Network Capacity Region of Multi-Queue Multi-Server Queueing System with Time Varying Connectivities

0.15 MB·English

by  Hassan Halabian

Network Capacity Region of Multi-Queue
Multi-Server Queueing System with Time Varying
Connectivities
Hassaan Halabian, Ioannis Lambadaris, Chung-Horng Lung
Department of Systems and Computer Engineering
Carleton University, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6 Canada
Email: {hassanh, ioannis.lambadaris, chung-horng.lung}@sce.carleton.ca
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1
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2 Abstract—Networkcapacityregionofmulti-queuemulti-server the capacity region of any policy is a subset of the network
queueing system with random ON-OFF connectivities and sta- capacity region. In fact, the network capacity region of a
n tionaryarrivalprocessesisderivedinthispaper.Specifically,the
system is the union of the capacity regions of all the possible
a necessaryandsufficientconditionsforthestabilityof thesystem
J resource allocation policies we can have for a network [8].
are derived under general arrival processes with finite first and

3 secondmoments.Inthecaseofstationaryarrivalprocesses,these A policy that achieves the network capacity region is called
1 conditions establish the network capacity region of the system. throughput optimal.
It is also shown that AS/LCQ (Any Server/Longest Connected The stability problem in wireless queueing networks was
] Queue) policy stabilizes the system when it is stabilizable.
T mainlyaddressedin[6],[7],[8],[9].In[6],authorsintroduced
Furthermore, an upper bound for the average queue occupancy
I is derived for this policy. the capacity region of a queueing network. They considered
s. a time slotted system in their work and assumed that arrival
c processes are i.i.d. sequences and the queue length process
[
I. INTRODUCTION is a Markov process. They also characterized the network
2 capacityregionof multi-queuesingle-serversystem with time
v Resource allocation is one of the main concerns in the varying ON-OFF connectivities which is described by some
4 design process of emerging wireless networks. Examples of
conditions on the arrival traffic [7]. They also proved that for
7 such networks are OFDMA and CDMA wireless systems in
a symmetric system (with the same arrival and connectivity
2
which orthogonal resources (OFDM subcarriers and CDMA
2 statisticsforallthequeues),LCQ(LongestConnectedQueue)
codes) must be allocated to multiple users. Research in this
. policy maximizes the capacity region and also provides the
1 areafocusesonfindingoptimalpoliciestoallocateorthogonal
optimalperformanceintermsofaveragequeueoccupancy(or
0
subchannelstotheusers.Therearestochasticarrivalsforeach
0 equivalentlyaveragedelay)[7].In[8],[9]and[13],thenotion
user which may be buffered to be transmitted in the future.
1 of network capacity region of a wireless network was intro-
v: Therefore, the resource allocation problem can be modeled duced for more general arrival and queue length processes.
as a multi-queue multi-server queueing system with parallel
i Furthermore, Lyapunov drift techniques were applied in [8]
X queues competing for available servers (which may model
and [9] to analyse the stability of the proposed policies for
r orthogonalsubchannels[1],[2],[3],[4],[11],[15]).However,
stochastic optimization problems in wireless networks.
a
because of users mobility, environmentalchanges, fading and
The problem of server allocation in multi-queue multi-
etc.,connectivityofeachqueuetoeachserverischangingwith
server systems with time varying connectivities was mainly
time randomly. Thus, we are faced to a multi-queue multi-
addressed in [1], [2], [3], [4], [11]. In [2], Maximum Weight
serversystem with time varyingchannelqualityforwhichwe
(MW)policy,athroughputoptimalserverallocationpolicyfor
have to design an appropriateserver allocation policy. One of
stationary connectivity processes was proposed. However, [2]
themainperformanceattributeswhichmustbeconsideredfor
doesnotexplicitlymentiontheconditionsonthearrivaltraffic
each policy is its capacity region and how much this region
toguaranteethestabilityofMW. References[1],[3],[4],[11]
coincides with the network capacity region [8]. The capacity
studytheoptimalserverallocationproblemintermsofaverage
region of a network is defined as the closure of the set of
delay. In [1], [3], [4], authors argue that in general, achieving
all arrival rate matrices for which there exists an appropriate
instantaneous throughput and load balancing is impossible in
policy that stabilizes the system [8]. This region is uniquefor
a policy. However, as they show this goal is attainable in the
eachnetworkandisindependentofresourceallocationpolicy.
