Large deviations the shap e of the loss curve and economies of scale
in large multiplexers
D D Botvich
School of Electronic Engineering
Dublin City University Dublin Ireland
N G Du eld
School of Mathematical Sciences
Dublin City University Dublin Ireland
and
Dublin Institute for Advanced Studies
Burlington Road Dublin Ireland
E mail duffieldn dcu ie
Revision March
Abstract
L
We analyse the queue Q at a multiplexer with L inputs We obtain a large deviation result
namely that under very general conditions
1 L
lim L log P Q Lb I b
L!1
provided the o ered load is held constant where the shap e function I is expressed in terms of the
cumulant generating functions of the input tra c This provides an improvement on the usual
L b L LI (b L)
e ective bandwidth approximation P Q b e replacing it with P Q b e
The di erence I b b determines the economies of scale which are to b e obtained in large
multiplexers If the limit lim t exists here is the nite time cumulant of
t!1 t t
the workload pro cess then lim I b b We apply this idea to a number of exam
b!1
ples of arrivals pro cesses heterogeneous sup erp ositions Gaussian pro cesses Markovian additive
pro cesses and Poisson pro cesses We obtain expressions for in these cases is zero for in
dep endent arrivals but p ositive for arrivals with p ositive correlations Thus economies of scale
are obtainable for highly bursty tra c exp ected in ATM multiplexing
Keywords Large deviations scaling limits ATM multiplexers heterogeneous sup erp ositions
Introduction
The problem of determining loss probabilities in queueing systems is crucial in the development
of emergent technology of telecommunications networks using the Asynchronous Transfer Mo de
ATM Much recent work has fo cused on the analysis of the single server queue with general
arrivals This enables one to analyse queues with correlated arrivals such as those which o ccur in
the bu er of an ATM multiplexer whose input is a sup erp osition of highly bursty sources
Consider a general single server queue For t T here T R or Z denote by A the amount
t
of work which arrives to b e pro cessed in the interval t and by S the amount which can b e
t
pro cessed in the same interval If more work arrived than can b e pro cessed the surplus waits in
the queue The workload pro cess W is de ned by W and
W A S
t t t
and the queue of unpro cessed work at time zero is
Q sup W
t
t
The relation b etween the tail of the queue length distribution and the large deviation prop erties
of the workload pro cesses has b een established in progressive degrees of generality Following a
heuristic prop osal by Kesidis et al see also for further bibliographical details Glynn
and Whitt showed for T Z that if the pair W t t satisfy a large deviation principle then
t
lim b log P Q b
b!1
where
supf j g
and is the cumulant generating function of the workload pro cess de ned by
W
t
lim t log E e
t!1
Alternatively can b e expressed through
inf t t
t
where the Legendre Fenchel transform of is de ned through
x sup x
We refer the reader to the b o ok Dembo and Zeitouni as a comprehensive reference for large
deviations that of Bucklew for a more heuristic approach and the article of Lewis and P ster
for a general introduction
Recently Du eld and O Connell have extended this result in two directions Firstly the same
result is shown to hold when T R sub ject to a lo cal growth condition on W Secondly an
analogous result holds with large deviation scalings more general than the linear scalings b and t
in and These are appropriate for treating for example the case where the workload is
fractional Brownian motion this has b een prop osed as a mo del for the workload by Leland et al
based on observations of Ethernet tra c
The relation is the basis of the e ective bandwidth approximation to the queue length distri
bution
b
P Q b e
See for example for development applications and further references
The motivation here is that for ATM multiplexers one wants to estimate exp onentially small loss
probabilities which in practice are to b e as small as However there is already theoretical
and numerical work indicating that is insu cient for this purp ose For a queue where the
input is an L fold sup erp osition of Markovian sources served at constant rate s Du eld proves
the upp er b ound
L b
P Q b e e
is as b efore and and dep end only the the tra c due to a single source and on the o ered load
through the ratio s L In the example of on o Markov sources with p ositive auto correlation is
