timestep stochastic simulation of computer networks using

untitledAndrzej Kochut ∗
Yorktown Heights, NY 10598, USA [email protected]
A. Udaya Shankar Department of Computer Science University of Maryland College Park, MD 20742 [email protected]
Abstract
Timestep stochastic simulation (TSS) is a novel method for generating sample paths of computer networks, with low computation cost independent of packet rates. It has accuracy adequate to evaluate general network and flow configurations, including arbitrary flow start times and durations, drop-tail queuing (i.e., does not require RED), and arbitrary state-dependent control mechanisms for congestion control and routing. TSS generates the evolution of the system state S(t) on a sample path in time steps of size δ. At each step, S(t+δ) is randomly chosen according to S(t) and the probability distribution Pr[S(t+ δ)|S(t)] obtained using the diffusion approximation. Because packet transmission and reception events are replaced by time steps, TSS generates sample paths at a fraction of the cost of packet-level simulation. Because TSS generates sample paths, control feedback can be based on sample path metrics, rather than ensemble metrics, thereby accurately cap- turing the effects of state-dependent control mechanisms.
1 Introduction
of computer networking. Yet existing evaluation techniques
are not adequate for handling the large heterogeneous ar-
chitectures and state-dependent control schemes that are in-
trinsic to modern computer networking. A state-dependent
control scheme is one where control (e.g., source rate or
window size, next hop, buffering threshold) is exercised
based on the observed state of the network (typically av-
eraged over some recent interval).
Packet-level simulation (e.g. [32, 30]), currently the most
popular technique, is accurate but does not scale with link
∗Work done when author was a graduate student at the University of Maryland, College Park.
bandwidth and network size. Purely analytical techniques
(e.g. [13, 14]) do not capture the effects of state-dependent
control or realistic traffic mix with reasonable accuracy.
This current state of affairs has motivated techniques
such as fluid approximation models (e.g. [25, 20, 3, 2, 1,
21]), henceforth referred to as “fluid” methods, where the
arrival and service times are defined by time-varyingmeans,
and a timestep computation yields an evolution of the net-
work state.
TSS can be viewed as a generalization of fluid approx-
imation methods, based on the first two moments rather
than only the first moment. In particular, arrival and service
times in TSS are defined by time-varying rate and variance, and a TSS computation yields an evolution of the network state.
As in stochastic packet-level simulation, repeated exe-
cutions of TSS on the same network (including the same
arrival and service parameters) yield different evolutions, each possible evolution corresponding to a particular choice
of random values (e.g., arrival or service times). The time
evolution of the first two moments of desired metrics can be
obtained by computing multiple evolutions and averaging
(as in stochastic packet-level simulation).
It is worthwhile to emphasize the distinction between
TSS and fluid approximation methods. Both TSS and fluid
methods compute an evolution. Fluid methods use deter-
ministic description of both inter-arrival and service pro-
cesses. Thus repeated executions of a fluid method on the
same network (including the same arrival and service pa-
rameters) yield the same evolution. Whereas in the case of TSS (and stochastic packet-level simulation), repeated exe-
cutions on the same network yield different evolutions.
Ultimately, we are interested not in individual evolu-
tions but in the evolution of instantaneous ensemble met-
rics, e.g., the evolution of mean queue size. In TSS and
in packet-level simulation, this is obtained by computing
multiple evolutions and averaging over them. Whereas a
Proceedings of the 14th IEEE International Symposium on Modeling, Analysis, and Simulation of Computer and Telecommunication Systems (MASCOTS '06) 0-7695-2573-3/06 $20.00 © 2006
fluid method computes only one evolution. So the question
arises as to when this one evolution adequately represents
all the different possible evolutions. Clearly, this would
hold if the different possible evolutions are all “close” to
each other. While this may be true of certain types of com-
puter networks, it is not true of most computer networks,
especially when the link scheduling is drop-tail as opposed
to a “per-flow” discipline like RED. The nonlinear control
mechanisms present in networks invariably cause small dif-
ferences to lead to vastly different evolutions over time; for
example, a small delay in a packet arrival may trigger a
timeout, after which the evolution would be very different
than if the timeout had not occurred. Section 2 illustrates
the diversity of queue size evolutions and its impact on fluid
approximations using a simple example network.