specialcase of ON-OFF connectivityprocesses. Theyalso in-
On the other hand, the capacity region of a specified policy,
troduced the MTLB (Maximum-ThroughputLoad-Balancing)
say π, is the closure of the set of all arrival rate matrices for
policyandshowedthatthispolicyisminimizingaclassofcost
which π results into the stability of the system. Obviously,
functions including total average delay for the case of two
symmetric queues (with the same arrival and connectivities
1ThisworkwassupportedbyMathematicsofInformationTechnologyand
statistics). [11] considers this problem for general number of
ComplexSystems(MITACS)andNaturalSciencesandEngineeringResearch
Council ofCanada (NSERC). symmetric queues and servers. Authors in [11] characterized
a class of Most Balancing (MB) policies among all work A server scheduling policy at each time slot should decide
conserving policies which are minimizing a class of cost on how to allocate servers from set K to the queues in set L.
functions including total average delay in stochastic ordering Thismustbeaccomplishedbasedontheavailableinformation
sense.Theyusedstochasticorderinganddynamiccouplingar- about the connectivities G (t) and also the queue length
ij
gumentsto show theoptimalityofMBpoliciesforsymmetric process X(t).
systems.
In this paper, we will characterize the capacity region X1(t)
of multi-queue multi-server queueing system with random A1(t) G (t)
11
ON-OFF connectivities and stationary arrival processes based
on the stochastic properties of the system. Toward this, the X2(t) G21(t)
A (t)
necessary and sufficient conditions for the stability of the 2
system is derived under a general arrival process with finite
G2K(t) G1K(t)
first and second moments. For stationary arrival processes,
these conditions establish the network capacity region of the G (t)
L1
system. We also showedthata simple serverallocationpolicy
X (t)
L
calledAS/LCQ maximizesthecapacityregioni.e.itscapacity A (t) G (t)
L LK
regioncoincidewiththenetworkcapacityregionandtherefore
Fig. 1: Multi-queue multi-server queueing system with time
it is a throughput optimal policy. It is worth mentioning that
varying connectivities
AS/LCQ acts exactly the same as MW policy proposedin [2]
when the connectivity process is ON-OFF in MW.
The rest of the paper is organized as follows. Section II
describes the model and notation required through the paper. III. STABILITY OF MULTI-QUEUEMULTI-SERVER
In section III we discuss about the strong stability definition SYSTEMWITH TIMEVARYING CONNECTIVITIES
in queueing networks and Lyapunov drift technique briefly. In this section, we will consider the stability problem of
Then, we will derive necessary and sufficient conditions for multi-queue multi-server system with ON-OFF connectivities
thestabilityofourmodelandalsofindanupperboundforthe for which we will find the necessary and sufficient con-
averagequeueoccupancy.InsectionIVwepresentsimulation ditions for its stability. We also show that AS/LCQ (Any
results andcomparestability and delayperformancesof some Server/Longest Connected Queue) policy will stabilize the
heuristic work-conservingpolicieswith those ofAS/LCQ and system as long as it is stabilizable. The details of this policy
theupperboundobtainedinsectionIII.SectionVsummarizes will be presented in part D of this section. At first, we will
the conclusions of the paper. have a review on the notion of strong stability in queueing
networks.
II. MODELDESCRIPTION
Our model in this paper is the same as the model used in A. Strong Stability
[1], [2], [3], [4], [11] with ON-OFF connectivity processes.
We begin with introducingthe definition of strong stability
We considera time slotted queueingsystem with equallength
for a queueingsystem [8], [9]. Other definitionscan be found
time slots and equal length packets. The model consists of
in [5], [6], [7], [14]. Consider a discrete time single queue
a set of parallel queues L and a set of identical servers K.
system with an arrival process A(t) and service process µ(t).
