost zero loss. The other QoS measures are delay-jitter (which is a measure of the variation in the delay) and bandwidth (which is the rate at which messages are processed). The message flow (will be called traffic henceforth) and the network conditions are ex- tremely stochastic in nature. Given the growth in the Internet as well as users demanding QoS for their applications, it is important to be able to predict the QoS measures as they will have to be guaranteed to the users. Also the QoS measures can be used for optimal design and admission control of the networks. Some of the main design aspects include buffer sizes, link capacities, network parameters, traffic shaping parameters, etc. While ex- ercising admission control, the network either rejects an incoming request for connection or accepts it (and provides the required QoS). As mentioned earlier, the main concentration of this paper will be on high-speed telecom-
1 munication networks. Other applications of stochastic processes in communications include coding theory, signal processing, image processing, pattern recognition, speech recognition, etc. Stochastic models using hidden Markov processes (for a thorough exposition of hidden Markov models, see Rabiner [68]), hidden semi-Markov models, Markov decision processes, etc are used in signal processing, image processing, pattern recognition and speech recogni- tion. Broadly there are three types of telecommunication networks – telephony (telephone networks for voice calls, fax, and also dial-up connections), cable-TV networks (cable, web- TV, etc), and high-speed networks such as the Internet. This paper focuses on high-speed networks with the motivation that in the very near future, internet telephony, video-on- demand, networked homes, multimedia applications, etc will possibly replace telephone and cable-TV networks due to low cost. However, unless the performance of high-speed networks improves greatly this will not be possible!
2 Traffic Models
Traffic flowing through the networks can be classified into several types. Two of the most common traffic types are ethernet packets/frames and ATM cells. Depending on the network segment, all messages are broken down into either packets or cells. The length or size of an ethernet packet ranges anywhere from 60 bytes to 1500 bytes and generally follows a bimodal distribution. The length of ATM cells is fixed at 53 bytes. Therefore the network traffic comprises of millions and billions of these little packets or cells! One of the most important tasks before evaluating the performance of telecommunication networks is to fit appropriate models for traffic to capture their stochastic nature. Data can be obtained by using “sniffers” on the network and analyzing a “dump” of all the packets or cells that were generated during the time the sniffer was used. The information that can be obtained about each packet or cell by sniffing include: its arrival time, its source, its destination, its length, its type, etc. To fit traffic models, only the time of arrival and packet size are sufficient.
2.1 Hierarchical Networks
Telecommunication networks are typically hierarchical in nature. Generally, traffic can be classified into four levels :
2 • Application Level : The traffic generated by an application, say, http or telnet or ftp which can vary significantly based on the protocols they follow.
• Source Level : Each workstation or computer can be thought of as a source that generates traffic. This traffic comprises of the traffic generated by different applications that are running on the source. Therefore the traffic that flows on a link that exits the computer is a mixture of the different applications. The process of mixing is known as multiplexing.
• Aggregate Level : Several computer, printers, etc are connected together to form a local area network (LAN). The traffic on a LAN pipe is the aggregated traffic that is multiplexed from all the sources.
• Backbone Level : The LANs are connected together by means of a backbone (say, the Internet backbone), and this forms the Metropolitan Arean Networks (MANs) or the Wide Area Networks (WANs). The traffic on a MAN/WAN pipe is the combination of the traffic from several LANs.
Appropriate traffic models can be used depending on the levels being considered. Some frequently used stochastic models for traffic flow are explained in the next section. Although, different researchers prefer to use different traffic models, the models can be broadly classified into two parts, discrete models and fluid models. In the discrete model each packet or cell is assumed to be a discrete entity that can be of varying sizes. In the fluid models it is assumed that the packets or cells are packed so close to each other that the traffic flow can be assumed to be a fluid flowing across a pipe, maybe at different rates.
2.2 Fluid-flow Traffic Models
In the fluid-flow models it is assumed that traffic is in the form of fluid which flows through a pipe at different rates at different times. For example, fluid flows at rate r(1) bytes per
second for a random amount of time t1,thenflowsatrater(2) bytes per second for a random
amount of time t2, and so on. This behaviour can be captured as a discrete stochastic process that jumps from one state to another whenever the traffic flow rate changes. This can be formalized as a stochastic process {Z(t),t≥ 0} that is in state Z(t)attimet. Fluid flows in thepipeatrater(Z(t)) at time t. Researchers have used different models for the {Z(t),t≥ 0} process that are summarized in the following description :
3 2.2.1 DTMC environmental processes
Consider a DTMC where a transition occurs every θ seconds. When the DTMC is in state
i, θr(i) bytes flow through the pipe. Let P =[pij] be the transition probability matrix,
where pij is the probability that the DTMC goes from state i to state j in one-step. For an irreducible, aperiodic and positive recurrent DTMC, let π be the steady-state distribution such that π = πP and πa =1witha being a column vector of ones.