Timestep Stochastic Simulation (TSS)
Our method computes the evolution of system state S(t) on a sample path at time instants t0, t1, t2, · · ·, where ti+1 − ti = δ for an appropriately small δ (smaller than
the time scale of the end-to-end control) assuming a start-
ing state S(t0). Note that S(t) contains the values of all important system variables at time t, for example queue
sizes of communication links, window sizes of TCP con-
nections, state of TCP connections (i.e., slow start or con-
gestion avoidance), etc. S(ti+1) can be computed based on S(t) and models of congestion control mechanisms. To illustrate the idea consider one of the variables in
S(t), the number of customers N(t) of a queue. We do the following for i = 0, 1, · · ·:
(1) Analytically obtain Pr[N(ti+1)|N(ti)], the probability distribution of N(ti+1) givenN(ti).
(2) Then randomly choose a value for N(ti+1) based on Pr[N(ti+1)|N(ti)].
Repeating these two steps for successive intervals generates
a sample path of the queue size N(t). Figure 1 depicts this procedure. Assume, that the initial queue size at time t0 equals N(t0) (marked by a black dot). Because of the randomness in arrival and service processes the time evolution
of N(t) in the interval [t0, t0 + δ] can follow a multiplicity of paths. Two of these paths originating from (t0, N(t0)) are shown. Each upward transition corresponds to the ar-
rival of the packet to the queue and each downward transi-
tion corresponds to the departure of the packet. The queue
sizeN(t0 + δ) is chosen based on the probability density of queue size at t0 + δ conditioned on N(t0). The probability density is shown as a solid line. The procedure is repeated
for step [t0 + δ, t0 + 2δ], as shown in the figure. The difficulty, of course, is in obtaining an analytical

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t0 + δ t0 + 2δ time
Figure 1: Computation procedure. At each step, the probability distribution of N(t + δ) given the current value of N(t) is obtained and a new value of N(t + δ) is randomly chosen based on this distribution.
S(ti) and models of control schemes (such as TCP). Even so, exact results for Pr[N(ti+1)|N(ti)] are not known for most queuing systems of interest. However, the diffusion
approximation, first proposed by Kolmogorov [18] and later
extended by Feller [7], provides a very accurate approxima-
tion of this distribution.
First, packet arrival and service distributions can be time-
varying stochastic processes characterized by the first two
instantaneous moments, i.e., mean rate at time t and vari-
ance at time t. This allows realistic modeling of traffic and
service times. In particular, the variation of a source at time
t represents howmuch jitter the source exhibits with respect
to its mean rate at time t. Second, because the diffusion ap-
proximation is based on the law of large numbers, its accu-
racy increases with increasing packet rates (and its compu-
tational cost is not affected).
TSS is a general-purpose network simulator whose com-
putational cost does not increase with increasing bandwidth.
It can be used as a general substitute for packet-level sim-
ulation of high speed networks, for example, not only to
examine special problems like the behavior of many TCP
flows sharing a bottleneck, but also to evaluate P2P net-
works, load balancing, interaction between dynamic routing
and flows, and so on. This paper focuses on handling an ar-
bitrary network topology with drop-tail queuing. However,
TSS can be easily extended to the statistically smoother
AQM disciplines. TSS can handle both the TCP and UDP
flows with arbitrary start times and durations (i.e., both TCP
elephants and mice [21]).
2 Diversity of Sample Paths and its Impact on Deterministic Fluid Approximations
We now present an example that illustrates the diversity
of evolutions found in even simple networks, show how
well TSS handles this diversity, and discuss what are the
implications for fluid modeling.