Each server can serve at most one packet at each time slot
Assume thatthe arrivalsare addedto the system atthe end of
and we do not allow server sharing by the queues. In other
eachtimeslot.We cansee thatthequeuelengthprocessX(t)
words, each server can serve at most one queue at each time
at time t evolves with time according to the following rule.
slot. Assume that |L| = L and |K| = K. At each time slot
t, the link between each queue i ∈ {1,...,L} and server j ∈ X(t)=(X(t−1)−µ(t))++A(t) (1)
{1,...,K} is either connected or disconnected. Assume that
where (·)+ outputs the term inside the brackets if it is
connectivityprocessbetweenqueueiandserverj ismodelled
nonnegative and is zero otherwise. Strong stability is given
byani.i.d.binaryrandomprocesswhichisdenotedbyG (t),
ij
by the following definition [8].
i.e. G (t)∈{0,1}. Suppose that p represents the expected
ij ij
Definition 1: A queue satisfying the conditions above is
value of this process, i.e. E[G (t)] = p . There are also
ij ij
called strongly stable if
exogenous arrival processes to the queues in set L. Assume
that the arrival process to each queue i at time slot t (i.e. the t−1
1
number of packet arrivals during time slot t) is represented limsup E[X(τ)]<∞ (2)
t
by A (t). For these processes we assume that E[A2(t)] < t→∞ τ=0
i i X
A2 <∞ for all t. Each queue has an infinite buffer space Naturally for a queueing system we have the following
max
i.e. we do not have packet drops. We assume that the new definition [8].
arrivals are added to each queue at the end of each time slot. Definition 2: A queueing system is called to be strongly
Let X(t) = (X (t),...,X (t)) be the queue length process stable if all the queues in the system are strongly stable.
1 L
vector at the end of time slot t after adding new arrivals to In our work we use the strong stability definition and from
the queues. Figure 1 shows the model used in this paper. now we use “stability” and “strong stability” interchangably.
The following importantpropertyof strongly stable queues C. Necessary Condition for the Stability of the System
gives an invaluable insight of the above definitions.
Leth (t)bethedepartureprocessattimeslottfromqueue
ik
Lemma 1 [8]: If a queue is strongly stable and either
i to server k. Then, we can have the following equation for
E[A(t)] ≤ A for all t or E[µ(t)−A(t)] ≤ D where A and
the queue length process which shows the evolution of queue
D are finite nonnegative constants, then
length process with time.
1
lim E[X(t)]=0 (3) K
t→∞ t Xi(t)=Xi(t−1)− hik(t)+Ai(t) (7)
A very important and useful mathematical tool used in Xk=1
network stability analysis and stochastic control/optimization To findthe necessaryconditionforthe stability ofthe system,
of wireless networks is Lyapunov Drift technique. We now we need to use the following lemma.
present a brief review of this technique. Lemma3:Ifthesystemisstronglystableundersomeserver
allocation policy π, then for each queue i
B. Lyapunov Drift 1 t 1 t K
lim E[A (τ)]= lim E[h (τ)], (8)
i ik
The basic idea behind the Lyapunov stability method is to t→∞ t t→∞ t
τ=1 τ=1k=1
defineanonnegativefunctionofqueuebacklogsinaqueueing X XX
i.e.forastablesystemtheaverageexpectedarrivalstoaqueue
systemwhichcanbeseenasameasureofthetotalaggregated
is equal to the average expected departure from that queue.
backloginthesystemattimet.Thenweevaluatethe“drift”of
Proof: See appendix A.
suchfunctionintwo successivetimeslots bytakingtheeffect
We now proceed to find the necessary condition for the
of our control decision (scheduling or resource allocation
stability of the system.
policy) into account. If the expected value of the drift is
Theorem 1: Ifthereexistsaserverallocationpolicyπunder
negative as the backlog goes beyond a fixed threshold, then
which the system is stable, then
the system is stable. This is the method used in [8], [9], [12],
[10], [13], [14] to prove the stability of the systems working t K
1
under their proposed policies. lim E[A (τ)]≤K− (1−p ) (9)
i ik
t→∞ t
For a queueing system with L queues and queue length τ=1i∈Q k=1i∈Q
XX XY
vector X(t) = (X1(t),...,XL(t)), the following quadratic ∀Q⊂{1,...,L}
Lyapunov function has been used in literature ([8], [9], [12],
Proof: See appendix B.
[13], [14]).
L Remark: If the arrival processes Ai(t)’s are stationary, then
V(X)= Xi2(t) (4) E[Ai(t)] = λi for all t and therefore the left hand side of
(9) will be equal to λ . Consequently, the necessary
Xi=1 i∈Q i
conditionforthestability ofthesystem withstationaryarrival
AssumethatE[Xi(0)]<∞,∀i=1,2,...,LandX(t)evolves processes would be P
with some probabilistic law (not necessarily Markovian).