strictly p ositive see Bu et and Du eld Thus in a large sup erp osition the loss probabilities
may b e exp onentially small even for small b the e ective bandwidth approximation can b e
extremely conservative through over estimating the loss probabilities On the other hand if were
negative then suggests that will under estimate the loss probabilities at large L even
for large b Moreover b oth types of error b ecome more severe as L increases at constant load
Both these types of b ehaviour have b een observed though numerical studies of queueing mo dels by
Choudhury et al in which the asymptotic approximation
L b
P Q b e e
is prop osed The e ective bandwidth approximation when compared with these results is shown
to over estimate the loss probabilities in examples of bursty sources and under estimate them in
examples of sub bursty sources Finally we note the work of Weiss for L fold sup erp ositions
of On O Markov uids There a sample path large deviation argument was used to identify the
most likely path to a rare event such as over ow from a large bu er and its probability with the
LJ b
asymptotics P Q Lb e where for large b J b b
Thus we are led to investigate the large deviation prop erties of the queue length distribution in
the size L Apart from the ab ove considerations we are motivated by the observation in examples
that the broad features of the queue length distribution remain roughly invariant when b oth the
size L and the queue length b are jointly scaled See simulation results in the thesis of Corcoran
heuristic arguments by Rasmussen et al and by Courcoub etis et al For example let
L
Q b e the queue due to a sup erp osition of L identical sources not necessarily Markovian served at
constant rate sL s xed and denote by the nite time cumulant due to a single source with
t
arrival pro cess A served at rate s
A st
t
t log E e
t
L
The workload W of the sup erp osition is the sum of L indep endent copies of W A st and
t t
t
L L
Q sup W As a consequence of Theorem of the next section it follows under convergence
t
t t
of as t and suitable lo cal regularity conditions on the workload that for b
t
L
lim L log P Q Lb I b
L!1
where I is the shap e function de ned by
I b inf t b t
t
t
L L
The heuristics b ehind are as follows By P Q Lb P fW Lbg The
t
t
probability of each event in the union is exp onentially small for large L and so the probability
L
is dominated by that of the most likely event sup P W Lb If for each xed t the fam
t t
ily of sup erp osed workloads satis es a large deviation principle with some rate function I i e
t
L LI b L inf I b
t t t
P W Lb e then altogether we get P Q Lb e The required large de
t t
viation principle for indep endent sup erp ositions follows from Cram er sTheorem and I t
t t
However indep endence is not a requirement for our pro of We note that this large deviation
principle was obtained for alternating renewal pro cesses by Simonian and Guib ert
The shap e function I can b e seen to give the large scale corrections to the e ective bandwidth
approximation is replaced by
L LI b L
P Q b e
Observe that if one replaces by in where lim then by I b reverts to
t!1 t
t
b I b b determines the error incurred by using the e ective bandwidth at large L or to make
a more p ositive statement it determines the economies of scale to b e obtained in multiplexers of a
large number of sources
We examine the initial value I and asymptotics as b of the shap e function I in Theorems
and For T Z a workload with stationary increments I This is just the
large deviation result for the loss probability in a bu erless resource as found by Hui The
asymptotics of I are
lim I b b
b!1
where
lim t
t
t!1
provided this limit exists and sub ject to some regularity conditions in the case T Z For large
b at least of order L this means we can make the approximation I b b This establishes
that the asymptotics found by Weiss for Markov uid sources hold far more generally and provides
a theoretical justi cation for
One sees from that for uncorrelated arrivals since then Thus there
t
are no economies of scale to b e gained at large rescaled bu er sizes for uncorrelated arrivals
since then b is asymptotic to I b for large b On the other hand a su cient condition for to
b e p ositive is that the workloads on disjoint time intervals are p ositively correlated Theorem
This is typically the case for highly bursty sources A generic con guration with I is
illustrated in Figure
Figure Economies of Scale with I
E ective bandwith approx b