2.2.2 CTMC environmental processes
Let {Z(t),t≥ 0} be an irreducible, finite state CTMC with generator matrix Q. When the
CTMC is in state i, traffic flows at rate r(i). Let R = diag[rii], where rii = r(i). Let p be the stationary distribution of the CTMC such that pQ =0andpa =1witha being a column vector of ones.
2.2.3 Alternating Renewal environmental processes
This is sometimes known as the on-off traffic where traffic either flows at the maximum link (or pipe) capacity r bytes per second or no traffic flows. The up times or on times (when traffic flows through the pipe) are distributed according to a general CDF U(·). The down or off times (when traffic does not flow through the pipe) are distributed according to a general CDF D(·).
2.2.4 SMP environmental processes
Consider a Semi-Markov Process (SMP) {Z(t),t ≥ 0} on state space {1, 2, ..., }. Fluid is generated at rate r(i)attimet when the SMP is in state Z(t)=i.LetSn denote the time of
the nth jump epoch in the SMP with S0 =0.DefineZn as the state of the SMP immediately after the nth jump, i.e.,
Zn = Z(Sn+).
Let
Gij(x)=P {S1 ≤ x; Z1 = j|Z0 = i}. (1)
The kernel of the SMP is
G(x)=[Gij(x)]i,j=1,..., .
4 Note that {Zn,n ≥ 0} is a discrete time Markov chain (DTMC) which is embedded in the SMP. Assume that this DTMC is irreducible and recurrent with transition probability matrix
P = G(∞).
Let Gi(x)=P {S1 ≤ x|Z0 = i)= Gij(x) j=1 and the expected time the SMP spends in state i be
τi = E(S1|Z0 = i).
Let
i { n } π =n lim→∞ P Z = i
be the stationary distribution of the DTMC {Zn,n ≥ 0}. Itisgivenbytheuniquenon- negative solution to π = πP and πi =1. i The stationary distribution of the SMP is given by
πiτi pi = lim P {Z(t)=i} = . (2) t→∞ πmτm m=1
2.2.5 MRGP environmental processes
Consider a regular Markov regenerative process {Z(t),t ≥ 0}. For the definition of MRGP see Cinlar [20], who calls it a semi-regenerative process. Also see Heyman and Sobel [42]
and Kulkarni [51]. Let {(Yn,Sn),n ≥ 0} be an embedded Markov renewal sequence in the
MRGP. Assume that {Yn,n≥ 0} has a finite state-space {1, 2, ..., }. From the definition of the Markov renewal sequences,
P {Yn+1 = j, Sn+1 − Sn ≤ x|Yn = i, Sn, ..., Y0,S0} = P {Y1 = j, S1 ≤ x|Y0 = i} (3)
for all x ≥ 0, and i, j =1, 2, ..., . Furthermore, given the history {Z(t), 0 ≤ t ≤ Sn}
and {(Yk,Sk), 0 ≤ k ≤ n}, the future of the Z(t) process, viz., {Z(t),t ≥ Sn}, depends on
the past only via Yn. Assume that the MRGP is regular, i.e., Sn →∞with probability
1asn →∞. It is clear from Eq. (3) that {Yn,n ≥ 0} is a discrete time Markov chain
5 (DTMC). Assume it is irreducible. Let µk = E[S1|Y0 = k], π =(πk) is a positive solution to π = πG(∞), and,
αkj = E( time spent by the Z(t) process in state j during [0,S1)|Y0 = k).
Then, k πkαkj pj = lim P {Z(t)=j} = . t→∞ k πkµk
2.3 Discrete Traffic Models
In the discrete traffic models it is assumed that the packet flow is in the form of discrete entities (like cars going on a highway) through the pipes. Therefore it is important to characterize the arrival process and the size of the packets. Some typical arrival processes, frequently used by researchers are explained below. The packet size distribution is usually a bimodal empirical distribution and is not discussed here.