Then, the following important lemma holds. K
Lemma2[8]:IfthereexistconstantsB >0andǫ>0such λi ≤ (1− (1−pik)) ∀Q⊂{1,...,L}. (10)
that for all time slots t we have i∈Q k=1 i∈Q
X X Y
L
D. Sufficient Condition for the Stability of the System
E[V(X(t+1))−V(X(t))|X(t)]≤B−ǫ X (t), (5)
i
i=1 Wecandividetheserverallocationpolicyinourmodelinto
X
twoschedulingproblems.First,weshoulddeterminetheorder
then the system is strongly stable and further we have
under which servers are selected for service and second, for
1 t−1 L B eachserverdecidetoallocateittoaparticularqueue.Consider
limsup E[Xi(τ)] ≤ (6) the policy that chooses an arbitrary ordering of servers and
t ǫ
t→∞
τ=0i=1 then for each server, allocates it to its longest connected
XX
The left hand side of expression (5) is usually called queue(LCQ).In otherwords, in this policywe do notrestrict
Lyapunovdriftfunctionwhichis a measure ofexpectedvalue ourselveswithaspecificorderingofserversandweacceptany
of changes in the backlog in two successive time slots. We permutation of the servers according to which servers will be
caneasilysee theideabehindLyapunovmethodinstabilizing selectedforservice.However,forthenextphaseofscheduling,
queueing systems from Lemma 1. It is not hard to show that, for each selected server we use the LCQ policy.We call such
when the aggregated backlog in the system goes beyond the a policy as AS/LCQ (Any Server/LongestConnected Queue).
bound B, then the Lyapunov drift in the left hand side of (5) We will now derive the sufficient condition for the stability
ǫ
will be negative, meaning that the system receives a negative of our model and prove that AS/LCQ stabilizes the system
drift on the expected aggregated backlog in two successive as long as condition (11) is satisfied. An upper bound is also
time slots. In other words the system tends toward lower derived for the time averaged expected number of packets in
backlogs and this results in its stability. the system.
Theorem 2: The multi-queue multi-server system is stable However, queue 1 has only two packets waiting for service
under AS/LCQ if for all t and therefore server 3 will be idle at this time slot (although
it could have been used to serve queue 2).
K
E[A (t)]<K− (1−p ) ∀Q⊂{1,...,L}. (11) Note that since AS/LCQ is a non-work conserving policy
i ik
it can not be delay optimal. However, it can achieve the
i∈Q k=1i∈Q
X XY network capacity region as explained previously in part D.
Furthermore, the following bound for the average expected
In fact, AS/LCQ will exhibit non-work conserving behaviour
“aggregate” occupancy holds.
in light arrival loads and as the load increases its behaviour
t−1 L will converge to work conserving. Since the capacity region
1
limsup E[Xi(τ)]≤ (12) of a system is mainly determined by its behaviour in heavy
t
t→∞ τ=0i=1 arrival loads, this property of AS/LCQ does not have conflict
XX
−L LA2 +K(2K−1) with its throughputoptimality.It is worth mentioningthat not
2 max
all work-conserving policies are throughput optimal. In the
(cid:0) K (cid:1)
max E[A (t)]−K+ (1−p ) following section by simulations we will observe that some
i ik
Q⊂{1,2,...,L},t  work-conservingpoliciescannotachievethe networkcapacity
Xi∈Q kX=1iY∈Q 
region.Inthefollowingsection,wewillalsoobservethathow
Proof: See appendix C.
  the service ordering of servers affects the average total queue
It is worth mentioning that AS/LCQ acts exactly the same
occupancy. However, as we showed in the previous part an
as MW policy proposed in [2] when the connectivity process
arbitraryorderingissufficienttoachievethenetworkcapacity
is ON-OFF in MW.
region.