Shap e function approx LI b L
L
log P Q b
b
It is interesting to note that dep ends on ner details of the workload pro cess than those which
determine the asymptotic slop e it dep ends not only on the limiting cumulant but rather on
the manner in which the approach as t To b orrow from the terminology of physics
t
is not a thermo dynamic quantity
The pap er is organised in the following way The basic large deviation result is stated and proved
in section The analysis of the shap e function I is done in section In section we apply them
to a number of examples The case of heterogeneous sup erp ositions is worked out in Gaussian
workload pro cess are covered in and the sp eci c example of Ornstein Uhlenbeck pro cesses in
including a calculation of the shap e function for a heterogeneous sup erp osition Markov Additive
Pro cesses are treated in In this case we can express in terms of the the maximal eigenfunction
of the Laplace transform of the Markov transition kernel Corollary Comparisons of the
approximation with simulation are made for sup erp ositions of Markovian on o sources
Finally in we apply the results to a very simple class of examples indep endent Poissonian
P
L
arrivals with general service distribution In light of the explicit distribution for M M L
it is not surprising that in this case I there are no economies of scale to b e obtained
for Poissonian arrivals at any bu er size
Large deviations
We b egin by stating our hypotheses concerning the workload pro cesses then give some examples
L
which satisfy the hypotheses For each L N W where T Z or R is a sto chastic
t2T
t
L
pro cess and W The queue length at time zero is
L L
sup W Q
t
t2T
L L
Note that if the increments of W are stationary then the distribution of Q is also stationary
For R de ne the cumulant generating function
L
L W
t
Lt log E e
t
Hyp othesis
i For each R the limits
L
lim and lim
t t
t
t!1
L!1
exist as extended real numbers Moreover the rst limit exists uniformly for al l t su ciently
large
ii and are essential ly smooth i e is di erentiable on the interior of its e ective
t
0
domain the region where it is nite and lim j j for any sequence in
n!1 n n
the e ective domain which converges to a point on its boundary
iii There exists for which for al l t T
t
L L L
iv T R For al l t r de ne W sup W W Then for al l R
0
0
r r
t r t
t r
L
W
t r
lim sup lim sup L sup log E e
r !
L!1 t
Remark if Hyp otheses i ii are satis ed then by the G artner Ellis theorem for each t the
L
pair W L L satis es a large deviation principle with go o d rate function given by the Legendre
t
Fenchel transform of t In other words for any Borel set
t
L
lim sup L log P W L inf t x
t
t
L!1 x2
and
L
x t lim inf L log P W L inf
t
t
L!1 x2
where the Legendre Fenchel transform of a function f is
f x sup f x f g
2R
from which it follows that t x t x t Let x b e the solution of x by Hyp othesis
t
t t t t
iii it is negative Then for x x
t
L
lim sup L log P W L x t x t
t t
L!1
and
L
lim inf L log P W L x t x t
t t
L!1
where indicates limit from ab ove Hyp othesis iii is a stability condition then there exists
a strictly p ositive solution of the equation which is the asymptotic decay rate of the
queue length distribution Hyp othesis iv is a lo cal regularity condition on the sample paths of
the workload
Examples
Homogeneous sup erp ositions There are L identical sources whose backward arrival pro cesses
are indep endent copies of A The sup erp osition is served at a constant rate sL the
t t2T
o ered load is indep endent of L Then
L A
t
s t log E e
t
t
indep endent of L Thus is the cumulant corresp onding to the workload W A st of a
t t t
single source served at rate s
Heterogeneous sup erp ositions Sources are classi ed by type j in some nite index set J There
P
are L sources of type j with L L sources in total The backward arrival pro cess for
j j
j
a source of type j is A All sources are indep endent Then
j t t2T
X
L L
p c s
j t
t j
j 2J
where
A L
j t
c t log E e and p L L
j t j
j
The limits c lim c are assumed to exist with c essentially smo oth Then
j t!1 j t j
L L
exist and for any R L is convergent provided the limits p lim p
j L!1
t
j
the convergence is uniform in t since J is nite Heterogeneous sup erp ositions have b een
previously analysed through the e ective bandwidth approximation see references ab ove
and through eigenfunction expansions for classes of Markovian uid mo dels by Kosten
building on the early work of Anick et al on homogeneous sup erp ositions and nite state
mo dels by Elwalid et al
