ACM/IEEE PADS 2008 1 Stochastic Process Models for Packet/Analytic-Based Network Simulations Robert G. Cole, George Riley, Derya Cansever and William Yurcik Index Terms|Hybrid Simulation Models, Event Driven Simulations, Network Models, Stochastic Queues and Dif- summed internal load fusion Equations. external arrival external departure Abstract network queue E present our preliminary work that develops a new W approach to hybrid packet/analytic network sim- network ulations for improved network simulation fidelity, scale, Fig. 1. An example network of queues with internal and external and simulation efficiency. Much work in the literature ad- arrival and departure traffic. dresses this topic, including [10] [11] [8] [12] [13] and others. Current approaches rely upon models, which we refer to in this paper as Deterministic Fluid Models [9] [12], to address foreground traffic, which is of primary interest, and back- the analytic modeling aspects of these hybrid simulations. ground traffic, which exists solely to provide competition Instead we draw upon an extensive literature on stochas- to the foreground traffic being measured. In the hybrid tic models of queues and (eventually) networks of queues approach discussed here, analytic models are used is to es- to implement a hybrid stochastic model/packet network timate the impact of the background traffic on the fore- simulation. We will refer to our approach as Stochastic ground traffic, which is handled explicitly via discrete- Fluid Models throughout this paper. We outline our ap- event handling with full packet{level detail. In the re- proach, present test cases, and present simulation results mainder of this paper, we will assume that the analytically comparing the measured queue metrics from our approach modeled traffic represents, in some sense, background traf- for hybrid simulation to those of a deterministic fluid model fic and that the explicitly handled packet traffic is fore- hybrid simulation and a full packet{level simulation. We ground traffic. We will use the terms background traffic also discuss plans for future areas of research on this ap- and foreground traffic to distinguish between the analyt- proach. ically modeled versus the explicitly handled event packet traffic. Of course other ways to divide up the analytically I. Introduction modeled and packet handled traffic are possible. Consider a network model as found in Figure 1. The There are two interesting aspects of the analytic model- network model is comprised of nodes and communica- ing in the context of a network of queues. One relates to tions links. Associated with each communications link is a the allocation and estimation of the intermediate load gen- queue. Traffic can arrive and depart the network model at erated by the traffic at the nodes within the network based each node in the network. Each node also carries internal upon the assumed routing patterns, total estimated exter- traffic which is forwarded throughout the network between nal traffic loads, finite queue sizes and associated network the traffic source and destination nodes. internal packet losses. The other aspect relates to meth- In hybrid simulation models, some of the traffic is han- ods of mixing, at a given queue, the analytically modeled dled via analytic methods and some of the traffic is han- background traffic with the explicitly handled foreground dled via more CPU and memory intensive discrete-event, traffic. Our focus in this paper is the later; hence we con- packet-level handling. The more traffic modelled analyt- centrate on methods to mix the analytically modeled traffic ically, the more efficient the simulation becomes (in the with the event-driven, packet-level traffic at a given queue. general case), and thus the simulation can scale to larger A future paper will concentrate on the network equations. networks. Typically, there is a distinction drawn between The remainder of this paper is organized as follows: In the next section we discuss previous methods for hybrid R. G. Cole is with the Applied Physics Lab and the Department of network simulation which rely upon deterministic fluid Computer Science, Johns Hopkins University, Baltimore, MD. Phone: +1 443 778{6951, e-mail: [email protected] model approximations to queue dynamics. In Section III G. Riley is with the School of Electrical and Computer En- we present our approach based upon methods in stochas- gineering, Georgia Institute of Technology, Atlanta, GA. Phone: tic queue dynamics. In Section IV we develop our hy- +1 404 463{1774, e-mail: [email protected] D. Cansever is the Program Director of Advanced Internet- brid equations for a specific instance of a stochastic queue working, SI International, Inc. Phone: +1 703 234{6960, e-mail: model, i.e., a Brownian Motion model based upon solutions [email protected] to the Fokker-Planck Equation. In