Discrete Event Dynamic Systems: Theory and Applications, 14, 267±307, 2004
# 2004 Kluwer Academic Publishers. Manufactured in The Netherlands.
Perturbation Analysis of Multiclass Stochastic
Fluid Models
GANG SUN
gsun@bu.edu
Department of Manufacturing Engineering and Center for Information and Systems Engineering, Boston
University, Brookline, MA 02446
CHRISTOS G. CASSANDRAS
cgc@bu.edu
Department of Manufacturing Engineering and Center for Information and Systems Engineering, Boston
University, Brookline, MA 02446
CHRISTOS G. PANAYIOTOU
christosp@ucy.ac.cy
Department of Electrical and Computer Engineering, University of Cyprus, Nicosia, Cyprus
Abstract. We use a stochastic ¯uid model (SFM) to capture the operation of ®nite-capacity queueing systems
with multiple customer classes. We derive gradient estimators for class-dependent loss and workload related
performance metrics with respect to any one of several threshold parameters used for buffer control. These
estimators are shown to be unbiased and directly observable from a sample path without any knowledge of
underlying stochastic characteristics of the traf®c processes. This renders them computable in on-line
environments and easily implementable in settings such as communication networks.
Keywords: perturbation analysis, stochastic ¯uid model, unbiased estimator, quality of service
1.
Introduction
In this paper, we use the framework of stochastic ¯uid models (SFM), together with
perturbation analysis (PA) techniques, in order to develop gradient estimators of various
performance metrics for queueing system with multiple customer classes. Such estimators
can be used for on-line control and performance optimization of communication networks
and manufacturing systems, as described in some of our previous work (Cassandras et al.,
2002; Yu and Cassandras, 2002). In particular, we are interested in using threshold-based
controllers and have applied in®nitesimal perturbation analysis (IPA) to obtain sensitivity
estimators for loss and workload metrics with respect to controllable threshold parameters.
In Cassandras et al. (2002), a single-node single-class SFM was analyzed. In Cassandras et
al. (2003), we considered a node with in®nite buffer capacity which processes two traf®c
streams: one traf®c stream is uncontrolled and the other is subject to threshold-based
buffer control. In Sun et al. (2002), we provided some results for a similar two-class model
with ®nite buffer capacity. In this paper, we analyze a ®nite-capacity model with M 2
traf®c classes and associated thresholds.
SFMs provide an alternative to queueing models when the latter become impractical
due to huge traf®c volumes and complex stochastic processes that cannot be handled by
tractable analytical derivations. The SFM paradigm allows the aggregation of multiple
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SUN ET AL.
events, associated with the movement of individual customers (e.g., packets or parts), into
a single event associated with a rate change. It foregoes the identity and dynamics of
individual customers and focuses instead on the aggregate ¯ow rate. SFMs have recently
been shown to be especially useful for analyzing various kinds of high-speed networks
(Cassandras et al., 2002, 2003; Wardi and Melamed, 2002; Kesidis et al., 1996; Kumaran
and Mitra, 1998; Miyoshi, 1998; Liu et al., 1999; Yan and Gong, 1999). As argued in
Cassandras et al. (2002, 2003), such models may not always be accurate for the purpose of
performance analysis, but they capture the salient features of the underlying ``real''
system in a way which is often suf®cient to solve control and optimization problems. In
this case, estimating the gradient of a given cost function with respect to key parameters
becomes an essential task. PA methods (Ho and Cao, 1991; Cassandras and Lafortune,
1999) are therefore suitable, if appropriately adapted to a SFM viewed as a discrete-event
system (Cassandras et al., 2002, 2003; Wardi et al., 2002; Liu and Gong, 2002). From a
technical standpoint, performance metrics associated with a SFM can often be shown to be
Lipschitz continuous with respect to controllable parameters of interest. This generally
results in unbiased IPA estimators associated with a SFM, whereas the same is usually not
true for the corresponding discrete event system. For example, IPA estimators have been
known to be biased when applied to queueing models with customer blocking or with
multiple customer classes (Cao, 1987). In addition, the estimators are generally simple to
implement and can be evaluated based on data observed on a single sample path of the
actual (discrete-event) system. Thus, we use the SFM to derive the form of a gradient
estimator, but can then implement it using actual system data. Even though using SFMbased estimators on an actual system may introduce a bias, simulation results indicate that
gradient estimators obtained in this fashion work well with gradient-based optimization
algorithms.
In a single-class SFM of a communication network node with threshold-based buffer
control, IPA was shown to yield simple nonparametric sensitivity estimators for packet
loss and workload metrics with respect to the threshold parameter (Cassandras et al.,
2002). In the in®nite-capacity two-class case studied in Cassandras et al. (2003), the
estimators generally depend on some traf®c rate information, but not on the stochastic
characteristics of the arrival and service processes involved. In this paper, we consider
a ®nite-capacity SFM with multiple …M 2† classes viewed as different traf®c streams
(see Figure 1). Each stream is associated with a threshold and is subject to thresholdbased control. The ®nite buffer capacity and the presence of M > 2 classes cannot be
handled as straightforward generalizations of Cassandras et al. (2002) and our analysis
leads to more complex estimator forms, as we shall see, although the basic theoretical
setting for IPA is similar to previous work such as Cassandras et al. (2002) and Yu and
Cassandras (2002). Thus, the main contributions of this paper are (i) the derivation of
IPA gradient estimators for multiple class-dependent loss metrics and a workload
metric with respect to the threshold parameters in the model and (ii) proving that these
estimators are unbiased. In the general case where M > 2, the IPA estimators we
derive are obtained through recursive equations. In the case where M ˆ 2, we are
further able to exploit the structure of the sample paths of the SFM and obtain explicit
closed-form expressions for the IPA estimators that involve readily observable sample
path data.
PERTURBATION ANALYSIS
269
Figure 1. Stochastic ¯uid model (SFM) with M ˆ 3 classes.
The paper is organized as follows. First, in Section 2, we present our model and de®ne
performance metrics and parameters of interest. We also present a partition of a sample
path into time intervals de®ned by speci®c events that facilitate the PA task. In Section 3,
we derive IPA estimators for the sensitivities of the traf®c loss volumes for all classes and
for the system workload with respect to any one controllable threshold parameter. In
Section 4, we analyze the simpler two-class model exploiting the structure of its sample
paths and derive closed-form (as opposed to recursively obtained) IPA estimators. In
Section 5, we establish the unbiasedness of the derived estimators and conclude with
Section 6.
2.
SFM of a Multiclass Queueing System
We consider a single node with a common FIFO buffer fed by M sources. Each source
de®nes a ``class'' m; m ˆ 1; . . . ; M, and is associated with a threshold Tm , where we
assume 0 < T1 < < Tm < < TM . By convention, we set T0 :0. If the buffer
content is above the value Tm , all traf®c from class m is rejected (which implies that all
traf®c from classes labeled i m are also rejected). In the context of communication
networks, it is worth noting that this model captures the operation of the differentiated
services (DS) protocol that has been proposed for supporting quality of service (QoS)
requirements (Blake et al., 1998; Heinanen et al., 1999; Panayiotou and Cassandras, 2001).
Class M corresponds to the top-priority traf®c and if all such traf®c is to be accepted, then
TM is simply the physical buffer capacity of the node.
The in¯ow rate of class m at time t is denoted by am …t† and the corresponding loss rate
by gm …t†. The service rate is denoted by b…t† and let x…t† be the buffer level at time t. For the
purpose of our analysis, we choose any one of the thresholds, say Tm, as the one with
respect to which we wish to carry out sensitivity analysis and denote this parameter by y.
We then write the m-th class loss rate as gm …y; t† and the buffer content as x…y; t† to express
their dependence on the choice of y. We also assume that the processes
fam …t†g; m ˆ 1; . . . ; M, and fb…t†g are independent of y and cannot take negative
values; other than that, these processes are only assumed to be bounded and rightcontinuous piecewise continuously differentiable w.p.1. The time variable t is con®ned to
an interval ‰0; TŠ for a given ®xed 0 < T < ?. It will also be convenient in our analysis to
make use of the following de®nition, for any m ˆ 1; . . . ; M:
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SUN ET AL.
Am …t†:
M
X
nˆm
an …t†
b…t†
1†
and observe that
Am …t† Am ‡ 1 …t†;
m ˆ 1; . . . ; M
1
2†
Figure 1 depicts the SFM described above with M ˆ 3 and T3 set to the physical capacity
of the buffer.
We assume that the parameter y is con®ned to a bounded (compact) interval
Y ˆ …Tm 1 ; Tm ‡ 1 † and let l…y† : Y ? R be any random function de®ned over the
underlying probability space …O; f; P†. Strictly speaking, we write l…y; o† to indicate
that this sample function depends on the sample point o [ O, but will suppress o unless it
is necessary to stress this fact. In what follows, we consider two performance metrics, the
m-th class loss volume Lm;T …y†; m ˆ 1; . . . ; M, and the cumulative workload (or just work)
QT …y†, de®ned on the interval ‰0; TŠ as follows:
Z
Lm;T …y† ˆ
T
0
Z
QT …y† ˆ
T
0
gm …y; t†dt;
x…y; t†dt
m ˆ 1; . . . ; M
3†
4†
Note that from the workload metric it is possible to obtain a delay metric using appropriate
forms of Little's law (see, for example, Wardi and Melamed, 2001).
Viewed as a discrete-event system, an event in a sample path of the above SFM may be
either exogenous or endogenous. An exogenous event is one that causes a change in the
dynamics of x…y; t† due to changes in one or more of the processes fam …t†g; m ˆ 1; . . . ; M,
and fb…t†g, which are independent of y. In particular, we will be interested in changes in
the sign of Am …t† in (1) for some m ˆ 1; . . . ; M and de®ne an associated exogenous event
e1 as one where the buffer content leaves the value x…y; t† ˆ Tm , for some m ˆ 0; . . . ; M,
after it has maintained it for some ®nite length of time. An endogenous event on the other
hand, denoted by e2, is de®ned to occur whenever the buffer content reaches the value
x…y; t† ˆ Tm , for any m ˆ 0; . . . ; M. Note that x…t; y† ˆ Tm for all t in some interval ‰s1 ; s2 Š
implies that Am …t† > 0 and Am ‡ 1 < 0. In other words, the arriving ¯ow rate from the
M m 1 highest priority classes is less than the capacity b…t†; thus, a portion of the
am …t† ¯ow is admitted, while there is some excess which is rejected. As a result, we
observe the period with constant x…t; y† ˆ Tm . At t ˆ s2 , if the sign of either Am …t† or
Am ‡ 1 …t† changes, then this interval will end with an e1 event.