For T R and A having stationary increments Hyp othesis iv is satis ed if for each
j t
j J
lim sup E exp sup jA j
j t
r ! t r
Time rescalings The single source arrival pro cess is A where the pro cess A is convergent
L t L Lt
in distribution to some pro cess A as L The sup erp osition is served at a xed rate s
t
P
L
denoting an L fold sup erp osition Thus with
P
L
L
sup Q A st
L t
t
t
P
L
sup A Lst
L Lt
t
P
L
L
We shall take the limit L and so assume Lt log E exp A st
L Lt
t
satis es Hyp othesis This class of mo dels is motivated by examples of rescaled Markovian
sources simulated by Corcoran and of rescaled renewal pro cesses examined by Sriram and
Whitt and Rasmussen et al
Theorem Under Hypothesis for each b
Upp er b ound
L
lim sup L log P sup W Lb I b
t
t
L!1
where I b inf t b t
t
t
Lower b ound
L
lim inf L log P sup W Lb I b
t
L!1
t
Pro of of Theorem Lower Bound
L L
lim inf L log P sup W Lb lim inf L sup log P W Lb
t t
L!1 L!1
t t
L
sup lim inf L log P W Lb
t
L!1
t
sup t b t I b
t
t
by where the last equality follows since by Hyp othesis iii is increasing on
t
x
t
0
0
t t Upp er Bound T Z For any t and
t
X
L L L
Lb Lb P W Lb t max P W P sup W
0 0 0
t t t
0
t t 0
t
0
t t
X
00 L 0 L
Lt t L Lb
Lb
0 0
00 0
t t
t t
e e t max e
0
t t
00
t t
L
by Chebychev s inequality Since uniformly in t and on
t t
t
L
we can nd and such that for all L t su ciently large This means
t
that for such L and t the geometric series in is summable yielding
0 L
t L Lb
Lb Lt L L
0 0
0
t t
t
e e Lb t max e P sup W
0
t
0
t t 0
t
Taking logarithms dividing by L taking the lim sup as L and nally taking the in mum over
0
we obtain the
t
0 0 L
b t b t t Lb max max lim sup L log P sup W
0 0
t t
0
t t 0
L!1 t
Recall so that taking the limit t we obtain in conjunction with the lower b ound the
stated result
Upp er Bound T R For any and n N de ne
L
L L L W
n
W sup W and nL log E e
n t n
n tn
By H older sinequality then for any p in
L
L L W p
n
n n p p p L log E e
n n
L
with W as in Hyp othesis iv According to Hyp othesis for any p we can make the
second term of the right hand side of as small as we like by choosing su ciently small then
L su ciently large Thus we can rep eat the steps and with and p xed take
the limits t then to obtain
L L
W Lb lim sup L log P sup W Lb lim sup L log P sup
n t
t n
L!1 L!1
p sup t b t
t
t
since p p p and nally let p to get the stated result
t
t
Analysis of the shap e function
In this section we analyse the initial value I of the shap e function and the asymptotics of I b
as b
For t R de ne t t
t t
Hyp othesis
L
i W has stationary increments
t
ii T R The limit lim exists as an extended real number for al l R
t! t
iii lies in the interior of the e ective domain of
Theorem Under Hypotheses and
T Z I
T R I
Remark Theorem says that a stable workload with stationary increments is asymptotically
most likely to exceed at the smallest times But this need not b e the case for non stationary
workloads For the pro of of Theorem we need the following Lemma
Lemma Under Hypothesis i for n N r T
r nr
Pro of This follows from H older s inequality and the stationarity of the increments of W For
P
n
with and
i in i i
i
n
Y
W t W t
t t
i i
i
E e E e
i
and hence
n
X
i t t
i
i
from which follows by taking n and t nr
i
Pro of of Theorem
t sup t sup t t
t t
t t
so that I inf
t
t
For T Z observe from Lemma that and hence Hence I
t
t
For T R since by Hyp othesis ii converges to then by Lemma of converges
t
t
to on the interior of the e ective domain of By Lemma n n is increasing for any
r
r and so n is decreasing Hence b inf b for any b in the interior of the
n
t
t
r
e ective domain of for any we can choose r such that
b inf b inf b b
n
t t r
r
t t
and lim b b The result now follows by Hyp othesis iii
n
n!1
r
The identi cation of the asymptotics of I requires some technical conditions as follows
Hyp othesis
i T Z or T R The fol lowing limit exists
lim t
t
t!1
0 0
ii T Z and are strictly convex and t t t is bounded above or
t
t
t
0 0 0
ii T Z and are strictly convex and are uniformly Lipschitz continuous on
t
t
0
some neighbourhood of and
p
0 0
t t t o
t t
0
Remark Hyp othesis i can b e understo o d as follows Let Then
t t t t
t
So the existence of a nite limit means that t for large t
t
0
Theorem Under Hypothesis i and with the addition of Hypotheses ii or ii for T Z
then
lim I b b
b!1
According to Hyp othesis ii and are di erentiable so the convergence of to implies the
t t
0 0
p ointwise convergence of to See for example Lemma IV of
t
Pro of of Theorem De ne