Figure 2 shows a two-link tandem network with TCP
flow from node N0 to node N2. Both links have equal ser-
NN0 1 2N
0
300
600
TSS sample path ns sample path
(a) pair of sample paths (b) pair of sample paths forN0 → N1 forN1 → N2
0
400
800
0
300
600
TSS sample path ns sample path
(c) pair of sample paths (d) pair of sample paths forN0 → N1 forN1 → N2
Figure 3: Evolutions of queue size generated by packet-level simulation and by TSS for tandem network in Figure 2.
vice rate of 1000 packets/sec, squared coefficient of vari-
ation of service time of 0.05, and buffer capacity of 500
packets. Figures 3a and 3c show two pairs of evolutions
as generated by packet-level simulation and TSS for link
N0 → N1. Similar pairs for link N1 → N2 are shown
in Figures 3b and 3d. Figures 4a and 4c present evolu-
tions of mean queue size and its standard deviation for link
N0 → N1, respectively. Figures 4b and 4d show corre-
sponding evolutions for link N1 → N2.
There are several points to note. First, the individ-
ual evolutions have great diversity, especially those of link
N1 → N2 (due to the first queue being nearly always non-
empty). There is no ”representative” evolution. Second,
TSS is accurate in computing individual evolutions as well
as the evolutions of ensemble metrics, for both links. TSS is
10 times faster than ns [32] for the presented configuration.
The difference increases to nearly 1000 times in case of two
1Gbps links. Third, the instantaneous mean queue size evo-
lution of a link is quite different from any of its individual
evolutions; the cyclical nature of the latter is not present
in the former. A fluid method would not have the random
time shifts that make the mean queue size evolution smooth
(as presented on Figure 4a). Instead, it would yield a cycli-
cal “saw-tooth” behavior that does not decay with time and
that corresponds to only one of the many possible evolu-
tions; the evolution would not be a representative evolution,
say in the example depicted on Figure 3a. This problem
stems from the simplification (used in all fluid approxima-
tions) that the behavior of traffic sources is driven by the
0
300
600
TSS mean ns mean
(a) mean queue size (b) mean queue size ofN0 → N1 ofN1 → N2
0
300
600
0
200
400
TSS std. dev. ns std. dev.
(c) standard deviation (d) standard deviation of queue size ofN0 → N1 of queue size ofN1 → N2
Figure 4: Mean queue size and its standard deviation for link N0 → N1 and for link N1 → N2 of network in Figure 2 as obtained by packet-level simulation and by TSS.
mean system evolution and not the sample system evolu-
tion. The deficiency of fluid models is even more visible
in case of the second queue in tandem. Fluid approxima-
tion predicts that this queue is empty all the time, because
arrival rate never exceeds the service rate of the link. In re-
ality however, randomness in inter-arrival and service pro-
cesses cause the queue to impact the flows’ round-trip time
significantly. TSS predicts this behavior very well as shown
on Figures 4b and 4d. Fourth, TSS yields the evolution not
only of mean values but also of second moments and even
distributions of metrics of interest.
3 TSS for One Queue
We model a communication link by a single-server
queue in the usual way, with service rate equal to the link
bandwidth (in packets/second) and maximum queue size
equal to the link buffer size (in packets). The state of the
link at time t is defined by the number of packets in the
queue at time t, denoted by N(t). We compute a time-stepped evolution of the stochastic
process N(t) given a starting state N(t0) as illustrated in Figure 1. Divide the time axis into intervals defined by time
instants t0, t1, · · · where ti+1 − ti = δ and δ is smaller than
the control time scale Δ. Then do the following iteratively for i = 0, 1, · · ·: compute Pr[N(ti+1)|N(ti)] and choose a randomN(ti+1) with this distribution. In order to compute Pr[N(ti+1)|N(ti)] we use diffusion approximation [18, 7] with holding barriers [10] (with Coxian distributions at the
barriers [4]).
for an unbounded queue, i.e., with only a lower barrier
N(t) = 0. Similar equations may be obtained for the two- barrier case (N(t) = 0 and N(t) = K), but the details are
omitted here for brevity.