For the purpose of our analysis, we partition the sample path into cycles de®ned by
successive exogenous events e1 (the term ``cycle'', however, should not be interpreted as
implying any sort of regenerative property). Suppose that a sample path includes K such
cycles, where K is a random number which is locally independent of y. Denote the
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PERTURBATION ANALYSIS
Figure 2. Typical sample path segment for a SFM with M ˆ 5 classes.
corresponding exogenous event times by xk ; k ˆ 1; . . . ; K and let xK ‡ 1 ˆ T, so we can
express cycles as intervals ck ˆ ‰xk ; xk ‡ 1 †; k ˆ 1; . . . ; K. We further denote all
endogenous event …e2 † times within ck by vk;i ; i ˆ 1; . . . ; Sk. By convention, let
vk;0 ˆ xk and vk;Sk ‡ 1 ˆ xk ‡ 1 , so ck is divided into periods
pk;i :‰vk;i ; vk;i ‡ 1 †;
i ˆ 0; . . . ; Sk
and the corresponding open interval …vk;i ; vk;i ‡ 1 † is denoted by pok;i. From our de®nition of
e1 and e2 , we can immediately see that throughout the last period, pk;Sk , of ck ; x…y; t† is
``¯at'', that is, x…y; t† ˆ Tm for some m ˆ 0; . . . ; M; we shall refer to this as a Type I
interval. During all other periods pk;i ; i ˆ 0; . . . ; Sk 1; x…y; t† takes values strictly
between any two adjacent thresholds; we shall refer to these as Type II intervals. A typical
sample path for the case of M ˆ 5 is shown in Figure 2.
During Type II intervals pk;i ; i ˆ 0; . . . ; Sk 1, if Tm 1 < x…y; t† < Tm for t [ pok;i , the
buffer content dynamics are
dx…t†
ˆ Am …t†
dt ‡
5†
On the other hand, during Type I intervals pk;Sk , the buffer content dynamics are
dx…y; t†
ˆ0
dt ‡
6†
With this discussion in mind, we may also view the SFM as a hybrid dynamic system in
which the buffer content satis®es the dynamics (5)±(6) with switches between M ‡ 1
``modes'' caused by the discrete events e1 and e2 we have de®ned above.
3.
IPA with Respect to Thresholds
Our objective here is to estimate the derivatives dE‰Lm;T …y†Š=dy; m ˆ 1; . . . ; M and
dE‰QT …y†Š=dy, through the sample derivatives dLm;T …y†=dy; m ˆ 1; . . . ; M and
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SUN ET AL.
dQT …y†=dy which are commonly referred to as the IPA estimators (Ho and Cao, 1991;
Cassandras and Lafortune, 1999). Henceforth, we shall use the ``prime'' notation to
denote derivatives with respect to y, so the sample derivatives above are denoted by
L0m;T …y† and Q0T …y†, respectively. For any sample function l…y†, an IPA-based estimate
l0 …y† of the performance metric derivative dE‰l…y†Š=dy is unbiased if
dE‰l…y†Š=dy ˆ E‰l0 …y†Š (see Ho and Cao, 1991; Cassandras and Lafortune, 1999).
Unbiasedness is the principal condition for making the application of IPA useful in
practice, for reliable sensitivity analysis purposes or for use in conjunction with control
and optimization methods that employ stochastic gradient-based techniques. We proceed
by ®rst evaluating the sample derivatives L0m;T …y† and Q0T …y† in terms of event time
derivatives dvk;i =dy, which we also denote by v0k;i, and then concentrate on the evaluation
of these event time derivatives based on observable quantities along a given sample path.
Similar to our de®nition of Am …t† in (1), let us also de®ne
Am;k;i :
M
X
nˆm
an …vk;i †
b…vk;i †
7†
Before proceeding, let us identify conditions under which the sample derivatives involved
exist. Recall that any exogenous event time is locally independent of y, whereas any
endogenous event time is generally a function of y. Excluding the possibility of the
simultaneous occurrence of exogenous and endogenous events, the only situation
preventing the existence of sample derivatives involves some t such that x…y; t† ˆ Tm ˆ y
and Am …t† ˆ 0 or Am ‡ 1 …t† ˆ 0; in such cases, the one-sided derivative still exists and can
be obtained through a ®nite difference analysis (as in Cassandras et al., 2002). However, to
keep the analysis simple, we focus only on the differentiable case by proceeding under the
following technical conditions:
Assumption 1:
a.
am …t† < ?; m ˆ 1; . . . ; M and b…t† < ? for all t [ ‰0; TŠ:
b.
For every y [ Y, w.p. 1, exogenous and endogenous events may not occur at the same
time.
c.
W.p. 1, there exists no interval …vk;i ; vk;i ‡ t†; t > 0, such that x…y; t† ˆ Tm for all
t [ …vk;i ; vk;i ‡ t†, and either Am …t† ˆ 0 or Am ‡ 1 …t† ˆ 0.
All three parts of Assumption 1 are mild technical conditions to ensure the existence of
sample derivatives. Regarding part c, as already pointed out, one-sided derivatives may
still be used if a sample path happens to contain a period with constant x…t; y† ˆ Tm for
some m ˆ 1; . . . ; M in which Am …t† ˆ 0 or Am ‡ 1 …t† ˆ 0. Speci®cally, suppose we change
y by some Dy > 0. Then, if Am …t† ˆ 0, the change in x…t†; Dx…t† ˆ
x…t; y ‡ Dy† x…t; y† ˆ 0. On the other hand, if Dy < 0, then Dx…t† < 0. As a result,
the left and right derivatives with respect to y are different.
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PERTURBATION ANALYSIS
3.1.
Class m Loss IPA Derivatives
Using the sample path partition into cycles, we may write (3) as follows:
Lm;T …y† ˆ
K Z
X
kˆ1
xk ‡ 1
xk
gm …y; t†dt
8†
Let us now de®ne
Z
lm;k …y† ˆ
xk ‡ 1
xk
gm …y; t†dt
9†
so that we can write
L0m;T …y† ˆ
K
X
kˆ1
l0m;k …y†
10†
and our objective is to evaluate l0m;k …y† for any k ˆ 1; . . . ; K. For simplicity, we will drop
the index k as we focus on a speci®c cycle ck in the following.
For the purpose of our analysis, a useful way of grouping periods pi within a typical
cycle is by de®ning sets associated with each class m ˆ 1; . . . ; M as follows:
1.
Partial Loss Period Set Um : For any pi [ Um, the buffer content is x…y; t† ˆ Tm for all
t [ pi , and class m traf®c experiences partial loss. In particular, the traf®c ¯ows satisfy
Am …t† > 0
and
Am ‡ 1 …t† < 0
11†
so
incoming ¯ow
PM
PMthat the processing capacity b…t† can accommodate the cumulative
a
t†
due
to
classes
m
‡
1;
.
.
.
;
M,
but
not
the
¯ow
n
nˆm‡1
n ˆ m an …t† that
includes the next lower priority class m. In this case, the system accepts only the
portion of the class m traf®c that can be accommodated and incurs a ``partial'' loss
rate
gm …y; t† ˆ Am …t†
12†
The dynamics of x…y; t† during this period are given by
dx…y; t†
ˆ0
dt ‡
Formally, we de®ne Um as follows:
Um :ˆ f pi : x…y; t† ˆ Tm ; t [ pi g:
13†
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SUN ET AL.
Note that the starting point vi of such a period corresponds to an endogenous event e2 ,
whereas the ending point vi ‡ 1 corresponds to an exogenous event e1 and is, therefore,
locally independent of y. Also note any one element of Um is always the last interval of a
cycle; conversely, the last interval of a cycle, pS , provided it is such that x…y; t† 6ˆ 0 for all
t [ pS , must be an element of Um for some m ˆ 1; . . . ; M.
2.
Full Loss Period Set Vm : During such periods, the buffer content is x…y; t† > Tm
( possibly excluding the starting point vi ; for instance, in Figure 2 …v3;1 ; v3;2 † [ V4 , but
‰v3;1 ; v3;2 † [ V3 ). In this case, all class m traf®c is lost:
gm …y; t† ˆ am …t†
14†
Formally, we de®ne Vm as follows:
Vm :ˆ f pi : x…y; t† > Tm ; t [ poi g
3.
15†
No Loss Period Set Wm : During such periods the buffer content is x…y; t† < Tm
( possibly excluding the starting point vi ) and no class m loss occurs:
gm …y; t† ˆ 0
16†
Formally, we de®ne Wm as follows:
Wm :ˆ f pi : x…t† < Tm ; t [ poi g
17†
Note that each of the sets above is locally independent of y (by a random function f …y†
being ``locally independent'' of y we mean that for a given y there exists Dy > 0 such that
for every y [ …y Dy; y ‡ Dy†, w.p.1 f …y† ˆ f …y†, where Dy may depend on both y and on
the sample path, and that for any particular m, Um [ Vm [ Wm ˆ ‰0; T† with all sets being
mutually exclusive). Moreover, recall that pS is the only possible period which belongs to
Um and vS ‡ 1 ˆ xk ‡ 1 . Thus, (9) can be written as
lm …y† ˆ
S
X
jˆ0
Z
1f pj [ Vm g
vj
vj ‡ 1
Z
am …t†dt ‡ 1f pS [ Um g
vS ‡ 1
vS
Am …t†dt
18†
where 1f ? g is the usual indicator function. By differentiating with respect to y, we obtain
l0m …y† ˆ
S
X
jˆ0
1f pj [ Vm g am …t†v0j ‡ 1
am …t†v0j
1f pS [ Um g ? Am;S v0S
19†
where the sample derivatives v0i …y† exist under Assumption 1. From (19), we observe that
the task of evaluating L0m;T …y† reduces to evaluating v0i …y†; i ˆ 1; . . . ; S, provided the ¯ow
rates involved in this expression are known.
275
PERTURBATION ANALYSIS
3.2.
Work IPA Derivative
Similar to the analysis in Section 3.1, we may write (4) as follows:
QT …y† ˆ
K Z
X
xk ‡ 1
xk
kˆ1
x…y; t†dt
20†
De®ne
Z
qk …y† ˆ
xk ‡ 1
xk
x…y; t†dt
21†
so that we can write
Q0T …y† ˆ
K
X
kˆ1
q0k …y†
22†
in which
q0k …y† ˆ
d
dy
Z
Z
xk ‡ 1
x…y; t†dt ˆ
xk
xk ‡ 1
xk
x0 …y; t†dt
23†
where we use the fact that xk ; xk ‡ 1 are independent of y. Again, we drop the index k in the
following analysis of a typical cycle ck .
Let us ®rst consider the last period pS in a cycle, which is a Type I period. If
x…y; t† ˆ Tm 6ˆ y for t [ pS , we have
x0 …y; t† ˆ 0
24†
If, on the other hand, x…y; t† ˆ y for t [ pS , we have
x0 …y; t† ˆ 1
25†
Next, let us consider the possible values x…y; t† may take in any other interval
pi ; i ˆ 0; . . . ; S 1 in the cycle. There are four cases.