0
t t
t
0 0
Since as t t t is increasing for t su ciently large and lim t
t!1
t
Set
b supft T j t bg
Ran b b is increasing and lim b By de nition of the Legendre Fenchel
b!1
transform of and
t
t b t b t t t t t
t
t t
We obtain upp er b ounds for lim sup I b b rst for T R then for T Z We then
b!1
show these are equal to a lower b ound for lim inf I b b
b!1
Upp er Bound T R Ran R and so for any b R
I b b inf t b t b
t
t
b b b b
b
b
b
the last equality following from b ecause b b for T R Since b as b
then by Hyp othesis
lim sup I b b
b!1
T Z In this case Ran is a discrete set but the conclusion holds provided we take
the limit along Ran since acts as the identity there But for any b R we have
b b b E b b b b b
b
b b
where
E b b b b b b b
b
b b
b b
b
where
0
b b
b
b
Here we have used the fact that since is strictly convex is di erentiable see Theorem
t
t
in
The pro of is complete if we can show that lim sup E since then lim sup I b b
b
b!1 b!1
lim sup b Note
b
b!1
b b b
for su ciently large b and the relations
0 0 0
b b b b b b
b
b b b
0
from which it follows that since is increasing This means E is non negative Combining
b b
t
these gives
b b b b
0 0
b
b b
b b
We now pro ceed under Hyp othesis ii Since from
0 0
lim b lim
b
b!1 b!1
then by
0 0
lim
b
b b
b!1
Hence
lim
b
b!1
0 0
for if not then for some and subsequence b so lim sup
b i b
i!1
i i
b b
i i
0 0 0 0
lim sup since is strictly convex in contradiction
i!1
b b
i i
with Finally
0 0
b b b b b b
b b
which is b ounded according to Hyp othesis ii Combining with then lim E as
b!1 b
required
0
Alternatively under Hyp othesis ii k b b for some k indep endent of b
b
and so
E k b b b
b
b b b
0 0
b b b
b b
which go es to as b by
Lower Bound T R or Z Supp ose rst that inf t b t is attained at b
t
t
inf t b t b b b b b
t
b
t
b
b
by and so
lim inf I b b
b!1
provided b as t But if this is not the case b is b ounded Thus we obtain a
contradiction with the upp er b ound if we can show that
lim t b t b
t
b!1
for any xed t But this is true since b t b t b is strictly convex and by achieves it
t
in mum at t
If inf t b t is not attained then we can rep eat the ab ove arguments replacing b with b
t t
for which the in mum is approximated to within uniformly in b then take at the end
We shall say more concerning the existence of in the context of Markov Additive Pro cesses in
section However we can make a general statement concerning the sign of
L
Theorem Let W have stationary increments and suppose for each L and for each t
t
t t t that W W and W W are non negatively correlated Then if exists it is
t t t t
4 3 2 1
non negative
w
0
W are non negatively correlated and w e is non decreasing Pro of Since W and W
t t
t t
for
W W W W W W W W W
0 0 0 0
t t t t t
t t+t t+t t+t
E e E e E e e E e E e E e
the last equality b eing due to the stationarity of the increments of W Thus t t is sup erad
t
ditive Applying the sub additivity theorem see Lemma of to t we obtain
t
lim sup
t t
t!1
t
Thus is non p ositive for all t and so lim t if it exists is also non p ositive
t t!1 t
Applications and Examples
Heterogeneous Sup erp ositions
We examine I and for the class of heterogeneous sup erp ositions describ ed in the section
P
Recall there are L L sources in total in the sup erp osition L of type j each having
j j
j
A
j t
backward arrival pro cess A We set c t log E e and assume the existence of the limits
j t j t
P
c lim c and p lim L L The service rate is sL so p c s
j t!1 j t j L!1 j t j j t
j
First we examine I Consider the case T R We assume the existence of the limits
C lim tc t Then
i t! i t
X
p C s
j j
j
Thus
X
p C s I
j
j
j
j
P
0 0
where s C b eing the unique solution of the equation p C s For T Z the
j
j j j
j
same formulae hold but with the C replaced by c
j j
Second we examine is the unique solution of the equation i e
X
p c s
j j
j
and so
X
p
j i
j
where
1
lim t c s
i j t
j
t!1
1 1 1
and s c Note s s due to the convexity of c since c s and s are
j j j
j j j j j
the usual e ective bandwidths of sources of type j for zero and in nite bu ers resp ectively See
for example for further details
Sup erp ositions of Gaussian Arrival Pro cesses
L L L
In this section we take W A sLt where for each L A is an L fold sup erp osition of inde
t t t
p endent copies of A a zero mean Gaussian pro cess with stationary increments and variance
t
t
We make the following Hyp otheses concerning A
Hyp othesis
i For some function k increasing to as t
t
lim sup k m where m E sup jA j and lim sup k
t t t r
t t
t! r t t!