Let us introduce the following notation:
• f(x, t|x0): conditional probability density of N(t) = x givenN(0) = x0.
• λ and cA: mean rate and squared coefficient of varia-
tion of the arrival process.
• μ and cS : mean rate and squared coefficient of varia-
tion of the service process.
• λi and ai: exit rate and probability of entering next
stage, for ith stage of Coxian holding time distribution.
• Pi(t): probability of being in ith stage of Coxian dis-
tribution at time t.
The time evolution of f(x, t|x0) is the solution to the diffusion equation, where f(x, t|x0) is replaced with f for
brevity:
∂f
∂t =
α
2
∂2f
where β = λ − μ
α = λcA + μcS . (2)
and initial and boundary conditions are as follows (δ∗(x) is the Dirac delta function):
f(0, t|x0) = 0 t ≥ 0 f(x, 0|x0) = δ∗(x − x0) 0 < x < ∞
∀i Pi(0) =
(3)
We assume that the input parameters (i.e., λ, cA, μ, cS ,
λi, ai) are constant on the interval of interest, which is
valid because δ is small enough compared to Δ. Then the Laplace transform f∗(x, s|x0) of f(x, t|x0) can be computed in terms of Laplace transform of the holding time dis-
tribution, denoted by h∗(s). The Stehfest [31] algorithm is used to invert the transform. TSS computational procedure
for a single communication link is presented in Figure 5.
Details are available in [17].
4 TSS for a Network
The previous section described timestep stochastic sim-
ulation of a single link driven by a single flow. We now
initialize internal state of the traffic source;
initialize queue size N(0); for t = 0 to StopT ime with step δ do obtain moments of the arrival process λ(t) and cA(t); obtain probability distribution Pr[N(t + δ)|N(t)] using diffusion approximation (Eqns. 1, 2, and 3);
choose new queue size N(t + δ) according to Pr;
write metrics for time t to the file;
end for;
Figure 5: TSS computational procedure for a single communication link and one traffic flow.
extend this to a network of queues driven by a set of flows.
Each flow represents the traffic generated by a UDP or TCP
source, and is defined by the following quantities: source
node; sink node; path through the network taken by its pack-
ets; and the first two moments of the source’s inter-packet
generation time during each timestep. (The second moment
corresponds to how much jitter the source exhibits with re-
spect to its mean rate.)
The model, of course, allows the source’s moments to
vary over different timesteps. For a time-dependent source,
the moments would depend only on time (e.g., a start and
stop time). For a state-dependent source, the moments
would depend on the evolution of the network state thus
far; for example, a TCP source’s rate would depend on the
recent history of the sizes of the queues along its path. A
model of TCP-Tahoe is discussed in simulation studies in
Section 5. We also implemented and evaluated a state-
dependent UDP source, but we do not report results here
because of the lack of space.
4.1 Departure Process, Splitting and Merging Processes
To extend TSS to a network of queues, we have to be able
to compute the first two moments of the departure process
of a queue as well as that of processes obtained by split-
ting and merging flows. We formulate equations for each
of these cases next. Throughout, we assume that the first
two moments of the arrival and service processes on each
link are constant within each δ interval. This assumption is
valid because we set δ to be smaller than the control time
scale Δ. Following [6, 5, 34] we approximate the transient de-
parture process using the Marshall’s formula [22] with π0
replaced with π0(t) as computed by the diffusion approximation. For inter-departure time φ(t), we obtain:
E[φ(t)] = 1
μ + π0(t)E[h]
E[h]
μ }
(4)
where Vs is the variance of the service process, μ is the
mean service rate, and E[h] and E[h2] are first two moments of the idle period.
At a node of a network, the incoming traffic flow may be
split into multiple flows (depending on the routing proba-
bilities). Let the inter-arrival time of the process being split
have mean rate λa and squared coefficient of variation ca.