Case 1:
x…y; vi † ˆ Tm 6ˆ y and Tm < x…y; t† < Tm ‡ 1 for t [ poi . In this case,
Z
x…y; t† ˆ Tm ‡
t
vi
Am ‡ 1 …t†dt
and, upon differentiating, we have
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SUN ET AL.
x0 …y; t† ˆ
Case 2:
Am ‡ 1;i v0i
x…y; vi † ˆ Tm 6ˆ y and Tm
Z
x…y; t† ˆ Tm ‡
t
vi
26†
1
< x…y; t† < Tm for t [ poi . We have
Am …t†dt
and, upon differentiating, we have
x0 …y; t† ˆ
Am;i v0i
27†
where Am;i was de®ned in (7).
Case 3:
x…y; vi † ˆ y:Tm and Tm < x…y; t† < Tm ‡ 1 for t [ poi . We have
Z
x…y; t† ˆ y ‡
t
vi
Am ‡ 1 …t†dt
and, upon differentiating, we have
x0 …y; t† ˆ 1
Case 4:
Am ‡ 1;i v0i
x…y; vi † ˆ y:Tm and Tm
Z
x…y; t† ˆ y ‡
t
vi
28†
1
< x…y; t† < Tm for t [ poi . We have
Am …t†dt
and, upon differentiating, we have
x0 …y; t† ˆ 1
Am;i v0i
29†
Summarizing this analysis, it is clear that the complete evaluation of x0 …y; t†, and hence of
Q0T …y† through (23), relies on the event time derivatives.
3.3.
Event Time Derivatives
As already discussed, the starting point of ck is independent of y because it is an
exogenous event, so x0k ˆ v0k;0 ˆ 0. Therefore, we next concentrate on a typical ck and,
dropping the index k, we shall show how an endogenous event time derivative
v0i ; i ˆ 1; . . . ; S, is related to v0i 1 so as to derive appropriate recursive relationships.
Recall that vi ; i ˆ 1; . . . ; S, are the event times when the buffer content reaches (and
possibly crosses) the thresholds T0 ; . . . ; TM . In addition, it is clear that the value of L0m;T …y†
277
PERTURBATION ANALYSIS
or Q0T …y† depends only on the event time derivatives v0i ; i ˆ 1; . . . ; S and the arrival and
processing rates at the instants vi . Therefore, we only need to focus on these particular
event instants and ignore all system activity in between them. This also explains why the
IPA estimators derived in this paper are not dependent on the stochastic characteristics of
the arrival and service processes; only arrival and service rates at selected event times vi
are involved. Furthermore; if these rates are known or they can be measured on line, then
evaluating L0m;T …y† and Q0T …y† reduces to the evaluation of the event time derivatives v0i
which is what we do next.
To do so, we consider all possible intervals pi :‰vi ; vi ‡ 1 †; i ˆ 0; . . . ; S 1. There are
eight possible cases.
Case 1:
Z
x…y; vi † ˆ Tm 6ˆ y and x…y; vi ‡ 1 † ˆ Tm ‡ 1 6ˆ y. In this case,
vi ‡ 1
vi
Am ‡ 1 …t†dt ˆ Tm ‡ 1
Tm
and, upon differentiating,
Am ‡ 1;i ‡ 1 v0i ‡ 1
Am ‡ 1;i v0i ˆ 0
so that
v0i ‡ 1 ˆ
Am ‡ 1;i
? v0
Am ‡ 1;i ‡ 1 i
30†
Case 2: x…y; vi † ˆ Tm 6ˆ y and x…y; vi ‡ 1 † ˆ Tm 1 6ˆ y. Recall that since x…t; y† > Tm 1 ;
t [ ‰vi; vi ‡ 1 †, ¯ows from all low priority classes 1; . . . ; m 1 are dropped and only the
higher priority classes m; . . . ; M are accommodated (see also (7)). Thus,
Z
vi
vi ‡ 1
Am …t†dt ˆ Tm
1
Tm
and, upon differentiating,
Am;i ‡ 1 v0i ‡ 1
Am;i v0i ˆ 0
so that
v0i ‡ 1 ˆ
Case 3:
Am;i
? v0
Am;i ‡ 1 i
x…y; vi † ˆ y ˆ Tm and x…y; vi ‡ 1 † ˆ Tm ‡ 1 . We have
31†
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SUN ET AL.
Z
vi ‡ 1
vi
Am ‡ 1 …t†dt ˆ Tm ‡ 1
y
and, upon differentiating,
Am ‡ 1;i ‡ 1 v0i ‡ 1
Am ‡ 1;i v0i ˆ
1
so that
v0i ‡ 1 ˆ
Case 4:
Z
Am ‡ 1;i
? v0
Am ‡ 1;i ‡ 1 i
1
x…y; vi † ˆ y ˆ Tm and x…y; vi ‡ 1 † ˆ Tm
vi ‡ 1
vi
Am …t†dt ˆ Tm
32†
Am ‡ 1;i ‡ 1
1
y
Am;i v0i ˆ
1
1.
We have
and, upon differentiating,
Am;i ‡ 1 v0i ‡ 1
so that
v0i ‡ 1 ˆ
Case 5:
Z
vi
Am;i
? v0
Am;i ‡ 1 i
x…y; vi † ˆ Tm
vi ‡ 1
1
Am;i ‡ 1
1
Am …t†dt ˆ y
33†
and x…y; vi ‡ 1 † ˆ y ˆ Tm . We have
Tm
1
and, upon differentiating,
Am;i ‡ 1 v0i ‡ 1
Am;i v0i ˆ 1
so that
v0i ‡ 1 ˆ
Case 6:
Am;i
1
? v0 ‡
Am;i ‡ 1 i Am;i ‡ 1
x…y; vi † ˆ Tm ‡ 1 and x…y; vi ‡ 1 † ˆ y ˆ Tm . We have
34†
279
PERTURBATION ANALYSIS
Z
vi ‡ 1
vi
Am ‡ 1 …t†dt ˆ y
Tm ‡ 1
and, upon differentiating,
Am ‡ 1;i ‡ 1 v0i ‡ 1
Am ‡ 1;i v0i ˆ 1
so that
v0i ‡ 1 ˆ
Am ‡ 1;i
1
? v0 ‡
Am ‡ 1;i ‡ 1 i Am ‡ 1;i ‡ 1
35†
Case 7: x…y; vi † ˆ x…y; vi ‡ 1 † ˆ Tm and x…y; t† > Tm for t [ poi . Note that it is possible to
have Tm ˆ y. We have
Z
vi ‡ 1
vi
Am ‡ 1 …t†dt ˆ 0
and, upon differentiating, we obtain (30).
Case 8: x…y; vi † ˆ x…y; vi ‡ 1 † ˆ Tm and x…y; t† < Tm for t [ poi . Once again, it is possible
to have Tm ˆ y. We have
Z
vi
vi ‡ 1
Am …t†dt ˆ 0
Upon differentiating, we obtain (31).
By combining (30)±(35), we can see that v0i ‡ 1 ; i ˆ 0; . . . ; S 1 is obtained through a
simple recursive relationship. We can summarize this analysis as follows:
THEOREM 3.1 For any endogenous event occurring at time vi ‡ 1 ; i ˆ 0; . . . ; S
have
v0i ‡ 1 ˆ Fi ? v0i ‡ Gi
1, we
36†
where Fi ; Gi are given in (30)±(35) depending on the values of x…y; vi † and x…y; vi ‡ 1 †.
Moreover, Gi 6ˆ 0 if and only if (i) x…y; vi † 6ˆ x…y; vi ‡ 1 † ˆ y or (ii)
x…y; vi ‡ 1 † 6ˆ x…y; vi † ˆ y.
Proof: The recursive relationship (36) follows directly from the analysis leading to
(30)±(35). The necessary and suf®cient condition for Gi 6ˆ 0 also follows from (30)±
(35).
j
280
SUN ET AL.
Note that if an exogenous event occurs at x, we get v0j ˆ 0 for all j 1 as long as the
condition Gi 6ˆ 0 above is not met. Thus, event time derivatives accumulate according to
(36) following an endogenous event; they are periodically reset to 0 and may maintain this
value for considerable portions of a sample path as long as Gi ˆ 0. Finally, based on (36),
we can return to (19) and (23) in order to fully evaluate the loss volume and work IPA
derivatives.
4.
IPA for the Two-Class Case
In the previous section, we saw that IPA estimators for the class-dependent loss volumes
and the workload with respect to a given threshold parameter among T1 ; . . . ; TM are
obtained through recursive relationships based on (36). For the special case where M ˆ 2,
we are able to obtain these estimators through closed-form expressions by exploiting the
speci®c structure of a sample path when only two classes are involved. Thus, we focus on
the SFM shown in Figure 3 and simplify notation by letting T2 ˆ b be the buffer capacity
and setting T1 ˆ Tm ˆ y, where 0 < T1 < b. In this case, there are three performance
metrics involved, the loss volume of class 1, L1;T …y†, the loss volume of class 2, L2;T …y†,
and the work, QT …y†.
A sample path example of this SFM is shown in Figure 4. As in the general M-class
case, the sample path of the two-class system is divided into cycles ck ; k ˆ 1; . . . ; K. In
the remainder of this section, we derive the IPA estimators L01;T …y†; L02;T …y†, and Q0T …y†.
Our main results may be summarized as follows: (i) We show that in this case these
estimators are given by closed-form expressions and derive these expressions; (ii) we
show that L02;T …y† is easy to obtain from L01;T …y† without requiring any additional
computation; and (iii) we obtain a simple upper and lower bound for L01;T …y†.
As seen in Section 3, L01;T …y†; L02;T …y†, and Q0T …y† are based on event time derivatives.
Theorem 3.1 shows how to evaluate vi ‡ 1 ; i ˆ 0; . . . ; S 1, for a particular cycle ck . For
the two-class case, there are several simpli®cations taking place in the eight cases
considered in Section 3.3. Speci®cally, Cases 1, 2 are clearly infeasible; Cases 3, 4 only
apply to period pS 1 ; Cases 5, 6 only apply to period p0 ; and only Cases 7, 8 are always
applicable. Taking advantage of these observations, (36) in Theorem 3.1 can be
considerably simpli®ed. In particular, the following lemma shows that all event time
derivatives that turn out to be of interest in evaluating L01;T …y†; L02;T …y†, and Q0T …y† are either
0 or can be expressed in terms of A1;i and A2;i (de®ned in (7) with the index k dropped). By
Figure 3. Stochastic ¯uid model (SFM) with M ˆ 2.