The fol lowing limits exist as extended real numbers
ii lim t
t!1
t
iii lim t
t!
t
iv lim t
t!1
1 t
Prop osition Under Hypothesis Theorems and hold with
s s
1
s I and
Pro of Hyp othesis ii means that exists and
t s and s
t
t
are clearly convex and di erentiable
To check that Hyp othesis iv holds we show that i implies By Borell s inequality see
Theorem
P sup jA j x x m
r t t
r t
for any x m where sup and is the canonical Gaussian distribution function
t t t
r t
From this is follows that for any
2 2 2
k m k
t t
t t
E exp k sup jA j e e k
t r t t
r t
With i we get that k m k and hence also k remain b ounded as t so that
t t t t t t
E exp k sup jA j is also b ounded as t Hence when k
t r t
r t
k
t
E exp sup jA j E exp k sup jA j
r t r
r t r t
as t
From the decay rate is s Under iii the limit
t
s s lim t t lim
t
t! t!
t
exists and b b s Thus Hyp othesis is satis ed and I s
Finally under iv the limit
lim t lim t s s
t
t 1
t!1 t!1
exists so Hyp othesis i is satis ed
The pro of go es through for heterogeneous sup erp ositions as describ ed in sections and where
L
A is a sup erp osition of sums of L copies of indep endent Gaussian pro cesses A with mean zero
j j t
t
P
L exist In and variance satisfying Hyp othesis provided the limits p lim L
j j L!1 j
j
j t
P
this case p
j
t j j t
We note that when A is Brownian motion take k t and so I
t 1
Compare with the discussion in section
Ornstein Uhlenbeck Arrival Pro cesses
An example where the workload is mo delled by a Gaussian pro cess with stationary increments is the
following Consider a queue with constant service rate for which the workload W is the p osition
t
comp onent of a stationary Ornstein Uhlenbeck pro cess with added negative drift Such an arrival
pro cess has b een prop osed by Norros et al as a mo del of continuous correlated arrivals It
arises as the heavy tra c limit of sup erp osed state Markov uid sources under suitable rescaling
of time and mean activity see
We consider the stationary Ornstein Uhlenbeck velocity pro cess V t R de ned to b e the
t
solution of the sto chastic di erential equation
p
dV V dt s dB t
t t
where V is normally distributed with zero mean and variance s Here B is standard Brownian
motion is a load parameter the case corresp onding to unit load and s can b e
viewed as a service rate The corresp onding p osition pro cess with zero initial condition is
Z
t
V ds A
s t
and the workload is
W A st
t t
W is Gaussian with mean st and variance
t
t
s t e
t
Hence
s
t
s and s
t
t
This gives s s s and s Thus items ii iii and iv of
1
Hyp othesis is satis ed and
I and
In item i we can take k t since as t st by and m const t as we
t t
t
shall now show We use the integral representation of
Z
t
p
s t t
e dB s s V e V e
t
Then for t
E sup jZ j E sup Z by symmetry
r r
r t r t
tE sup V
r
r t
Z
r
p
0
dB r s tE sup
r t
p
s tE sup B r
r t
const t
In fact we can easily calculate I b numerically Normalizing b by s a routine calculation yields
s b t t b
t sb t
t
t
t e
t
We can also p erform the same calculations for heterogeneous sup erp ositions Arrivals of type j are
Ornstein Uhlenbeck p osition pro cesses with mean and variance
r t
j
s r t e
j j
j t
and o ccur with limiting prop ortion p in the sup erp osition Here we have included p ossible time
j
P
p and the analysis rescalings r on each pro cess Thus the sup erp osition has variance
j j
j t
j t
of the previous section gives
X
s p r
j j
j
j
X
p r s
j j
j
j
X
A
I p r
j
j
j
j
P
p r
j
j
j j
P
p r
j j
j
j
b t
t sb t
P
t
r t
j
p r t e
j j
j
j
In Figure the curve of b I sb is plotted for two types with r r
and p p The curve lies b etween those obtained for homogeneous arrivals of each type
separately these are also plotted
Figure Ornstein Uhlenbeck sup erp ositions
Homogeneous p p
Heterogeneous p p
I sb
Homogeneous p p
b
Markov Additive Arrival Pro cesses