Note that for all the servers we only use the backlog
information at the beginning of each time slot, i.e. during the
implementation of AS/LCQ policy at each time slot we do IV. SIMULATION RESULTS
notupdatethe queuelengthsuntilall the serversareallocated Simulation is used to show the validity of our analysis
at which point we update the queue lengths. It is interesting in the previous section and also to compare performance
to note that this policy can be non-work conserving at some of AS/LCQ to some heuristic work conserving policies in-
time slots. In other words, there may exist some idle servers cluding LCSF/LCQ (Least Connected Server First/Longest
at a time slot while they could have served other backlogged Connected Queue), MCSF/LCQ (Most Connected Server
queues. We will discuss about it through an example in part First/Longest Connected Queue), LCSF/SCQ (Least Con-
E of this section. nected Server First/Shortest Connected Queue), MCSF/SCQ
Remark: Note that by considering stationary assumption on (MostConnectedServer First/ShortestConnected Queue)and
the arrival processes, the condition (11) would be a Randomized policy [11].
K The LCSF (MCSF) policy at the first phase of scheduling
λ <K− (1−p ) ∀Q⊂{1,...,L}. (13) (i.e. determination of servers order) will sort the servers for
i ik
i∈Q k=1i∈Q service according to their number of connectivities in an
X XY
ascending (descending) order. The LCQ (SCQ) policy will
Accordingtothedefinitionofsystemcapacityregionand(10)
assign the selected server it to its longest connected queue
and (13), equation (10) characterizes the network capacity
(shortest connected queue). Note that in order to make SCQ
regionofmulti-queuemulti-serversystemwithstationaryON-
policies work conserving, we only serve the shortest non-
OFF connectivities and stationary arrivals.
emptyqueues.TheRandomizedpolicyateachtimeslotmakes
random server selections and for each server random non-
E. Discussion empty queue selection.
As mentioned earlier, AS/LCQ may exhibit non-work con- We have simulated a system consisting of 16 queues
servingbehaviorduringsome time slots. This can be clarified (L = 16) and 4 servers (K = 4). First, we considered a
by the following example. symmetric system in which all the arrivals to all the queues
ConsiderasystemwithL=2andK =3withqueuelength arethesameindistribution.Wealsoassumedthatconnectivity
vector X(t) = (2,1) at time slot t. For the connectivities at variables have the same distribution (the same connectivity
this time we have the following matrix. probabilities).Inthissystem,arrivalsareassumedtohavei.i.d.
Bernoulli distributions. The capacity region for these special
1 1 1
G(t)= cases would be an n dimensional cube whose side size is
(cid:20) 0 0 1 (cid:21) equalto K−K(1−p)L whichis0.243forp=0.2andisalmost
L
Assume that the ordering of server selection is server 1 0.25 for p = 0.9. Figures (2) and (3) show the average total
first and then server 2 and finally server 3. Servers 1 and 2 occupancy of different policies for connectivity probabilities
both are allocated to queue 1 according to LCQ rule. Server 0.2, and 0.9 versus arrival rate per queue. In these figures,
3 is connected to both of the queues. Since in AS/LCQ all it is observed that in all the cases if the arrivals are inside
the servers are allocated first and then the queue lengths are the capacity region,AS/LCQ can stabilize the system andhas
updated afterwards, queue 1 is the longest connected queue average total occupancy below the bound we derived in the
for server 3. Thus, server 3 is allocated to queue 1 as well. previous section.
T
0.2 0.72 0.86 0.3 0.66 0.21 0.84 0.03

0.1 0.65 0.65 0.15 0.58 0.32 0.69 0.12

103 0.02 0.42 0.94 0.35 0.9 0.16 0.96 0.21
0.8 1 0.7 0.09 0.1 0.45 0.13 0.07 
Average Total Occupancy (Packets)110012 ALMMRLBCCCSoaaCCuSnSp/SSLndaFFCFFdoc////LSQimSLtCyCCCi zQRQQQeedgion Ibcdthnaueipstsitcaohrccniiiasbtsyweeex.iirtnpehIngeatrihoitcmhenoienssinacnfitmi,sgteeahurimrrrseaia,vtcenawasnlsseeferoa.orriFbseesiagnefcourovhlrtleeosew(ltao4hsit)anytagsnhtRotdohawnecesdhaPcoathohrmaiescqizstrueoeeerdnsuiuzepldet.oissTlatifnrhcoidye-r
100
could not capture the capacity region wholly. However, LCQ
X: 0.243 policies (AS/LCQ, LCSF/LCQ and MCSF/LCQ) performs
Y: 0.1014
10−1 10−1 similarly to each other from stability point of view.