281
PERTURBATION ANALYSIS
Figure 4. Typical sample path segment with M ˆ 2.
convention, we shall set A1;0 :1 and A2;0 :1. We will also use the notation dxe to denote
the smallest integer greater than or equal to x.
LEMMA 4.1 Within a cycle that starts with x, for all vi ; i ˆ 1; . . . ; S, which satisfy
x…y; vi † ˆ y:
1.
If x…y; x† ˆ 0:
v01 ˆ
v02n ˆ
A1;0
A1;1
n
Y
A2;2i
iˆ1
37†
1
A2;2i
?
where 1 n dS
v02n ‡ 1 ˆ
2
38†
1
1=2e and S > 1; and
n
A1;2n Y
A2;2i 1 A1;2i
?
A1;2n ‡ 1 i ˆ 1 A2;2i
A1;2i
where 1 n bS
2.
A1;2i
A1;2i
2
1
39†
1=2c, and S > 2.
If x…y; x† ˆ b:
v01 ˆ
v02n ˆ
A2;0
A2;1
n
Y
A1;2i
iˆ1
40†
1
A1;2i
where 1 n dS
?
A2;2i
A2;2i
2
1
1=2e and S > 1; and
41†
282
SUN ET AL.
v02n ‡ 1 ˆ
n
A2;2n Y
A1;2i 1 A2;2i
?
A2;2n ‡ 1 i ˆ 1 A1;2i
A2;2i
where 1 n bS
3.
2
42†
1
1=2c and S > 2.
If x…y; x† ˆ y:
v0i ˆ 0
Proof:
4.1.
43†
See Appendix.
j
Class 1 Loss Derivatives
Based on the analysis in Section 3.1, to evaluate L01;T …y† we need to evaluate l01;k …y† for any
k ˆ 1; . . . ; K. Focusing on a typical cycle ck and dropping the index k in the following
analysis, we get from (19):
l01 …y† ˆ
S
X
iˆ0
1f pi [ V1 g a1 …vi ‡ 1 †v0i ‡ 1
a1 …vi †v0i
1f pS [ U1 g ? A1;S v0S
44†
Regarding the last period pS , we next make an observation which simpli®es our analysis
because it allows us to proceed without needing to evaluate v0S beyond the case already
covered by Lemma 4.1. In particular, since the last partial loss period of each cycle can
only take the values y; b, and 0, this allows us to identify three possible cases:
i.
ii.
If pS [ U1 , then x…y; vS † ˆ y and Lemma 4.1 applies allowing us to obtain v0S .
If pS [ V1 then x…y; vS † ˆ b and we cannot obtain v0S via Lemma 4.1. However, by the
continuity of the sample path, we must have pS 1 [ V1 . Thus, the last two terms
contributed by the sum on the right-hand-side of (44) are due to pS 1 in the amount
a1 …vS †v0S
a1 …vS
0
1 †vS
1
and by pS in the amount
a1 …vS †v0S
where we use the fact that v0S ‡ 1 ˆ 0. Therefore, the combined contribution of pS
and pS is
a1 …vS
0
1 †vS
1
and we observe that v0S is not needed for evaluating l01 …y†.
1
45†
283
PERTURBATION ANALYSIS
iii.
If pS [ W1 , then x…y; vS † ˆ 0 and Lemma 4.1 does not apply. However, again by the
continuity of the sample path, we must have pS 1 [ W1 . It is clear from (44) that v0S is
again not involved in the evaluation of l01 …y†.
The next four lemmas provide the expressions for the derivative l01 …y† in (44) over a cycle,
depending on the state of the SFM when the cycle starts and ends. In the ®rst lemma, we
show that l01 …y† ˆ 0 whenever the cycle starts from y (e.g., ck ‡ 1 in Figure 4) regardless of
how it ends. In the second lemma, we show that l01 …y† ˆ 1 whenever the cycle ends with
y (e.g., ck in Figure 4) if it starts with x…y; x† 6ˆ y.
LEMMA 4.2 For a cycle that starts with x, if x…y; x† ˆ y, then:
l01 …y† ˆ 0
Proof:
46†
See Appendix.
j
LEMMA 4.3 For a cycle that starts with x and contains events occurring at vi ; i ˆ
1; . . . ; S, if x…y; x† 6ˆ y and x…y; vS † ˆ y, then:
l01 …y† ˆ
Proof:
1
47†
See Appendix.
j
The next two lemmas consider the cases where a cycle starts with x…y; x† ˆ 0 and b,
respectively, and does not end with y (e.g., ck 2 ; ck 1 , and ck ‡ 2 in Figure 4). In these
cases, l01 …y† is given by ¯ow rate information contained in A1;i and A2;i (de®ned in (7) with
the index k dropped).
LEMMA 4.4 For a cycle that starts with x and contains events occurring at vi ; i ˆ
1; . . . ; S:
1.
If x…y; x† ˆ 0; x…y; vS † ˆ 0:
For S ˆ 1: l01 …y† ˆ 0
For S > 1: l01 …y† ˆ
2.
48†
1‡
S Y
1†=2
iˆ1
A1;2i A2;2i
?
A2;2i A1;2i
1
1
:
49†
If x…y; x† ˆ 0, x…y; vS † ˆ b:
For S ˆ 2 : l01 …y† ˆ
1‡
A2;1
A1;1
For S > 2 : l01 …y† ˆ
1‡
A2;S
A1;S
50†
1
S Y
2†=2
1
iˆ1
A1;2i A2;2i
?
A2;2i A1;2i
1
1
:
51†
284
SUN ET AL.
Proof:
See Appendix.
j
LEMMA 4.5 For a cycle that starts with x and contains events occurring at
vi ; i ˆ 1; . . . ; S:
1.
2.
If x…y; x† ˆ b, x…y; vS † ˆ 0:
For S ˆ 2: l01 …y† ˆ
1‡
A1;1
A2;1
For S > 2: l01 …y† ˆ
1‡
A1;S
A2;S
52†
1
S Y
2†=2
1
iˆ1
A2;2i A1;2i
?
A1;2i A2;2i
1
1
:
53†
If x…y; x† ˆ b; x…y; vS † ˆ b:
For S ˆ 1: l01 …y† ˆ 0
For S > 1: l01 …y† ˆ
Proof:
54†
1‡
S Y
1†=2
iˆ1
A2;2i A1;2i
?
A1;2i A2;2i
1
1
:
55†
See Appendix.
j
Finally, we can also establish some simple bounds on l01 …y† as shown in the following
lemma.
LEMMA 4.6 The class 1 loss over any cycle satis®es:
Proof:
See Appendix.
1 l01 …y† 0.
j
Motivated by our analysis thus far, let us partition the set of all cycles as follows. First,
de®ne
F ˆ fk [ f1; . . . ; Kg : x…y; xk † ˆ yg
to be the set of cycles that start from y, corresponding to Lemma 4.2. Let F…y†
denote the
complement set of F and de®ne
y
F
F00
0b
F
b0
F
x…y; xk ‡ 1 † ˆ yg
ˆ fk : k [ F;
x…y; xk † ˆ 0; x…y; xk ‡ 1 † ˆ 0g
ˆ fk : k [ F;
x…y; xk † ˆ 0; x…y; xk ‡ 1 † ˆ bg
ˆ fk : k [ F;
x…y; xk † ˆ b; x…y; xk ‡ 1 † ˆ 0g
ˆ fk : k [ F;
bb ˆ fk : k [ F;
x…y; xk † ˆ b; x…y; xk ‡ 1 † ˆ bg
F
285
PERTURBATION ANALYSIS
which cover the remaining cases, corresponding to Lemmas 4.3±4.5. Moreover, note that
all the above sets are locally independent of y. The expressions in Lemmas 4.4±4.5 can
also be simpli®ed by de®ning for any k ˆ 1; . . . ; K,
Rk ; n ˆ
Rk;n ˆ
n=2
Y
A1;k;2i A2;k;2i
?
A
; k; 2i A1;k;2i
iˆ1 2
n=2
Y
A2;k;2i
A
i ˆ 1 1;k;2i
?
A1;k;2i
A2;k;2i
1
1
1
1
;
n ˆ 2; 4; . . .
;
n ˆ 2; 4; . . .
and Rk;0 ˆ 1
Rk;0 ˆ 1
and
56†
57†
Then, using Lemmas 4.2±4.5 and the de®nitions above, we get
l01;k …y† ˆ
1‡
8
0
>
>
>
>
Rk;Sk
>
>
>
>
< A2;k;Sk
A1;k;Sk
A1;k;Sk
>
>
>
A
>
>
> 2;k;Sk
>
>
: Rk;Sk
1
y
if k [ F
if k [ F00
0b
if k [ F
1
1
1
1
1
Rk;Sk
2
Rk;Sk
58†
b0
if k [ F
bb
if k [ F
if k [ F
2
1
We may now summarize our results as follows.
THEOREM 4.1 The sample derivative L01;T …y† is given by
L01;T …y† ˆ
K ‡
X
00
k[F
‡
X A1;k;S
k
A
2;k;Sk
k [ Fb0
Rk;Sk
1
1
1
Rk;Sk
‡
X A2;k;S
k
A
1;k;S
k
k [ F0b
2
‡
X
bb
k[F
Rk;Sk
1
1
Rk;Sk
2
1
59†
y contained in ‰0; TŠ, including a possibly
where K is the (random) cardinality of the set F
incomplete last cycle.
Proof:
The result follows immediately from (10) and (58).
j
The expression in (59) provides the IPA estimator for the loss metric L1;T …y†. Unlike the
M > 2 case studied in Section 3, here we are able to obtain a closed-form expression in
terms of A1;k;i and A2;k;i . Again, we note that L01;T …y† does not depend on any distributional
information regarding the arrival and service processes and involves only ¯ow rates at
event times vk;i which may be estimated on line.
As far as implementation is concerned, the evaluation of l01;k …y† in Lemmas 4.2±4.5
requires observing events e1 and e2 and the corresponding rates of a1 ; a2 , and b at their
occurrence times, so that we can evaluate A1;k;i and A2;k;i . If a cycle starts from an empty
buffer or full buffer and ends with a partial loss period at y, then the only implementation
286
SUN ET AL.
requirement is that such a cycle be detected and the contribution of this cycle is simply 1.
It is also worth mentioning the following simple bounds we can obtain:
COROLLARY L01;T …y† is bounded as follows:
K L01;T …y† 0
Proof:
4.2.
Follows directly from (59) and Lemma 4.6, in view of (10).
j
Class 2 Loss Derivatives
Since T2 ˆ b, class 2 experiences partial loss only during the period through which
x…y; t† ˆ b, and the loss rate is g2 …y; t† ˆ a2 …t† b…t†. From (19) we have
l02 …y† ˆ
1f pS [ U2 g ? A2;S v0S
60†
The following lemma greatly simpli®es the task of evaluating the sample derivative
L02;T …y†, since it can be obtained as a byproduct of the evaluation of L01;T …y† in Theorem 4.1.