The packets may enter one of n communication links with
probabilities q1, q2, . . . , qn. Let the inter-arrival time of the
ith resulting process have mean rate λi and squared coeffi-
cient of variation ci. Following [13] and [19], the transition
rates and squared coefficients of variation of the ith compo-
nent process are given by:
λi = λa ∗ qi
Consider the merge of independent renewal processes. It
turns out that the first two moments of the merged pro-
cess are not obtainable from just the first two moments of each of the constituent renewal processes. One has to work
with the distributions of the inter-arrival times. We use
the approach described in [19], which substitutes a hypo-
or hyper-exponential distribution (depending on variances
of processes being merged) leading to a closed-form ap-
proximate expression. Details are omitted for space reasons
(available in [19]).
4.2 Network Flows
TSS computes a sample path of the queue size evolution
for each communication link in the network. In order to
complete the description of the system and to model con-
gestion control schemes, we need to specify for each com-
munication link the number of packets of each flow that tra-
versed the link in each timestep.
For each flow j and communication link i we introduce
a series of flow-conservation equations. Denote by N j i (t)
the number of packets of flow j in queue i at time t, by
D j i (t) the number of packets of flow j departing from queue
i during [t, t + δ], by λ j i (t) mean arrival rate of packets of
flow j to queue i during [t, t+δ], and byA j i (t) arrival count
of packets of flow j to link i during [t, t + δ]. Under the assumption that the routing is properly configured and flows
do not have cycles, we have:
A j i (t) =

λ j i (t) ∗ δ if i is the first-hop of flow j
D j k(t) otherwise, where k is the
previous link on the path
of flowj (6)
Denote by Ai(t) = ∑
Ak i (t) the total arrival count to
the link i during [t, t + δ], and by P full i (t) the probability
that a packet arriving during [t, t + δ] finds the queue full.
P full i (t) is computed using the diffusion equation. Total number of packets lost due to the overflow of queue iwithin
time interval [t, t + δ], denoted by Li(t), is determined as:
Li(t) = Ai(t)P full i (t) (7)
Losses are assigned to flows proportionally to the share of a
flow in the aggregate traffic. The number of packets lost by
flow j on link i during [t, t + δ], denoted by L j i (t), is:
L j i (t) = Li(t)
A j i (t)
Denote the mean arrival rate and squared coefficient of vari-
ation of interarrival times to link i during [t, t + δ] by λi(t) and cA,i(t), respectively. These values are computed using techniques presented at the beginning of this section. Mean
service rate and squared coefficient of variation of service
time of queue i during [t, t + δ] are denoted by μi(t) and cS,i(t), respectively. Queue size N(t + δ) is randomly chosen according to the probability distribution computed as
described in Section 3 based on N(t), λi(t), cA,i(t), μi(t), and cS,i(t). Departure count of flow j from link i within
[t, t + δ] is determined according to:
D j i (t) = N
j i (t) − N
j i (t) − L
j i (t) (9)
5 TCP Network Simulations
paring its results with that of packet level simulations. We
studied a broad variety of network topologies with various
traffic flow configurations with both TCP and UDP traffic.
The TSS achieves high accuracy while computing the re-
sults orders of magnitude faster than packet level simula-
tor (several thousand times faster for topologies with 1Gbit
links). Due to the lack of space we report on only two ex-
ample network topologies.
We use TCP sources with slow-start, congestion avoid-
ance, and retransmissions driven by timeout (i.e., TCP-
Tahoe sources). To represent a TCP-Tahoe source in TSS,
we augment the state of a TCP source with the following:
(1) congestionwindow size; (2) slow start threshold; (3) last
acknowledged sequence number; (4) RTO timer value, i.e.,
time before which the last sent data segment has to be ac-
knowledged or loss will be assumed. We also extend the
information maintained at each queue for a TCP flow. In
addition to the number of packets, TSS also keeps track of
the lowest and highest sequence numbers, and updates them
based on the arrival, departure, and loss count for the last
hop of the return path. The data arrival count and variability
N2 N4
(a) 14-node network topology (b) Large network topology
Figure 6: 14-node network topology with 40 TCP flows (a) and 35-node network topology with 180 TCP flows (b).
at the TCP sink is used by the TCP source as representing
the acknowledgment stream. We make the assumption that
the acknowledgments encounter a fixed delay.