Arrival Rate Per Queue (Packets/Time Slot)
Fig. 2: Average Total Occupancy for p=0.2

103
Occupancy (Packets)111100001234 ALMMRLBCCCSoaaCCuSnSp/SSLndaFFCFFdoc////LSQimSLtCyCCCi zQRQQQeedgion Average Total Occupancy (Packets)110012 ALMMRLBCCSoaCCuSnS/SSLndFFCFFdo////LSQmSLCCCCizQQQQed
Average Total 100 100
10−1
10−1 Arrival Rate Per Queue (Packets/Time Slot)
XY:: 00..2051 Fig. 4: Average total occupancy for an asymmetric scenario
10−2
10−1
Arrival Rate Per Queue (Packets/ Time Slot)
From the above simulations, we can also observe that
Fig. 3: Average Total Occupancy for p=0.9
AS/LCQ performs slightly worse as compared with other
policies in light arrival loads. This behaviour is because the
factthatAS/LCQmayexhibitnon-workconservingbehaviour
We can further conclude that as the connectivity variable more frequently for light arrival loads. However, as the load
increases the performance of the work conserving policies increases AS/LCQ will be work conserving with high prob-
become the same. This agrees with intuition since when the ability. We can also observe that the obtained bound is not
system is close to full connectivity, any work conserving tight.
algorithm will be optimal in terms of average occupancy
and of course better than any non-work conserving policy V. CONCLUSIONS
like AS/LCQ. Although AS/LCQ has larger average total
In thispaperwe derivedthe necessaryandsufficientcondi-
occupancy compared to other policies, it still stabilizes the
tions for the stability of multi-queuemulti-serversystem with
system as long as arrivals are inside the capacity region and
random connectivities and characterized the capacity region
hasboundedaveragetotaloccupancy.However,thisis notthe
of this system for stationary arrivals. We also introduced
case forLCSF/SCQ andMCSF/SCQ policiesandtheycannot
AS/LCQ policyand arguedthatalthoughthis policyis a non-
stabilize the system for certain arrivals inside the capacity
work conserving policy, it can stabilize the system for all
region.Fromthese figureswe alsosee thatrandomizedpolicy
the arrivals inside the capacity region and therefore it is a
performsveryclosetotheotherpoliciesinthesespecialcases
throughput optimal policy. Then, we derived an upper bound
and this is due to existence of symmetry (in arrivals and
of the average queue occupancy for this policy. Finally, we
connectivities) in these cases.
used simulations to validate our analysis and compare this
We have also simulated an asymmetric system in which
policy to some work conserving policies in terms of average
connectivity variables comes from the following matrix in
queue occupancy.
which p =E[G (t)]. This matrix was chosen randomly.
ij ij If we modify the policy such that the queue lengths are
updatedaftereachserverisallocated,wecanestablishawork
0.9 0.2 0.2 0.8 0.2 0.1 0.5 0.6 conservingpolicy.However,thisdoesnotincreasethecapacity
0.8 0.1 0.1 0.9 0.02 0.5 0.8 0.8 regionfora systemwithstationaryarrivals.However,wemay
p=
0.9 0.02 0.5 0.99 0.3 0.8 0.78 0.99 help us to obtain a tighter bound than we obtained in this
 0.8 0.03 0.9 0.87 0.5 0.98 0.62 0.4 work.


APPENDIX A Note that P[B2(τ)]≥0 and for P[B1(τ)], we have
k k
PROOF OF LEMMA3
P[B1(τ)]= (1−p ) (22)
Proof: If we write equation (7) for τ = 1,2,...,t and then k ik
adding them up, we will have iY∈Q
Finally, from (16), (21) and (22) we conclude that
t K t
X (t)=X (0)− h (t)+ A (t) (14)
i i ik i t K
1
τX=1kX=1 τX=1 lim E[Ai(τ)]≤ (1− (1−pik)) (23)
Takingtheexpectationfrombothsides,dividingbytandthen t→∞ t
τ=1i∈Q k=1 i∈Q
XX X Y
takingthelimitastgoestoinfinity,wewillhavethefollowing.
and the theorem follows. (cid:3)
E[X (t)] E[X (0)]
i i
lim = lim
t→∞ t t→∞ t
t K t
1 1
− lim E[hik(t)]+ lim E[Ai(t)] (15) APPENDIX C
t→∞ t t→∞ t
τ=1k=1 τ=1 PROOF OF THEOREM2
XX X
According to Lemma 1 and the assumption that
Proof: We will start with the Lyapunov function evaluation.