LEMMA 4.7 For a cycle that starts with x and contains events occurring at vi ; i ˆ
1; . . . ; S:
8
<0
l02 …y† ˆ 1
:
l01 …y†
if pS =[ U2
if pS [ U2 and x…y; x† ˆ y
if pS [ U2 and x…y; x† 6ˆ y
61†
where l1 …y† is the class 1 loss incurred over the cycle.
Proof:
4.3.
See Appendix.
j
Work Derivative
As we saw in Section 3.2, Q0T …y† requires the evaluation of
q0k …y† ˆ
Z
xk ‡ 1
xk
x0 …y; t†dt
62†
for every k ˆ 1; . . . ; K. Again, we drop the index k for convenience, as we consider a
typical cycle ck . Regarding the ®rst period, p0 , in the cycle, there are two possibilities. If
287
PERTURBATION ANALYSIS
x…y; x† 6ˆ y, Cases 1 and 2 in Section 3.2 apply. In view of the fact that x0 ˆ 0, by (26) and
(27) we have
x0 …y; t† ˆ 0
63†
If, on the other hand, x…y; x† ˆ y, then either Case 3 or Case 4 of Section 3.2 applies. Either
way, from (28) and (29) we get
x0 …y; t† ˆ 1
64†
If S > 1, observe that all endogenous events contained in a cycle are constrained to be such
that x…y; vi † ˆ y (see also Figure 4). Thus, for all pi ; i ˆ 1; . . . ; S 1, either Case 3 or Case
4 of Section 3.2 applies. If x…y; t† > y for t [ poi ; i > 0, by (28) we have
x0 …y; t† ˆ 1
A2;i v0i
65†
Otherwise, if x…y; t† < y for t [ poi ; i > 0, by (29) we have
x0 …y; t† ˆ 1
A1;i v0i
66†
Finally, consider the last period, pS , in the cycle. From (24) and (25), if pS =[ U1 then we
have
x0 …y; t† ˆ 0
67†
and, otherwise,
x0 …y; t† ˆ 1
68†
We can summarize this analysis in the following:
LEMMA 4.8 For a cycle that starts with x and contains events occurring at vi ; i ˆ
1; . . . ; S:
q0 …y† ˆ 1fx…y; x† ˆ yg ? …v1
x† ‡
SX1
iˆ1
vi ‡ 1
vi †fi ‡ 1f pS [ U1 g ? …vS ‡ 1
vS †
69†
where
fi ˆ
Proof:
1
1
A1;i v0i ;
A2;i v0i ;
if x…y; t† > y;
if x…y; t† < y;
t [ poi
t [ poi
The proof follows from (62) by combining (63)±(68).
70†
j
288
SUN ET AL.
Note that if x…y; x† ˆ y, by (43) we have v0i ˆ 0 for i ˆ 1; . . . ; S 1, and (70) implies
fi ˆ 1 for i ˆ 1; . . . ; S 1. Therefore, in this case, by combining the ®rst two terms of
(69) the total contribution of the ®rst S periods of a cycle becomes vS x. The third item of
(69) is the contribution of pS which is its length if it happens to be a class 1 partial loss
period; in this case, the total contribution of the cycle is simply its full length.
THEOREM 4.2 The sample derivative Q0T …y† is given by
Q0T …y† ˆ
K
X
kˆ1
q0k …y†
71†
where K is the (random) number of cycles contained in ‰0; TŠ, including a possibly
incomplete last one.
Proof:
(70).
The result follows from (22) and Lemma 4.8, using the de®nitions in (69) and
j
The expression in (71) provides the IPA estimator for the work metric de®ned in (4). Its
implementation requires the same information as that for the loss metric with the addition
of timers to measure the duration of the periods pi ; i ˆ 0; . . . ; S.
5.
IPA Estimator Unbiasedness
We now return to the general M-class case in order to prove the unbiasedness of the IPA
derivatives L0m;T …y†; m ˆ 1; . . . ; M and Q0T …y†. In general, the unbiasedness of an IPA
derivative l0 …y† is ensured by the following two conditions (see Rubinstein and Shapiro
[20], Lemma A2, p. 70):
Condition 1. For every y [ Y, the sample derivative l0 …y† exists w.p.1.
Condition 2. W.p.1, the random function l…y† is Lipschitz continuous throughout Y, and
the (generally random) Lipschitz constant has a ®nite ®rst moment.
We have already discussed the mild technical conditions required to ensure the existence
of L0m;T …y†; m ˆ 1; . . . ; M and Q0T …y†. Consequently, establishing the unbiasedness of
L0m;T …y†; m ˆ 1; . . . ; M and Q0T …y† as estimators of dE‰Lm;T …y†Š=dy; m ˆ 1; . . . ; M and
dE‰QT …y†Š=dy, respectively, reduces to verifying the Lipschitz continuity of
Lm;T …y†; m ˆ 1; . . . ; M and QT …y† with appropriate Lipschitz constants.
The buffer content over the nominal sample path is denoted by x…y; t†, while the buffer
content in a perturbed sample path is denoted by x…y ‡ Dy; t† resulting when y is replaced
by y ‡ Dy. We superimpose two such sample paths and denote the combined
superimposed event times (e1 and e2 ) by t0 < t1 < < tJ…T†, where t0 ˆ 0; tJ…T† ˆ T.
Thus, J…T† is the random number of all superimposed events (exogenous and endogenous)
289
PERTURBATION ANALYSIS
in ‰0; TŠ. The interval ‰0; TŠ can, therefore, be divided into time intervals
Ij :‰tj 1 ; tj †; j ˆ 1; . . . ; J…T†; we shall use Ijo :…tj 1 ; tj † to denote the corresponding
open intervals. Recalling the de®nitions of Type I and Type II intervals in Section 2, in
each Ijo , the nominal sample path is in either a Type I or a Type II interval and the same is
true for the perturbed sample path. Moreover, if Ij is a Type I interval, for t [ Ij we have
dx… ? ; t†
ˆ0
dt ‡
72†
while if Ij is a Type II interval and Tm < x… ? ; t† < Tm ‡ 1 for t [ Ijo , we have
dx… ? ; t†
ˆ Am ‡ 1 …t†
dt ‡
73†
Then, under the assumption that E‰J…T†Š < ?, we shall establish next that
L0m;T …y†; m ˆ 1; . . . ; M and Q0T …y† are indeed unbiased estimators.
At this point it is worth recalling that x…y; t† and x…y ‡ Dy; t† are continuous functions of
t due to Assumption 1a. Next, we show that 0 Dx…t† Dy for all t [ ‰0; TŠ, where
Dx…t† ˆ x…y ‡ Dy; t† x…y; t† and Dy > 0 (the case Dy < 0 is similarly handled). To do
so, we ®rst need the following result (Lemma 5.1). In order to maintain the notation as
simple as possible, we de®ne:
Bm …t†:
M
X
an …t†
b…t†
and Cm …t†:
n ˆ n0
n0m
M
X
an …t†
b…t†
n ˆ n0
n0m
Recalling the de®nition of Am …t† in (1), observe that
Bm …t† Am …t† Cm …t†
LEMMA 5.1 Consider an interval Ij ˆ ‰tj
Then
1 ; tj †,
and assume that 0 Dx…tj
1†
Dy.
0 Dx…t† Dy for all t [ Ij
Proof:
See Appendix.
74†
j
We can now use this result to establish the same bound for Dx…t† over all t [ ‰0; TŠ.
LEMMA 5.2 For all t [ ‰0; TŠ,
0 Dx…t† Dy
Proof:
See Appendix.
75†
j
290
SUN ET AL.
Finally, we can establish the unbiasedness of L0m;T …y†; m ˆ 1; . . . ; M and Q0T …y†.
THEOREM 5.1 The IPA derivatives L0m;T …y†; m ˆ 1; . . . ; M and Q0T …y† are unbiased
estimates of dE‰Lm;T …y†Š=dy; m ˆ 1; . . . ; M and dE‰QT …y†Š=dy, respectively. In other
words,
dE‰Lm;T …y†Š
;
E L0m;T …y† ˆ
dy
Proof:
6.
See Appendix.
m ˆ 1; . . . ; M
and
E‰Q0T …y†Š ˆ
dE‰QT …y†Š
dy
j
Conclusions
We have considered in this paper a SFM of a queueing system with multiple customer
classes subject to threshold-based buffer control. We have developed IPA estimators for
class-speci®c loss metrics and for a work metric with respect to one of the threshold and
shown them to be unbiased. We have also shown that in the two-class case it is possible to
obtain closed-form expressions for the estimators (as opposed to evaluating them
recursively using Theorem 3.1). The simplicity of the estimators derived and the fact they
are not dependent on knowledge of the arrival or service processes makes them attractive
for on-line control and optimization in applications such as manufacturing and
communication networks. We emphasize again that the use of a SFM allows us to
obtain the form of a particular gradient estimator, but the actual implementation is carried
out using system data (e.g., observing events such as an actual buffer content crossing
some threshold Tm ) from the actual (discrete-event) system.
One of the limitations of IPA applied to SFMs to date is the absence of feedback
mechanisms in the models analyzed, i.e., the assumption that arrival and service processes
are not dependent on controllable parameters. Recent work (see Yu and Cassandras, 2003)
has shown that it is possible to still obtain relatively simple and unbiased IPA estimators in
such cases. This is a positive step towards the development of a more general theory for
applying PA techniques to SFMs, but the level of generality currently available is limited
to certain classes of models. The key elements of a general approach, however, would be
typi®ed by the main steps seen in this paper: (i) Decomposing a sample path through
appropriately de®ned events that re¯ect changes in the time-driven dynamics of an
underlying system; (ii) deriving sample derivatives (assuming they exist) for the event
times and the state of the underlying system (in our case, the buffer content); (iii)
translating these derivatives into derivatives of performance metrics of interest (in our
case, loss and workload); and (iv) establishing the Lipschitz continuity of the performance
metrics if that holds, which, under generally mild conditions, leads to proving
unbiasedness of the IPA derivatives obtained in the previous step.
PERTURBATION ANALYSIS
291
Appendix
Proof of Lemma 4.1: The proof is based on Theorem 3.1 applied to the two-class
system as a special case, as well as the observation that all endogenous events occurring at
vi within a cycle must satisfy x…y; vi † ˆ y (see also Figure 4). We begin with the ®rst period
p0 ˆ ‰x; v1 † where x…y; v1 † ˆ y, for which there are three possibilities:
1.
x…y; x† ˆ 0. Since x is an exogenous event, x0 ˆ 0, therefore by (36) and (34),
v01 ˆ G0 ˆ
2.