2.3.1 Poisson Process
If the interarrival distribution of the packets can be modeled using an independent and identically distributed exponential distribution, then the packet arrival process can be char- acterized as a Poisson process with parameter λ,whereλ is the mean arrival rate (in terms of number of packets per unit time).
2.3.2 MMPP (Markov-Modulated Poisson Process)
Whenever a new connection is established through a pipe or an existing connection is ter- minated, the mean packet flow rate (number of packets per unit time) increases or decreases respectively. To account for this behavior, the traffic arrival process is modeled as an MMPP. Consider a CTMC {Y (t),t ≥ 0} with generator matrix Q. When the CTMC is in state i, packets flow according to a Poisson process with mean rate λ(i). This captures the change in mean arrival rates effectively.
2.3.3 BMAP (Batch Markovian Arrival Process)
The batch Markovian arrival process (BMAP) is explained in detail in Lucantoni [58]. The BMAP is a generalization of the Markovian arrival process (MAP) which was introduced by Lucantoni et al [57]. A special case of the MAP are the phase type renewal processes and
6 the MMPP. The following definition and properties of BMAP is reproduced from Lucantoni et al [59].
Consider a series of m × m matrices Dk, k ≥ 0, such that D0 has negative diagonal
elements and nonnegative off-diagonal elements and for k ≥ 1, Dk are nonnegative. Define an irreducible infinitesimal generator D such that ∞ D = Dk. k=0
To assure that arrivals will occur assume that D = D0. Consider a two-dimensional Markov process {N(t),J(t),t≥ 0} on the state space {(i, j): i ≥ 0, 1 ≤ j ≤ m} with an infinitesimal generator Q given by D D D D ... 0 1 2 3 D0 D1 D2 ... Q = D D ... 0 1 D0 ... ... Here, N(t) counts the number of attivals in time t and J(t) represents a state or phase. For example, a transition from state (i, j) to state (i + k, l), k ≥ 1, 1 ≤ j, l ≤ m denotes a batch
arrival of size k and thus the batch size can depend on j and l. The matrix D0 is nonsingular and the sojourn time in the set of states {(i, j):1≤ j ≤ m} is finite w.p. 1. Thus the arrival process does not terminate. Let π denote the stationary probability vector of the Markov process with generator D such that πD =0,πe=1.
The mean arrival rate of the process is hence ∞ λ = π kDke = πd. k=1
One can think of D0 as governing transitions in the phase process which do not generate
arrivals and Dk as the rate of arrivals of size k (with the appropriate phase change). As a simple example, for Poisson arrivals with mean arrival rate λ, m =1,D0 = −λ, D1 = λ and
Dk = 0 for all k ≥ 2.
2.3.4 Fractals or Self-similar Arrival Process
Willinger et all [79] describe the latest developments and advances in using self-similar traffic for performance modeling of high-speed telecommunication networks. Here, the notations
7 and descriptions follow Willinger and Paxson [78]. Experimental traces of traffic processes exhibit high spatial variability and long-range dependence (autocorrelations with a power law decay). Heavy-tailed distributions (such as Pareto distributions) with infinite variance are used to model the extreme spacial variability. Typical probability distributions [F (·)] are of the form
−β 1 − F (x)=κ1x , where κ1 is a positive (finite) constant independent of x and the tail index β is such that 0 <β<2. A fractional Gaussian noise is used to model the fractal or long-range dependent or self-similar behavior. A covariance-stationary Gaussian process X =(Xk : k ≥ 1) is called a fractional Gaussian noise with Hurst parameter H ∈ [0.5, 1) if the autocorrelation between Xn and Xn+k, k ≥ 0, is given by
2H 2H 2H cor(Xn,Xn+k)=0.5 (k +1) − 2(k) +(k − 1) .
The Hurst parameter H quantifies the strength of the fractal scaling.
A discrete-time, covariance-stationary, zero-mean stochastic process X =(Xk : k ≥ 1) is called exactly self-similar or fractal with scaling parameter H ∈ [0.5, 1) if for all levels of aggregation (or resolution), m ≥ 1,
X(m) = mH−1X, where the aggregated processes X(m) are defined by
X m− k + ...+ Xkm X(m)(k)= ( 1) +1 ,k≥ 1. m For an exactly self-similar process with scaling parameter H,
(m) 2H−2 VarX = κ1m .