In this section we obtain an expression for in the case that the increments of the workload W
o ccur at integer times and are distributed according to the state of an underlying Markov pro cess X
describing the con guration of the source of the arrivals Sp eci cally one could consider W to b e
the single source workload in a homogeneous sup erp osition describ ed in section the corresp onding
results for heterogeneous sup erp ositions follow from section A convenient description for this
is that of a Markov Additive Pro cess
To b e precise let X X b e a irreducible ap erio dic Markov pro cess on a state space E with
t t2Z
+
eld E and adjoin to it an additive comp onent W W with W such that X W is
t t2Z
+
a Markov pro cess on the state space E R Furthermore for each t N the joint distribution of
0 0 0
dep ends only on X W the increment Z W W and X conditioned on X
t t t t t t t t t
This dep endence can b e expressed through the kernel
P x G B P X G Z B j X x
t t t
for G E and B a Borel set of R
For R de ne the transformed kernel P by
Z
z
P x G P x G dz e
t
and denote by P its t fold convolution A technical recurrence condition on the kernel P see
is required for what follows
Hyp othesis i There exists a probability measure on E R an integer m and real numbers
a b such that
m
0
a G B P x G B b G B
for al l x E G E and Borel sets B of R
ii The convex hul l of the support of E has non empty interior
iii The set f R j E g is open
Note that Hyp othesis is automatically satis ed in the case that E is nite and P x E dz has
compact supp ort for all x E
The main technical result we require concerning the kernel P is an extension of the standard Perron
Frobenious to non discrete state spaces Lemma and Lemma of and Theorem I I I
of Let q denote the stationary distribution of X
Prop osition is strictly convex and essential ly smooth For al l in the e ective domain of
e is the simple maximal eigenvalue of P The corresponding right eigenfunction r and
Radon Nikodym derivative d dq of the left eigenmeasure are uniformly bounded and
R
positive With the normalization dx r x
t t t
x G r x G e P O
where
Corollary
Z
lim t log E q dx r x
t
t!1
R
t
Pro of This follows since t log q dx P x E and
t
To calculate I we note that
Z
z
log q dx P x dy dz e
In the sp ecial case frequent in mo delling that the increment Z is a non random function of X
t t
when b oth are conditioned on X we have
t
P x dy dz R x dy dz
y
where R is the transition kernel for X and so reduces to
Z
y
log q dy e
Finally note that the complexity of the calculation of and I and indeed the whole curve of
I b is indep endent of the number of sources L in the sup erp osition
Application two state Markov chain We consider a L fold homogeneous sup erp osition of
arrivals streams each of which is generated by a stationary discrete time Markov chain on two
states on and o In the on state an arrival of unit length is generated in the o state no
arrival is generated Transitions from o to on o ccur with probability a the reverse transition with
probability d The sup erp osition is serviced are serviced at constant rate sL with s
Within the general framework ab ove we have E f g Z X with s s
t t
so that P x dy dz R x dy dz where R is the transition matrix of X
y
a a
R
d d
a d
the stability condition is a a d s The stationary distribution of R is q
a d a d
The eigenvector eigenvalue analysis of P yields the following e is the maximal eigenvalue of
the matrix
a ae
s
P R e e
d d e
Let y e and set
q
x a y d a a y d ady
s s
The eigenvalues of P are v y ax a Hence log y for y such that y ax
a The unnormalised eigenmeasures and eigenfunctions are
d ax and r r y x
resp ectively Thus
Z
t t
q r v q r v
t t
t
e q du P u E
r r
and
q r ax dy ax d
log log
r ax dy a d
using the values for x obtained from with y e