76†
x…y; x† ˆ b. By (36) and (35) and in view of x0 ˆ 0, we get
v01 ˆ G0 ˆ
3.
A1;0
1
:
A1;1 A1;1
A2;0
1
:
A2;1 A1;1
77†
x…y; x† ˆ y. By (36) and the facts that G0 ˆ 0 (see Theorem 3.1) and x0 ˆ 0, we get
v01 ˆ 0
78†
Next, consider any pi ; 1 i S 1. Since we must have x…y; vi † ˆ x…y; vi ‡ 1 † ˆ y, by
Theorem 3.1 we get Gi ˆ 0. We then have two possible cases:
1.
pi [ V1 . In this case, x…y; t† > y for t [ poi , and from (36) and (30) we get
v0i ˆ
2.
A2;i 1 0
v
A2;i i
1
79†
pi [ W1 . In this case, x…y; t† < y for t [ poi , and from (36) and (31) we get
v0i ˆ
A1;i 1 0
v
A1;i i
1
The proof is completed by combining (76)±(80).
80†
j
Proof of Lemma 4.2: For a cycle with x…y; x† ˆ y, Lemma 4.1 asserts that all event time
derivatives of interest are 0, so by (44) we immediately obtain (46).
j
Proof of Lemma 4.3: Let li …y† be the class 1 loss over an interval pi :
Z vi ‡ 1
li …y† ˆ
g1 …y; t†dt
vi
81†
Since x…y; vS † ˆ y, we have pS [ U1 . Suppose S > 1 (the special case where S ˆ 1 will
292
SUN ET AL.
easily follow). There are two possible cases regarding x…y; x†, that is, x…y; x† ˆ b or 0.
First, suppose x…y; x† ˆ b. Then, the periods pi ; i ˆ 0; . . . ; S 1 belong to either W1 or V1
and appear alternately with the ®rst one p0 ˆ ‰x; v1 † [ V1 . Thus, from (81) and in view of
g1 …y; t† ˆ a1 …t† from (14) we get
l00 …y† ˆ a1 …v1 †v01
since v00 ˆ x0 ˆ 0. In light of (7) the above equation can be rewritten as
l00 …y† ˆ A1;1 v01
A2;1 v01
82†
Also from (81), for all other full loss periods p2n ; n 1, we have
l02n …y† ˆ a1 …v2n ‡ 1 †v02n ‡ 1
a1 …v2n †v02n
83†
By (79) in the proof of Lemma 4.2,
A2;2n ‡ 1 v02n ‡ 1
A2;2n v02n ˆ 0
Adding the left-hand-side above to the right-hand-side of (83) and recalling (7), we get
l02n …y† ˆ A1;2n ‡ 1 v02n ‡ 1
A1;2n v02n
84†
For the loss over the partial loss period pS , from (81) and g1 …y; t† ˆ A1 …t† in (12), we have
l0S ˆ
A1;S v0S
85†
In addition, we know that p2n 1 [ W1 for n 1, therefore l02n
in the proof of Lemma 4.2, we have:
A1;2n
0
1 v2n
1
1 …y†
ˆ 0. Using (7) and (80)
ˆ A1;2n v02n
86†
The cumulative contribution of all full loss periods is given by combining (82), (84), and
(85). If S is odd (including S ˆ 1), then pS 1 [ V1 and the total contribution is
l01 …y† ˆ
S
X
iˆ0
l0i …y† ˆ
A2;1 v01 ‡
S
X
iˆ1
On the other hand, if S is even, then pS
l01 …y† ˆ
S
X
iˆ0
l0i …y† ˆ
A2;1 v01 ‡
1†i ‡ 1 A1;i v0i
1
SX1
[ W1 and the total contribution is
1†
iˆ1
A1;S v0S
i‡1
A1;i v0i
A1;S v0S
293
PERTURBATION ANALYSIS
The last two terms of the right-hand-side of the last two equations are equal to 0 by using
(86), so for both cases we get
l01 …y† ˆ
A2;1 v01
87†
By Lemma 4.1, v01 ˆ 1=A2;1 , so the last equation yields l01 …y† ˆ 1, which is precisely
(47).
The second case is x…y; x† ˆ 0, for which a similar argument also gives l01 …y† ˆ 1,
completing the proof of (47).
j
Proof of Lemmas 4.4 and 4.5: We combine the proofs of these two lemmas since they
are similar in nature. First, note that if S ˆ 1 the buffer content never reaches y, therefore it
is easy to see that any class 1 loss is unaffected by y over this interval, which implies
l01 …y† ˆ 0. This establishes (48) and (54). Next, for S > 1; p1 ; . . . ; pS 1 belong alternately
to V1 and W1 . Depending on the value of x…y; t† at x and vS , there are four cases to consider:
Case 1: x…y; x† ˆ 0; x…y; vS † ˆ 0. Since the buffer content never reaches b in this cycle,
this case is identical to the in®nite buffer case studied in Cassandras et al. (2003), and (48),
(49) are precisely the results obtained in Lemma 3.3 of Cassandras et al. (2003).
Case 2: x…y; x† ˆ 0; x…y; vS † ˆ b. In this case, there is at least one no loss and one full
loss period before pS and S must be even with p2n 1 ; 1 n S=2, all being full loss
periods. It follows from (45) that the contribution of pS 1 and pS together is
0
1 †vS
a1 …vS
1
which we can rewrite, using (7), as
A1;S
0
1 vS
1
‡ A2;S
0
1 vS
88†
1
Using (81) and (7), the loss derivative contribution of any other full loss period is
A1;2n
0
1 v2n
1
‡ A1;2n v02n ;
Adding (89) for all n ˆ 1; . . . ; …S
the proof of Lemma 4.3 gives
l01 …y† ˆ
A1;1 v01 ‡ A2;S
0
1 vS
1n
S
2
2
89†
2†=2 and (88) together and then using (86) obtained in
1
We can now replace v01 and v0S 1 by (37) and (39) with n ˆ S
respectively, which immediately yields (51).
90†
2=2 in Lemma 4.1,
Case 3: x…y; x† ˆ b; x…y; vS † ˆ 0. As in the previous case, S must be even and
p2n ; 0 n S 2=2, are all full loss periods. The loss derivative contribution of p0 is
given in (82). The contribution of any other full loss period is given by (84):
294
SUN ET AL.
A1;2n v02n ‡ A1;2n ‡ 1 v02n ‡ 1 ;
1n
Adding (82) and (91) for all n ˆ 1; . . . ; …S
l01 …y† ˆ
A2;1 v01 ‡ A1;S
Replacing v01 and v0S
yields (53).
1
0
1 vS
S
2
91†
2
2†=2 together and then using (86) gives
1
by (40) and (42) with n ˆ …S
2†=2 in Lemma 4.1, respectively,
Case 4: x…y; x† ˆ b; x…y; vS † ˆ b. In this case, S must be odd and
p2n ; 0 n …S 1†=2, are all full loss periods. The loss derivative contribution of the
®rst full loss period ‰x; v1 † is given by (82). The contribution of the last full loss period
‰vS 1 ; vS †, together with the contribution of pS 1 , is given by (88). The loss derivative
contribution of any other full loss period is as in (91):
A1;2n v02n ‡ A1;2n ‡ 1 v02n ‡ 1 ;
1n
S
3
Adding (82), (88), and (92) for all n ˆ 1; . . . ; …S
get
l01 …y† ˆ
A2;1 v01 ‡ A2;S
0
1 vS
92†
2
3†=2 together and then using (86) we
1
Finally, replacing v01 and v0S 1 by (40) and (41) with n ˆ …S
respectively, yields (55). The proof is now complete.
93†
1†=2 in Lemma 4.1,
j
Proof of Lemma 4.6: Let us focus on a cycle with x…y; x† 6ˆ y; x…y; vS † 6ˆ y and S > 1,
since in all remaining cases l01 …y† ˆ 1 or 0 from Lemmas 4.2 and 4.3. For such a cycle,
at some vi ; 1 i S 1, the sample path crosses y from either below or above. If at vi the
sample path crosses y from below, this event corresponds to the start of a full loss period,
therefore A2;i > 0. Then, using (2), we have
A1;i A2;i > 0
since a1 …vi † 0. Therefore,
0<
A2;i
1
A1;i
94†
If, on the other hand, at vi the sample path crosses y from above, this event corresponds to
the start of a No Loss period, implying that A1;i < 0. Thus, recalling (2), we get
A2;i A1;i < 0
295
PERTURBATION ANALYSIS
so that
0<
A1;i
1
A2;i
95†
For Cases 1 and 2 in the proof of Lemma 4.4, the cycle starts with x…y; x† ˆ 0, thus event
times v2n 1 correspond to crossing y from below, while event times v2n …y† correspond to
crossing y from above. For Cases 3 and 4 in the proof of Lemma 4.5, the cycle starts with
x…y; x† ˆ b, so event times v2n 1 correspond to crossing y from above and event times v2n
correspond to crossing y from below. By using (94) and (95) in (49)±(55) it is easy to see
that 1 < l01 …y† 0, which completes the proof.
j
Proof of Lemma 4.7: First, if pS =[ U2 , then the ®rst part of (61) immediately follows
from (60). Next, consider a cycle with pS [ U2 , so x…y; vS † ˆ b. If x…y; x† ˆ y, we must
have x…y; vS 1 † ˆ y and, by (43), v0S 1 ˆ 0. Then by (32) we have
1
A2;S
v0S ˆ
It follows from (60) that l02 …y† ˆ 1, which establishes the second part of (61).
It remains to consider the case x…y; x† 6ˆ y. If S ˆ 1, we must have x…y; x† ˆ b, which is
Case 8 in Section 3.3, and by (31) we have v0S ˆ 0, since v00 ˆ x0 ˆ 0. It follows from (60)
that
l02 …y† ˆ 0
96†
This case also corresponds to (54) in Lemma 4.5, giving l01 …y† ˆ 0, and we see that (61) is
satis®ed.
On the other hand, if S > 1, during the cycle the buffer content hits y at least once; thus,
pS 1 is a full loss period and the buffer content evolves from y to b with
Z
vS
vS
A2 …t†dt ˆ b
y
1
Upon taking derivatives, we get
A2;S v0S
A2;S
0
1 vS
1
ˆ
1
Therefore, by (60) we get
l02 …y† ˆ 1
A2;S
0
1 vS
1
97†
The cycle satis®es either x…y; x† ˆ 0 or x…y; x† ˆ b. In the former case, (90) in the proof of
Lemma 4.4 applies and using Lemma 4.1 to get A1;1 v01 ˆ 1, we have
296
SUN ET AL.
l01 …y† ˆ
0
1 vS
1 ‡ A2;S
1 …y†
ˆ
l02 …y†
where the second equality follows from (97). In the latter case, (93) in the proof of Lemma
4.5 applies and using Lemma 4.1 to get A2;1 v01 ˆ 1, we get the exact same expression for
l01 …y† and we have
l01 …y† ˆ
1 ‡ A2;S
0
1 vS
1
ˆ
l02 …y†
which satis®es (61) and completes the proof.
j
Proof of Lemma 5.1: Recalling the de®nition of Ij , we know that there is no event
occurring in Ijo . Thus, within Ijo , both the original sample path and the perturbed sample
path are either at one of the thresholds or bounded between two adjacent thresholds. Next
we consider all possible combinations of different types which Ij belongs to in nominal
sample path and in the perturbed sample path. There are four cases.