E[X (0)] < ∞, the left hand side term and the first
i we will use the quadratic function (4) as our Lyapunov
term in the right hand side term are equal to zero and
function. The Lyapunov drift for two successive time slots
therefore the result is proven. (cid:3)
has the following form.
E[V(X(t+1))−V(X(t))|X(t))]
APPENDIX B L
PROOF OF THEOREM1 =E Xi2(t+1)−Xi2(t)|X(t)
" #
Sincethesystemisstronglystable,(8)mustbesatisfiedfor Xi=1
L
any subset of queues Q⊂{1,...,L}, i.e. =E (X (t+1)−X (t))2 |X(t)
i i
1 t 1 t K "Xi=1 #
lim E[A (τ)]= lim E[h (τ)] (16) L
i ik
t→∞ t τ=1i∈Q t→∞ t τ=1i∈Qk=1 + 2E Xi(t)(Xi(t+1)−Xi(t))|X(t) (24)
XX XXX
" #
i=1
We now define the sets B (τ) as X
k
For the the first term we have:
B (τ)={G (τ),X (τ −1),ℓ∈Q} (17)
k ℓk ℓ
L
For Bk(τ), three disjoint cases are imaginable. E (Xi(t+1)−Xi(t))2 |X(t)
Bk1(τ)={Gik(τ)=0,i∈Q} "Xi=1 #
Bk2(τ)={Gik(τ)=0,i∈Q}c∩{Xi(τ −1)=0,i∈Q} L K
Bk3(τ)={Gik(τ)=0,i∈Q}c∩{Xi(τ −1)=0,i∈Q}c =E (Ai(t+1)− hik(t))2 |X(t)
By conditioning each term in the right hand side summation "i=1 k=1 #
X X
in (16) to the event B (τ) we have L
k
=E A2(t+1)|X(t)
i
K K "i=1 #
X
E[hik(τ)]= EBk(τ)E hik(τ)|Bk(τ) L K
kX=1Xi∈Q Xk=1 Xi∈Q −2E Ai(t+1)hik(t)|X(t)
  (18) "i=1k=1 #
XX
We can easily see that L K 2
+ E h (t) |X(t) (25)
ik
E hik(τ)|Bkj(τ)=0 j =1,2 (19) Xi=1 Xk=1 ! 
i∈Q
X K
 
and Using the the fact that hik(t) ≥ 0 we get the following
k=1
E h (τ)|B3(τ) ≤1 (20) inequality X
 ik k 
Xi∈Q L K 2 L K 2
Using (19) and (20), equation (18) can be simplified to the h (t) ≤ h (t) ≤K2 (26)
ik ik
following. i=1 k=1 ! i=1k=1 !
X X XX
K K L K
E[hik(τ)]≤ (1−P[Bk1(τ)]−P[Bk2(τ)]) (21) Since Ai(t+1)hik(t) ≥ 0, the first term in (24) can
Xk=1Xi∈Q Xk=1 Xi=1kX=1
be bounded by summation can be rewritten as
L
L E X (t)h (t+1)|X(t)
E (Xi(t+1)−Xi(t))2 |X(t) "i=1 i isk #
" # X
i=1 L
X
L =E X (t)h (t+1)|X(t)
≤ E[A2(t+1)]+K2 (27) " qi qisk #
i i=1
X
i=1 L L
X
= E X (t)h (t+1)|X(t),Dsk P(Dsk)
Now assume that we select the servers for service according " qi qisk l # l
l=0 i=1
to an arbitrary order s1,s2,...,sK. Thus, for the second term X X (31)
in (24) we have
Note that
L L
E"Xi=1Xi(t)(Xi(t+1)−Xi(t))|X(t)# E"Xi=1Xqi(t)hqisk(t+1)|X(t),Dlsk#≥(Xql(t)−(k−1))+.