2.3.5 Fractional Brownian Motion vs. Levy Processes
The following is directly adapted from Konstantopoulos and Lin [48]. A Levy motion {Zt} is a process with stationary independent increments, and its marginal distribution is a stable random variable (invariant under affine transformations). It is a self-similar process with D 1/α Hurst constant 1/α : {Ztx,t≥ 0} = {x Zt,t≥ 0}. Here, {Zt} is a zero mean process and
Z1 has finite β-moments for any 0 <β<α.Sinceα>1, {Zt} is a martingale, in fact an
8 Lβ martingale for any 1 <β<α. Note that the fractional Brownian motion is not even a semi-martingale. Konstantopoulos and Lin [48] consider multiplexing of multiple number of sessions such that the sessions arrive according to a Poisson process with mean rate λ and each session remains active for a random time T with a heavy-tailed distribution
1 − F (x)=P {T>x}∼κx−α, as x →∞
for a fixed 1 <α<2. Also, when each session is active, traffic is generated at the rate of 1 per unit time. Konstantopoulos and Lin [48] show that (i) in the limit the arrival processes converge to a Levy motion with stable non-Gaussian independent increments, and, (ii) the autocorrelation function is asymptotically that of a fractional Brownian motion and not a Levy motion!
3 Network Performance Using Traffic Models
When an application (also called source) sends a message to a destination, the message tra- verses several nodes (also called hops or network interfaces) before reaching the destination. The messages are stored in buffers at the nodes briefly before proceeding on to the next hop. At these nodes, traffic from other applications are either superposed along with or split from this application’s traffic. In this section the traffic models considered in Section 2 will be used to model buffer content processes and evaluate performance measures.
3.1 Single Class Single Buffer Content Processes
To begin the analysis, first concentrate on the simplest model: a network with a single node, a single class of traffic, hence a single buffer. The main aim is to obtain the probability distribution of the buffer contents in the long-run given a traffic model of the input to the buffer and the buffer emptying scheme.
3.1.1 Effective bandwidths of all the traffic model types
First consider the source generating traffic into the buffer. The concept of effective bandwidth of traffic generated by a source or traffic stream flowing through a telecommunication pipe or link is explained. Let A(t) be the total amount of traffic (fluid or discrete) generated by a source or flowing through a pipe over time (0,t]. For the following analysis consider a fluid model. Note that it is straightforward to perform similar analysis for discrete models as well.
9 Consider a stochastic process {Z(t),t≥ 0} that models the traffic flow. Also let r(Z(t)) be the rate at which the traffic flows at time t.Then
t A(t)= r(Z(u))du. (4) 0 The asymptotic log moment generating function (ALMGF) of the traffic is defined as 1 h(v) = lim log E{exp(vA(t))}. (5) t→∞ t Using Equation (5) one can show that h(v) is an increasing, convex function of v and for all v>0,
rmean ≤ h(v) ≤ rpeak. (6)
where rmean = E(r(Z(∞))),
rpeak =sup{r(z)}, z and h(v) denotes the derivative of h(v) with respect to v. The Effective Bandwidth of the traffic is defined as
eb(v)=h(v)/v. (7)
It can be shown that eb(v) is an increasing function of v and
rmean ≤ eb(v) ≤ rpeak,
and, lim eb(v)=rmean and lim eb(v)=rpeak. v→0 v→∞ It is not easy to calculate the effective bandwidths using Equation (7). However, when {Z(t),t ≥ 0} is a Continuous Time Markov Chain (CTMC), or a regenerative process, or a Markov Regenerative Process (MRGP), one can compute the effective bandwidths more easily. The methods are illustrated briefly.
1. CTMC Source : Elwalid and Mitra [27], and Kesidis et al [47], use eigenvalue tech- niques to show how to compute the effective bandwidths of sources that are modulated by CTMCs as follows. Let {Z(t),t ≥ 0} be an irreducible, finite state CTMC with generator matrix Q. When the CTMC is in state i, the source generates fluid at rate
10 r(i). Let R = diag[rii], where rii = r(i). Let e(M) denote the largest real-eigenvalue of a square matrix M. Then,
h(v)=e(Q + vR). (8)
2. MRGP Source : Kulkarni [50] and Gautam [33] show how to compute the effec- tive bandwidths of sources that are modulated by Markov Regenerative Processes and Regenerative Processes. Let {Z(t),t ≥ 0} be an m-state Markov Regenerative Pro-
cess (MRGP). Let {(Yn,Sn),n ≥ 0} be an embedded Markov renewal sequence in
the MRGP. Assume that {Yn,n ≥ 0} is an irreducible Discrete Time Markov Chain (DTMC) with a finite state-space {1, 2,...,m}.Let
S1 F1 = r(Z(t))dt 0
be the total fluid generated by the source during [0,S1]. Define
−uS1+vF1 Λij(u, v)=E{e ; Y1 = j | Y0 = i}, (9)
for i, j =1, 2,...,m and −∞
Λ(u, v)=[Λij(u, v)]
be an m × m matrix. Let e(Λ(u, v)) be the largest real-positive eigenvalue of Λ(u, v).