Case 1: For both sample paths, Ij is in a Type I interval. By (72) we have
0 Dx…t† ˆ Dx…tj 1 † Dy for t [ Ij .
Case 2: For the nominal sample path, Ij is in a Type I interval, but for the perturbed
sample path, it is in a Type II interval. Assuming x…y; t† ˆ Tm , we have
dx…y; t†
ˆ0
dt ‡
Am …t† > 0 and
98†
Am ‡ 1 …t† < 0
There are two possibilities regarding Tm :
1.
If Tm ˆ y, then x…y; t† ˆ y over Ij . By the assumption 0 Dx…tj 1 † Dy, we have
y x…y ‡ Dy; tj 1 † y ‡ Dy. Next we prove that x…y ‡ Dy; t† > y ‡ Dy and
x…y ‡ Dy; t† < y for t [ Ijo are both impossible. First, assuming x…y ‡ Dy; t† >
y ‡ Dy; t [ Ijo , we have
dx…y ‡ Dy; t†
ˆ Bm ‡ 1 …t† Am ‡ 1 …t† < 0
dt ‡
but since x…y ‡ Dy; tj
1†
y ‡ Dy, the above implies that
x…y ‡ Dy; t† y ‡ Dy; t [ Ijo
99†
Next, suppose x…y ‡ Dy; t† < y; t [ Ijo . Since Dy > 0, we have x…y ‡ Dy; t† < y ‡ Dy;
t [ Ijo . It follows that
dx…y ‡ Dy; t†
ˆ Cm …t† Am …t† > 0
dt ‡
297
PERTURBATION ANALYSIS
and since x…y ‡ Dy; tj
x…y ‡ Dy; t† y;
1†
y, we conclude that
t [ Ijo
100†
Combining (99) and (100) we have y x…y ‡ Dy; t† y ‡ Dy; t [ Ijo . In view of
x…y; t† ˆ y over Ij and the assumption 0 Dx…tj 1 † Dy we immediately get
0 Dx…t† Dy; t [ Ij .
2.
If Tm 6ˆ y, we have x…y; t† ˆ Tm 6ˆ y. The perturbed sample path cannot cross Tm in Ijo ,
therefore, Ij is Type II means the perturbed sample path is either strictly above or
strictly below Tm throughout Ijo . We consider each of these two cases next.
a.
If x…y ‡ Dy; t† > x…y; t† ˆ Tm for t [ Ijo , we have Dx…t† > 0. In addition, since
x…y ‡ Dy; t† > Tm we must have
dx…y ‡ Dy; t†
ˆ Bm ‡ 1 …t† Am ‡ 1 …t† < 0
dt ‡
Combining (98) and (101) we have Dx…t† < Dx…tj
0 Dx…t† Dy; t [ Ij .
b.
101†
1†
Dy; t [ Ijo . Thus, we get
If x…y ‡ Dy; t† < x…y; t† ˆ Tm for t [ Ijo , we have Dx…t† < 0. Since we assume
Dx…tj 1 † 0, by the continuity of the sample path we must have Dx…tj 1 † ˆ 0.
In addition, since x…y ‡ Dy; t† < Tm , we have
dx…y ‡ Dy; t†
ˆ Cm …t† Am …t† > 0
dt ‡
102†
Combining (98) and (102) we have Dx…t† > Dx…tj 1 † ˆ 0. This contradicts our
previous conclusion that Dx…t† < 0. As a result, this case is impossible to occur.
Case 3: For the nominal sample path, Ij is in a Type II interval, but for the perturbed
sample path, it is in a Type I interval. Assuming x…y ‡ Dy; t† ˆ Tm , we have
dx…y ‡ Dy; t†
ˆ0
dt ‡
Am …t† > 0 and Am ‡ 1 …t† < 0
103†
There are two possibilities regarding Tm :
1.
If Tm ˆ y, then in the perturbed path we have x…y ‡ Dy; t† ˆ y ‡ Dy over Ij . By the
assumption 0 Dx…tj 1 † Dy, we have y x…y; tj 1 † y ‡ Dy. Next, we prove
that x…y; t† > y ‡ Dy and x…y; t† < y for t [ Ijo are both impossible. First, suppose that
x…y; t† > y ‡ Dy; t [ Ijo . Since Dy > 0, we have x…y; t† > y; t [ Ijo . Thus,
298
SUN ET AL.
dx…y; t†
ˆ Bm ‡ 1 …t† Am ‡ 1 …t† < 0
dt ‡
but since x…y; tj
1†
y ‡ Dy, this implies that
t [ Ijo
x…y; t† y ‡ Dy;
104†
Next, suppose x…y; t† < y; t [ Ijo , in which case we must have
dx…y; t†
ˆ Cm …t† Am …t† > 0
dt ‡
but since x…y; tj
x…y; t† y;
1†
y, we conclude that
t [ Ijo
105†
Combining (104) and (105) we have y x…y; t† y ‡ Dy; t [ Ijo . In view of
x…y ‡ Dy; t† ˆ y ‡ Dy over Ij and the assumption 0 Dx…tj 1 † Dy we get
0 Dx…t† Dy; t [ Ij .
2.
If Tm 6ˆ y, then in the perturbed sample path we have x…y ‡ Dy; t† ˆ Tm 6ˆ y ‡ Dy.
The nominal sample path cannot cross Tm in Ijo , therefore, Ij is Type II means the
nominal sample path is either strictly above or strictly below Tm throughout Ijo . We
consider each of these two cases next.
a.
If x…y; t† < x…y ‡ Dy; t† ˆ Tm for t [ Ijo , then Dx…t† > 0. In addition, since
x…y; t† < Tm we must have
dx…y; t†
ˆ Cm …t† Am …t† > 0
dt ‡
Combining (103) and (106) we have Dx…t† < Dx…tj
get 0 Dx…t† Dy; t [ Ij .
b.
106†
1†
Dy; t [ Ijo . Therefore we
If x…y; t† > x…y ‡ Dy; t† ˆ Tm for t [ Ijo , we have Dx…t† < 0. In addition, by the
assumption Dx…tj 1 † 0 and the continuity of the sample path we must have
Dx…tj 1 † ˆ 0. Since x…y; t† > Tm we must have
dx…y; t†
ˆ Bm ‡ 1 …t† Am ‡ 1 …t† < 0
dt ‡
107†
Combining (103) and (107) we have Dx…t† > Dx…tj 1 † ˆ 0. This yields a
contradiction. As a result, this case is impossible to occur.
Case 4: For both sample paths, Ij is in a Type II interval. Assume Tm < x…y; t† <
Tm ‡ 1 ; t [ Ijo . Then,
PERTURBATION ANALYSIS
299
dx…y; t†
ˆ Am ‡ 1 …t†
dt ‡
If Tm < x…y ‡ Dy; t† < Tm ‡ 1 ; t [ Ijo too, then clearly the dynamics of both sample paths are
identical and the result is trivial. Thus, let us consider the cases that the perturbed sample
path is either above Tm ‡ 1 or below Tm. Note that if either Tm or Tm ‡ 1 is the threshold being
perturbed, then its value is y in the nominal and y ‡ Dy in the perturbed path. Thus, we
have two cases to consider.
1.
If x…y ‡ Dy; t† > Tm ‡ 1 for t [ Ijo , we have
dx…y ‡ Dy; t†
ˆ Bm ‡ 2 …t†
dt ‡
Then, since Am ‡ 1 …t† Am ‡ 2 …t† Bm ‡ 2 …t†, we get
dx…y; t† dx…y ‡ Dy; t†
dt ‡
dt ‡
Therefore, Dx…t† Dx…tj 1 † Dy; t [ Ij . Next we prove that Dx…t† 0; t [ Ij . If Tm ‡ 1
is not the threshold being perturbed, then from x…y; t† < Tm ‡ 1 and
x…y ‡ Dy; t† > Tm ‡ 1 we immediately get x…y; t† < x…y ‡ Dy; t† for t [ Ijo , so
Dx…t† > 0. If Tm ‡ 1 is the threshold which is perturbed, then we have x…y; t† < y
and x…y ‡ Dy; t† > y ‡ Dy, so that again we get x…y; t† < x…y ‡ Dy; t† for t [ Ijo in
view of Dy > 0, so that Dx…t† > 0. Therefore, by combining this with Dx…t† Dy;
t [ Ij and Dx…tj 1 † 0 we get 0 Dx…t† Dy; t [ Ij .
2.
If x…y ‡ Dy; t† < Tm for t [ Ijo , we have
dx…y ‡ Dy; t†
ˆ Cm …t†
dt ‡
108†
Then, from Am ‡ 1 …t† Am …t† Cm …t† we get
dx…y; t† dx…y ‡ Dy; t†
dt ‡
dt ‡
109†
Therefore, Dx…t† Dx…tj 1 † 0, t [ Ij . Next, if Tm is not the threshold being
perturbed, then x…y ‡ Dy; t† < Tm < x…y; t† for t [ Ijo , which implies Dx…t† < 0. This,
however, contradicts Dx…t† 0, t [ Ij which we just established, therefore this case is
impossible to occur. Thus, Tm must be the threshold which is perturbed if this case
arises, so we have x…y; t† > y, and x…y ‡ Dy; t† < y ‡ Dy for t [ Ijo , which implies
Dx…t† < Dy for t [ Ijo . Therefore, by combining with Dx…t† 0, t [ Ij and Dx…tj 1 † y
we get 0 Dx…t† Dy, t [ Ij .
300
SUN ET AL.