We check that all the required hypothesis are satis ed Prop osition gives Hyp otheses i ii
L
coupled with the fact that for for homogeneous sup erp ositions One veri es by explicit
t
t
0
di erentiation that is b ounded negatively away from zero for all t if the stability condition
t
s a a d is satis ed this gives iii Hyp othesis i follows from the assumed stationary
of the arrival streams As seen previously Hyp othesis i follows since exists and is nite
0
One sees that from the decomp osition that t t A tB where A is b ounded
t t t
t
and t B B as t Hence Hyp othesis ii is satis ed in this mo del and indeed
t t
in any Markovian mo del for which such a di erentiable decomp osition exists Hyp othesis is
automatically satis ed for a mo del with deterministic arrivals and E nite
At b we nd
ae d ax d
I inf log inf log
s s
a d e a d x
where x sd a s This agrees with the large deviation upp er b ound according to Hui
for the probability of over ow at a bu erless resource i e with b
The sign of can b e related to the sign of the correlations of the arrivals pro cess One sees
from that sgn sgn x x y But it is shown from Prop osition of that
a d x y while a d y x and a d x
y Furthermore the covariance of successive arrivals Cov Z Z ad a d a d
t t
Summarizing
sgn a d sgn sgn Cov Z Z
t t
Bursty sources will mo delled with a d successive arrivals are p ositively correlated A sub
bursty Markov mo del i e with negatively correlated arrivals has b een studied numerically by
Choudhury et al It is found that the log loss curves are concave and asymptotic to a straight
line with p ositive intercept at b corresp ondingly our value for will b e negative
Comparisons and estimates Theorem can b e used as a basis for approximation of sup erp o
sitions of nitely many lines we take
L LI b L
P Q b e
The are using with y e The resulting approximation is compared with simulations
t
in three cases Figure takes a d L and s a sup erp osition of
highly bursty sources In Figure the parameters a d L and s
are chosen to make I the curve is very close to linear In Figure a sub bursty case
a d L and s is shown In these examples the shap e of the log loss
curve is closely repro duced by the approximation but with a shift which makes the approximation
conservative in these cases This may well b e a limitation of the rst order large deviation metho d
In fact the discrepancy is well within the O log L re nements to the large deviation estimate of
L
P W in so some improvement may b e p ossible with further work involving these
re nements to the rst order large deviation result As a nal numerical example we take a large
sup erp osition of extremely bursty sources a d s L For these
parameters LI log and L log the desired loss probabilities of are
already obtained at b
Figure a d with a d s and L
Approximation LI b L
Simulation
log P Q b
b
Figure I with a d s and L
Approximation LI b L
Simulation
log P Q b
b
Figure a d with a d s and L
Approximation LI b L
Simulation
log P Q b
b
Poissonian Arrivals
We conclude with a brief discussion of Poissonian arrivals In this case one sees easily that
t
indep endent of t and hence from
I b inf t b t b
t
I we draw the conclusion that there are no economies of scale to b e obtained from a
sup erp osition of Poissonian arrivals at any bu er size b In contrast Bernoulli arrivals will generally
give I take a d in the on o mo del as an example The di erence in I b etween
the two cases can b e shown to go to zero if one constructs the Poissonian arrivals pro cess as a
continuum limit of Bernoulli arrivals
Acknowledgements The authors thank John Lewis Neil O Connell Ad Ridder Fergal Toomey
and particularly Raymond Russell for their comments Raymond Russell wrote the software used
in the simulations of the on o Markov mo del
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