L K
Therefore, equation (31) can be bounded by
=E X (t)(A (t+1)− h (t+1))|X(t)
i i ik
"i=1 k=1 # L
X X
L E Xi(t)hisk(t+1)|X(t)
=E Xi(t)Ai(t+1)|X(t) "Xi=1 #
"i=1 # L L
X
L K ≥ (Xql(t)−(k−1))pqlsk (1−pqjsk)
−E Xi(t)hisk(t+1)|X(t) (28) Xl=1 u=Yl+1
"i=1k=1 # L L
XX
= X (t)p (1−p )
ql qlsk qjsk
The first term in (28) can be written as follows. l=1 j=l+1
X Y
L L
L L −(k−1) p (1−p ) (32)
qlsk qjsk
E X (t)A (t+1)|X(t) = E[A (t+1)]X (t) (29)
i i i i l=1 j=l+1
" # X Y
i=1 i=1
X X For the second term in (32) we have
For the second term in (28) we have L L
(k−1) p (1−p )
qlsk qjsk
L K l=1 j=l+1
X Y
E X (t)h (t+1)|X(t)
i isk L
" #
Xi=1kXK=1 L =(k−1)1− (1−pqjsk) (33)
j=1
= E X (t)h (t+1)|X(t) (30) Y
i isk  
" # and for the first term
k=1 i=1
X X
L L
Now, we introduce the following notation. We sort the X (t)p (1−p )
ql qlsk qjsk
queue length process at time slot t in an ascending order l=1 j=l+1
X Y
X ,X ,....,X , i.e. X (t)≥X (t) for all i=2,...,L
q1 q2 qL qi qi−1 L L
caonndsiidferXthqie(tf)oll=owiXngqi−d1e(cto)m, pthoesnitioqni o≥f thqei−c1o.nnFeucrttihveitrymporroe-, = (Xqj(t)−Xqj−1(t))1− (1−pqlsk)
j=2 l=j
X Y
cesses for each server k.
 
L
+X (t) 1− (1−p ) (34)
Dk ={G (t+1)=0, for all i∈{1,...,L}} q1  qjsk 
0 ik j=1
Y
Dik ={Gqik(t+1)=1,Gqℓk(t+1)=0, Equation (29) also can be written as follows.
for i<ℓ≤L and all i∈{1,...,L}}
L L
E[A (t+1)]X (t)= E[A (t+1)]X (t)
The probability of events Dk is given by i i ql ql
i i=1 l=1
X X
L L
L L = (X (t)−X (t)) E[A (t+1)]
P(Dk)= (1−p ) , P(Dk)=p (1−p ) qj qj−1 ql
0 ik i qik quk j=2 l=j
X X
iY=1 u=Yi+1 L
+X (t) E[A (t+1)] (35)
In the second term of equation (30), each term in the q1 ql
l=1
X
Using equations (28)-(30) and (32)-(35) we have the fol- where ǫ=−2m. According to condition (11), m is negative,
L
lowing bound for the second term in (24). therefore ǫ > 0. Since L X (t) ≤ LX (t), therfore
i=1 i sL
L −ǫ(LXsL(t)) ≤−ǫ Li=1PXi(t). Consequently the Lyapunov
E X (t)(X (t+1)−X (t))|X(t) drift (38) is bounded by
i i i
" # P
i=1
X E[V(X(t+1))−V(X(t))|X(t))]
L L
≤ (X (t)−X (t)) E[A (t+1)] L
qj qj−1 ql ≤LA2 +2K2−K−ǫ X (t)
j=2 l=j max i
X X
L Xi=1
+X (t) E[A (t+1)] L
q1 ql =B−ǫ X (t) (40)
i
l=1
X
i=1
L K L X
− (X (t)−X (t)) 1− (1−p ) in which B that has positive value is defined as
qj qj−1  qlsk 
Xj=2 kX=1 Yl=j B =LA2 +2K2−K (41)
max
 
K L
Therefore, according to Lemma 2, the multi-queue multi-
−X (t) 1− (1−p )
q1  qjsk  server system is stable under AS/LCQ as long as condition
k=1 j=1
X Y (11)issatisfiedandalsothetimeaverageexpectedcongestion
 
K L
in the system is bounded by
+ (k−1) 1− (1−p )
 qjsk 
k=1 j=1 1 t−1 L B
X Y
limsup E[X (τ)]≤ (42)
L   t i ǫ
t→∞
≤ j=2(Xqj(t)−Xqj−1(t)) which is equal to (12)Xτ.=0Xi=1 (cid:3)
X
L K L
· E[A (t+1)]− 1− (1−p )
 ql  qlsk 
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