Since we have shown that 0 Dx…t† Dy, t [ Ij for all feasible cases, the proof is
complete.
j
Proof of Lemma 5.2: The proof is by induction over all Ij ; j ˆ 1; . . . ; J…T†. We have
Dx…t0 † ˆ 0, therefore, using Lemma 5.1, we get 0 Dx…t† Dy for all t [ I1 . Next, we
assume that
0 Dx…t† Dy;
for all t [ Ij
110†
We have already shown that the result holds for I1 , so it remains to show that
0 Dx…t† Dy for all t [ Ij ‡ 1 . Using the continuity of the sample path, from (110) we
have 0 Dx…tj † Dy. In view of Lemma 5.1, we get 0 Dx…t† Dy for all t [ Ij ‡ 1 and
complete the proof.
j
We start with L0m;T …y† and recall that
Proof of Theorem 5.1:
Z
Lm;T …y† ˆ
T
gm …y; t†dt
0
Using our de®nition of time intervals Ij ˆ ‰tj
Lm;T …y† ˆ
J…T† Z
X
tj
jˆ1
tj
J…T† Z
X
tj
1 ; tj †; j
ˆ 1; . . . ; J…T† as before, we have
gm …y; t†dt
1
so that
DLm;T ˆ
jˆ1
tj
Dgm …t†dt
111†
1
where DLm;T ˆ Lm;T …y ‡ Dy† Lm;T …y†, and Dgm …t† ˆ gm …y ‡ Dy; t† gm …y; t†.
For any time interval Ij , the nominal and perturbed sample paths can each be (i) above
Tm , (ii) at Tm , and (iii) below Tm. So there are nine cases to consider regarding the joint
state of the sample paths in Ij :
Case 1: Both nominal and perturbed sample paths are below Tm. There is no class m loss,
so we have
Z
tj
tj
Dgm …t†dt ˆ 0
112†
1
Case 2: Both nominal and perturbed sample paths are above Tm . Class m experiences full
loss in both sample paths, so we also have
301
PERTURBATION ANALYSIS
Z
tj
tj
Dgm …t†dt ˆ 0
113†
1
Case 3: Both nominal and perturbed sample paths are at Tm . We have gm …y; t† ˆ
gm …y ‡ Dy; t† ˆ Am …t†, so that
Z
tj
tj
Dgm …t†dt ˆ 0
114†
1
Case 4: The nominal sample path is below Tm while the perturbed sample path is above
Tm . Class m experiences no loss in the nominal sample path and full loss in the perturbed
sample path, so we have gm …y† ˆ 0 and gm …y ‡ Dy; t† ˆ am …t†. Therefore
Z
tj
Z
tj
Dgm …t†dt ˆ
1
tj
tj
am …t†dt 0
115†
1
In this case, we also have
dx…y; t†
ˆ Cm …t†
dt ‡
dx…y ‡ Dy; t†
ˆ Bm ‡ 1 …t†
dt ‡
Therefore,
dx…y; t†
dt ‡
dx…y ‡ Dy; t†
ˆ Cm …t†
dt ‡
Bm ‡ 1 …t† am …t†
so that
Z
Dx…tj
1†
Dx…tj † tj
tj
am …t†dt
1
and it follows that
Z
Dy Dx…tj
1†
Dx…tj † ‡
Z
tj
tj
am …t†dt ˆ Dx…tj † ‡
1
Z
tj
tj
Dgm …t†dt 1
tj
tj
Dgm …t†dt
1
116†
where the last inequality is due to Dx…tj † 0 from Lemma 5.2. Combining (115) and (116)
we get
302
SUN ET AL.
Z
0
tj
tj
Dgm …t†dt Dy
117†
1
Case 5: The nominal sample path is above Tm while the perturbed sample path is below
Tm . In view of 0 Dx…t† Dy for t [ ‰0; TŠ in Lemma 5.2, the only possibility for this case
is that Tm is the threshold being perturbed, that is, x…y; t† > y and x…y ‡ Dy; t† < y ‡ Dy.
In this case, class m experiences full loss in the nominal sample path and no loss in the
perturbed sample path, so we have gm …y† ˆ am …t† and gm …y ‡ Dy; t† ˆ 0. Therefore
Z
tj
Z
tj
Dgm …t†dt ˆ
tj
tj
1
am …t†dt 0
118†
1
In addition, we have
dx…y; t†
ˆ Bm ‡ 1 …t†
dt ‡
dx…y ‡ Dy; t†
ˆ Cm …t†
dt ‡
Therefore,
dx…y ‡ Dy; t†
dt ‡
dx…y; t†
ˆ Cm …t†
dt ‡
Bm ‡ 1 …t† am …t†
so that
Z
Dx…tj †
Dx…tj
1†
tj
tj
am …t†dt
1
therefore we get
Z
Dy Dx…tj † Dx…tj
1† ‡
Z
tj
tj
am …t†dt ˆ Dx…tj
1
1†
Z
tj
tj
Dgm …t†dt 1
tj
tj
Dgm …t†dt
1
119†
where the last inequality is due to Dx…tj
(119) we get
Z
Dy tj
tj
Dgm …t†dt 0
1†
0 from Lemma 5.2. Combining (118) and
120†
1
Case 6: The nominal sample path is at Tm while the perturbed sample path is above Tm .
Class m experiences partial loss in the nominal sample path and full loss in the perturbed
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PERTURBATION ANALYSIS
sample path, so we have gm …y† ˆ Am …t†; gm …y ‡ Dy; t† ˆ am …t† and Am ‡ 1 …t† < 0.
Therefore
Z
Z
tj
tj
Dgm …t†dt ˆ
1
Z
tj
tj
am …t†dt
tj
1
Z
tj
Am …t†dt ˆ
tj
1
tj
Am ‡ 1 …t†dt > 0
121†
1
In this case, we also have
dx…y ‡ Dy; t†
ˆ Bm ‡ 1 …t†
dt‡
dx…y; t†
ˆ0
dt‡
so we get
Z
Dx…tj †
Dx…tj
1†
ˆ
tj
tj
Bm ‡ 1 …t†dt
1
In view of
Z
tj
Z
tj
Bm ‡ 1 …t†dt 1
tj
tj
Am ‡ 1 …t†dt
1
we get
Dy Dx…tj 1 † Dx…tj 1 † Dx…tj †
Z tj
Z tj
Z
ˆ
Bm ‡ 1 …t†dt Am ‡ 1 …t†dt ˆ
tj
1
tj
1
tj
tj
Dgm …t†dt
122†
1
where the second inequality is due to Dx…tj † 0 from Lemma (5.2). Combining (121) and
(122) we get
Z
0<
tj
tj
Dgm …t†dt Dy
123†
1
Case 7: The nominal sample path is at Tm while the perturbed sample path is below Tm. In
view of 0 Dx…t† Dy for t [ ‰0; TŠ in Lemma 5.2, the only possibility for this case is that
Tm is the threshold being perturbed, that is, x…y; t† ˆ y while y x…y ‡ Dy; t† < y ‡ Dy. In
this case, class m experiences partial loss in the nominal sample path and no loss in the
perturbed sample path, so we have gm …y† ˆ Am …t†; gm …y ‡ Dy; t† ˆ 0 and Am …t† > 0.
Therefore
304
SUN ET AL.
Z
Z
tj
tj
Dgm …t†dt ˆ
tj
tj
1
Am …t†dt < 0
124†
1
In this case we also have
dx…y ‡ Dy; t†
ˆ Am …t†
dt‡
dx…y; t†
ˆ0
dt‡
so we get
Z
Dx…tj †
Dx…tj
1†
ˆ
tj
tj
Am …t†dt
1
and it follows that
Z
Dy Dx…tj † Dx…tj †
Dx…tj
1†
ˆ
where the second inequality is due to Dx…tj
(125) we get
Z
Dy tj
tj
Z
tj
Am …t†dt ˆ
Dgm …t†dt
125†
tj
1
1†
0 from Lemma 5.2. Combining (124) and
Dgm …t†dt < 0
tj
tj
1
126†
1
Case 8: The nominal sample path is above Tm while the perturbed sample path is at Tm .
In view of 0 Dx…t† Dy for t [ ‰0; TŠ in Lemma 5.2, the only possibility for this case is
that Tm is the threshold being perturbed, that is, x…y ‡ Dy; t† ˆ y ‡ Dy while y < x…y; t† y ‡ Dy. In this case, class m experiences full loss in the nominal sample path and partial
loss in the perturbed sample path, so we have gm …y† ˆ am …t†; gm …y ‡ Dy; t† ˆ Am …t† and
Am ‡ 1 …t† < 0. Therefore
Z
tj
Z
tj
Dgm …t†dt ˆ
1
tj
Z
ˆ
Z
tj
Am …t†dt
1
tj
tj
In this case, we also have
tj
tj
am …t†dt
1
Am ‡ 1 …t†dt < 0
1
127†
305
PERTURBATION ANALYSIS
dx…y ‡ Dy; t†
ˆ0
dt‡
dx…y; t†
ˆ Am ‡ 1 …t†
dt‡
so we get
Z
Dx…tj †
Dx…tj
1†
ˆ
tj
tj
Am ‡ 1 …t†dt
1
and it follows that
Z
Dy Dx…tj † Dx…tj †
Dx…tj
1†
ˆ
where the second inequality is due to Dx…tj
(128) we get
Z
Dy tj
tj
Dgm …t†dt < 0
tj
1†
Z
tj
Am ‡ 1 …t†dt ˆ
1
tj
tj
Dgm …t†dt
128†
1
0 from Lemma 5.2. Combining (127) and
129†
1
Case 9: The nominal sample path is below Tm while the perturbed sample path is at Tm .
Class m experiences no loss in the nominal sample path and partial loss in the perturbed
sample path, so we have gm …y† ˆ 0; gm …y ‡ Dy; t† ˆ Am …t† and Am …t† > 0. Therefore
Z
tj
Z
tj
Dgm …t†dt ˆ
1
tj
tj
Am …t†dt > 0
1
In this case, we also have
dx…y ‡ Dy; t†
ˆ0
dt‡
dx…y; t†
ˆ Cm …t†
dt‡
so we get
Z
Dx…tj
therefore,
1†
Dx…tj † ˆ
tj
tj
Cm …t†dt
1
130†
306
SUN ET AL.
Z
Dy Dx…tj
1 † Dx…tj
1†
Dx…tj † ˆ
Z
tj
tj
Cm …t†dt 1
tj
Z
tj
Am …t†dt ˆ
1
tj
tj
Dgm …t†dt
1
131†
where the second inequality is due to Dx…tj † 0 from Lemma 5.2. Combining (130) and
(131) we get
Z
0<
tj
tj
Dgm …t†dt Dy
132†
1
We may now combine the nine cases above, i.e., the inequalities (112)±(114), (117), (120),
(123), (126), (129), (132) with (111) and obtain
J…T† Z tj
X
DLm;T Dgm …t†dt J…T†jDyj
t
j 1
jˆ1
Next, we consider Q0T …y† for which we can write
Z
QT …y† ˆ
T
0
x…y; t†dt
therefore by Lemma 5.2 we have
Z
jDQT …y†j ˆ 0
T
Dx…y; t†dt T jDyj
This completes the proof.
j
Acknowledgments
Gang Sun and Christos Cassandras were supported in part by the National Science
Foundation under Grants EEC-00-88073 and DMI-0330171, by AFOSR under contract
F49620-01-0056, and by ARO under under grant DAAD19-01-0610.
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