Section V we report on W. Yurcik is with the Department of Computer Science, Univer- sity of Texas-Dallas, Dallas, TX. Phone: +1 309 531{1570, e-mail: our initial simulations investigations comparing three test [email protected] cases; one based upon pure event-driven packet-level sim- 2 ACM/IEEE PADS 2008 ulations, one based upon hybrid deterministic fluid/packet ωmax simulation and one based upon our hybrid stochastic pro- ω+ (t i ) slope = cess/packet simulation. In Section VI we discuss conclu- λd − µ sions and future investigations. work ω− (t i ) ωi ( work carried by packet i ) II. Deterministic Models time Most current approaches to performance speedup for i i+1 i+2 i+3 network simulations involving hybrid event-analytic sim- packet arrivals ulation rely on analytic models based upon deterministic Fluid Flow Approximations (FFAs), e.g., [7] [4] [10] [11] [8] Fig. 2. An illustration of the hybrid deterministic modeling approach [13]. We refer to these approaches as Deterministic Mod- for mixing traffic at a queue. els because the evolution of the queue dynamics modeling the background analytic traffic is assumed to be a deter- ministic process captured in the form of differential equa- rate at the queue, i.e., the inverse of the link bandwidth. tions. The fluid dynamics is derived from the integration Let w(t) be the total workload in the queue at time t, and of these equations. In some cases the differential equations w−(ti) and w+(ti) be the total workload in the queue just are solvable, e.g., fixed arrival rate models, and numeri- prior to and just following (respectively) a distinguished cal integration of the differential equations is not neces- time event, e.g., the arrival of packet i at time ti. Let wi sary. Both cases address variable mean arrival rates; one be the work carried by the ith packet to the queue. Finally, through time dependent variables in the differential equa- let wmax be the maximum amount of work (or backlog) in tions [9] [13] and one through discrete event fluid models the queue due to its finite size. where rate changes are propagated throughout the network Figure 2 gives a pictorial representation of the method via events [4] [11] [7]. used to mix analytically modeled background traffic with Two approaches to using deterministic analytic models explicitly handled packet traffic. Individual packet arrival for hybrid simulations are found in the literature. One events are indicated along the bottom axis of the figure. approach, used in [13], divides the network into a hierarchy These packets carry with them a given workload which is a consisting of a high speed core and a lower speed edge. function of their packet size and the link bandwidth serv- In the core, all traffic is modeled analytically and in the ing the queue in question. If the packet is allowed to enter edge all traffic is modeled via packet-level, event-driven the queue, then the workload in queue jumps from w−(ti) simulation. Packet traffic traversing the core is converted to w+(ti) where the difference is the workload, wi, carried to fluid load on the core and is then converted back to by the packet. In between packet arrivals, the workload packet-level traffic at the far edge. evolves deterministically. As mentioned, we assume the Another approach, used in [4] [11] and [8], is to treat queue dynamics evolve at a constant rate between packet some of the traffic throughout the network analytically. arrivals (as long as the workload does not exceed the max- Then develop methods to explicitly mix both analytic and imum wmax or drops below the minimum of zero) given by packet traffic at each multiplexing point throughout the the difference λd −µ which is fixed over the averaging time network. Our interests are in this later approach. interval δ. Then the explicit handling of the packet traffic and the A. Deterministic/Packet Mixing Equations process for updating the impact of the background traffic Here, we develop the simplest, least assuming method are as follows: to mix deterministic traffic with packet traffic. There exist • If, during the current δ time averaging period, λd < µ, methods to improve the fidelity of our example determinis- i.e., the background traffic does not overflow the server tic model, e.g., [4] [11] [8], but these improvements require rate. For each packet arrival bringing wi work to the a priori knowledge of the behavior of the foreground traf- queue, do the following tasks: fic, which in general is not known. This choice is made { Increment the packet arrival counter i at the queue. here for several reasons. First, we wish to make an equiva- { Update the queue backlog just prior to the packet lent comparison to our stochastic models presented below arrival time ti, as in order to access its ability to negate a reliance on a priori w (t ) max[w (t )+(λ −µ)(t −t ); 0] (1) traffic knowledge.