e(Λ (u,v))
e* (v)
1
u* (v) u
Figure 1: e(Λ(u, v)) vs u
Define e∗(v) = sup {e(Λ(u, v))} {u>0:e(Λ(u,v))<∞}
11 and u∗(v)=inf{u>0:e(Λ(u, v)) < ∞}.
Then for a given v,
(a) if e∗(v) ≥ 1(seeFigure1),h(v) is a unique solution to e(Λ(h(v),v)) = 1,
(b) if e∗(v) < 1(seeFigure2),h(v)=u∗(v).
e(Λ (u,v))
1
e* (v)
u* (v) u
Figure 2: e(Λ(u, v)) vs u
3. Regenerative Source : Assume that {Z(t),t ≥ 0} is a regenerative process with
regeneration epochs {Sn,n≥ 0},withS0 =0.Let
S1 F1 = r(Z(t))dt 0
be the total fluid generated by the source during [0,S1]. Define
Λ(u, v)=E{e−uS1+vF1 }. (10)
Since Λ(u, v) is a scalar, e(Λ(u, v)) = Λ(u, v). Following the technique in the MRGP case above, define e∗(v) = sup {e(Λ(u, v))} {u>0:e(Λ(u,v))<∞} and u∗(v)=inf{u>0:e(Λ(u, v)) < ∞}.
For a given v,
12 (a) if e∗(v) ≥ 1, h(v) is a unique solution to e(Λ(h(v),v)) = 1,
(b) if e∗(v) < 1, h(v)=u∗(v).
Table 1 summarizes the effective bandwidths of some discrete ATM traffic in the forms of cells (see Krishnan et al [49] for the calculation of effective bandwidths for traffic modeled by fractional brownian motion).
SOURCE TYPE EFFECTIVE BANDWIDTH constant arrival rate of R cells/second R Poisson source with intensity R(eδ−1) R cells/second δ irreducible and aperiodic discrete-time Markov source with transition probability matrix P, rate matrix ψ = diag(ψ1,...,ψm) where m is the size of the state space of of the DTMC, ψi is the number of cells that arrive when in state i,and R log[ρ(eδψP)] ρ(A) is the spectral radius of matrix A δ Markov Modulated Poisson Process (MMPP) with intensity ψ = diag(ψi), where ψ is a function of a CTMC with infinitesimal generator Q,andµ(A)isthelargest µ(Q+(eδ−1)ψ) eigenvalue of the matrix A δ
Table 1: Effective Bandwidth of Input Sources.
3.1.2 Approximate Methods And Bounds (via martingales)
Z 1 (t) 2 Z (t) X(t) C Z K (t)
Figure 3: Single Buffer Fluid Model
Consider a single buffer that admits a single-class traffic from K independent sources, each driven by a random environment process {Zk(t),t≥ 0} (see Figure 3). Note that Zk(t) can be thought of as the state of the kth input source (k =1, 2,...,K)attimet.When
13 source k is in state Zk(t), it generates fluid at rate rk(Zk(t)) into the buffer. Let X(t)be the amount of fluid in the buffer at time t. The buffer has infinite capacity and is serviced by a channel of constant rate c. The dynamics of the buffer-content process {X(t),t≥ 0} is
described by K k k − dX(t) k=1 r (Z (t)) c if X(t) > 0 = { K k k − }+ (11) dt k=1 r (Z (t)) c if X(t)=0. where {x}+ =max(x, 0). The solution is given by (see Kulkarni and Rolski [54]) t K X(t) = sup Y (t), rk(Zk(s)) − c ds , ≤u≤t u 0 k=1
where t K Y (t)=X(0) + rk(Zk(s)) − c ds. 0 k=1 It has been shown in Kulkarni and Rolski [54] that the buffer-content process {X(t),t≥ 0} is stable if