The source adjusts its congestion window based on the
statistics of the arriving acknowledgment stream. In par-
ticular, if the congestion window is smaller than the slow-
start threshold, the congestion window gets increased by
the number of received acknowledgments. Otherwise, the
source is in congestion avoidance mode and the congestion
window is expanded by the number of arriving acknowledg-
ments divided by the current size of the congestion window.
5.2 14-node Network
13 communication links (as shown in Figure 6a). The cen-
tral link (N12 → N13) has service rate of 2000 packets/sec
and all other links have service rates of 1000 packets/sec.
Buffering capacity of all links is 1500 packets and squared
coefficient of variation of service time is 0.05. Traffic is
generated by 40 TCP flows with time-dependent start and
stop times. In particular, 10 TCP flows from nodeN0 toN4
and 10 TCP flows from node N2 to N6 are active from the
beginning of the simulation until 300 seconds. Moreover,
10 TCP flows from node N1 to N5 and 10 TCP flows from
node N3 to N7 are active from 100 seconds until 200 sec-
onds. Figure 7 presents comparison of mean queue size and
its deviation as computed by packet-level simulation and
TSS for links N2 → N9, N9 → N12, and N12 → N13.
TSS correctly predicts periods of large queuing for all links
of interest. It also approximates variability of the queue size
very well.
We now examine the quality of TSS approximation for
a 35-node network with 35 communication links and 180
TCP flows shown in Figure 6b. All links have service rate
0
800
1600
(a) mean queue size (b) standard deviation of queue size ofN2 → N9 ofN2 → N9
0
900
1800
(c) mean queue size (d) standard deviation of queue size ofN9 → N12 ofN9 → N12
0
150
300
(e) mean queue size (f) standard deviation of queue size ofN12 → N13 ofN12 → N13
Figure 7: Comparison of mean queue size and its standard deviation as computed by packet-level simulator and TSS for links N2 → N9, N9 → N12, and N12 → N13 of network in Figure 6a.
of 5000 packets/sec, which corresponds to (assuming 1KB
packets) approximately 40 Mbps. Buffer capacities vary
from 1000 to 5000 packets. 10 bulk TCP flows are contin-
uously active for each of the following source-destinations:
N14 → N17, N15 → N18, N16 → N19, N14 → N23,
N15 → N24, N16 → N25, N20 → N23, N21 → N24,
N22 → N25, N20 → N26, N21 → N27, N22 → N28,
N29 → N32, N30 → N33, N31 → N34, N29 → N17,
N30 → N18, N31 → N19. Because this configuration has
multiple interacting TCP flows on a non-trivial topology, it
gives raise to various behaviors and is a good configuration
for presenting the quality of TSS approximation. Figure 8
presents comparison of mean queue size and its deviation
as computed by packet-level simulation and TSS for links
N0 → N1, N12 → N5, and N2 → N3. TSS results match
very closely those of packet-level simulation.
6 Related Work
of the most widely used is packet-level simulation [32, 30],
in which an event is simulated for every packet transmission
and reception. This technique, if used carefully, can provide
0
500
1000
(a) mean queue size (b) standard deviation ofN0 → N1 ofN0 → N1
0
3000
6000
(c) mean queue size (d) standard deviation of queue size ofN12 → N5 ofN12 → N5
0
500
1000
(e) mean queue size (f) standard deviation of queue size ofN2 → N3 ofN2 → N3
Figure 8: Comparison of mean queue size and its standard deviation as computed by packet-level simulator and TSS for links N0 → N1, N12 → N5, and N2 → N3 of network in Figure 6b.
very high accuracy. However, it becomes prohibitively ex-
pensive for high-speed communication links and large net-
work architectures.
There are many analytical methods to solve for steady-
state metrics of queuing networks, for example, [13, 14],
and, more relevant to our approach, [34, 33, 9, 15, 16, 10,
19]. There is also work on computation of transient met-
rics in networks with time-dependent control schemes, for
example [6, 5]. However, these methods are not effective
for solving networks with complex state-dependent control
schemes (such as TCP’s congestion control and dynamic
adaptive routing).
equations, which are then solved in timestep fashion to ob-
tain time evolutions of the network state. So TSS is similar
to fluid methods in this respect. However, fluid methods
consider the differential equations in only the first moment,
whereas TSS considers them in the first two moments. In
particular, in one class of fluid methods (e.g., [24, 20, 25]),
the set of differential equations is reduced to a set of deter-
ministic differential equations. Consequently, the approach
yields only a single evolution, intended to correspond to
the ensemble-averaged mean of the metrics. The second
group of fluid approximation techniques [2, 1, 21], describe
time evolution of TCP congestion window again assuming
deterministic network behavior. However, they introduce
stochastic description of start and stop times of TCP flows.
There are hybrid approaches that combine stochastic
fluid approximations with packet-level simulation, as in
[11, 3]. These approaches inherit the limitations of stochas-
tic fluid approximation that makes it difficult to properly
capture the effect of state-dependent control causing diverse
sample paths.
directly yield the time evolution of an instantaneous
ensemble-averaged metric of a network. One example
is the Z-iteration [23, 29], which computes instantaneous
ensemble-average metrics of interest (e.g., queue size, loss
rate, source rate) by approximating the relationship be-
tween instantaneous metrics by their steady-state counter-
parts. But it is restricted to networks of M(t)/M(t)/*/*
queues, and so cannot capture the effect of state-dependent
control schemes.
cesses that form the foundation of our TSS approach. There
is a large body of literature devoted to diffusion approxima-
tions for obtaining queue size distributions. The approaches
date back to Kolmogorov [18], who first proposed diffusion
equations. Feller [7] extended his ideas and provided the
framework for solving various problems using diffusion. In
a series of articles devoted to the analysis of road traffic
[27, 26, 28], Newell proposed a set of approximation tech-
niques applicable in low, mild, and heavy traffic conditions.
Similar work also came from Gaver [8] and Kingman [12].
Gelenbe [9, 10] and Kobayashi [15, 16] proposed the use
of diffusion approximations with holding barriers to model
queue size evolution for equilibrium and non-equilibrium
cases. (There are many types of barriers, including absorb-
ing and reflecting.)
Whitt [34], Duda [6], Kuehn [19] and others have studied
the problem of obtaining the statistics of the departure pro-
cess of a queue, building upon the work by Marshall [22]
relating the Laplace transforms of the distributions of ser-
vice time, idle period, and inter-departure time. Whitt [35]
and Kuehn [19] have studied the problem of determining
the statistics of traffic flows obtained by splitting and merg-
ing other traffic flows in the context of communication net-
works.
We presented a novel technique, called timestep stochas-
tic simulation, for fast performance evaluation of computer
networks. Our method generates sample paths of the sys-
tem with very high accuracy. In each step the new state of
the system is chosen randomly based on the current state
and a probability distribution obtained using diffusion ap-
proximation. The method is much faster than packet-level
simulation and has almost the same accuracy. Because
state-dependent control gets feedback based on sample path
metrics, our method is more suitable for modeling state-
dependent control schemes (such as TCP’s) than fluid ap-
proximations. Our future research plans include extending
the TSS with RED queue model. We also plan to apply it
to various networking problems, such as dynamic routing,
load balancing, or optimal cache placement.
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timestep stochastic simulation of computer networks using

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