Achieving Delay Rate-function Optimality in OFDM Downlink with Time-correlated Channels

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by  Zhenzhi Qian

Achieving Delay Rate-function Optimality in
OFDM Downlink with Time-correlated Channels
Zhenzhi Qian∗, Bo Ji†, Kannan Srinivasan∗, Ness B. Shroff∗‡
∗Department of Computer Science and Engineering, The Ohio State University, Columbus 43210, OH
†Department of Computer and Information Sciences, Temple University, Philadelphia 19122, PA
‡Department of Electrical and Computer Engineering, The Ohio State University, Columbus 43210, OH
Abstract—Therehavebeenrecentattemptstodevelopschedul- twospecialcasesofON/OFFchannelswithatwo-usersystem
6 ing schemes for downlink transmission in a single cell of a or a system that allows fractional server allocation. However,
1 multi-channel(e.g.,OFDM-based)cellularnetwork.Theseworks
this problem becomes much harder in general cases. On the
0 have been quite promising in that they have developed low-
2 complexity index scheduling policies that are delay-optimal (in other hand, in cellular networks, minimizing average delay
a large deviation rate-function sense). However, these policies may cause a large delay for certain users that have stringent
n
require that the channel is ON or OFF in each time-slot with delay requirements.
a
J a fixed probability (i.e., there is no memory in the system), Another line of works focus on designing scheduling poli-
whilethereality isthat duetochannel fadingand dopplershift,
9 cies that maximize the rate-function of the steady-state prob-
channelsareoftentime-correlatedinthesecellularsystems.Thus,
2 ability that the largestqueue length exceedsa given threshold
animportantopenquestioniswhetheronecanfindsimpleindex
schedulingpoliciesthataredelay-optimalevenwhenthechannels when the number of channels and users both go to infinity.
]
T are time-correlated. In this paper, we attempt to answer this In [4] and [5] the authors showed that their proposed policy
I questionfortime-correlatedON/OFFchannels.Inparticular,we canachieveboththroughputoptimalityandqueuelengthrate-
s. showthattheclassofoldestpacketsfirst(OPF)policiesthatgivea function optimality. However, simulations in [6] - [8] show
higherprioritytopacketswithalargedelayisdelayrate-function
c thatgoodqueuelengthperformancedoesnotnecessarilyimply
[ optimal under two conditions: 1) The channel is non-negatively
correlated,and2)ThedistributionoftheOFFperiodisgeometric. good delay performance. In fact, queue-length-based policies

3 We use simulations to further elucidate the theoretical results. usuallysufferfromthesocalled“lastpacket”problem,which
v occursin the situationwhere a certain queuehas a verysmall
1 I. INTRODUCTION number of packets. Hence, this queue is rarely scheduled
4
Orthogonal frequency division multiplexing (OFDM) is a by the queue-length-based policies, resulting in large packet
2
6 digital multi-carrier modulation method that has been widely delays.
0 used in wideband digital communications. A practical and To that end, a delay-based metric has been investigated in
. importantapplicationisthedownlinkphaseofa singlecellof recent works in [7], [10] and [11]. The authors developed
1
0 OFDM-based cellular networks, where the wideband can be several policies that achieve both throughput optimality and
6 divided into a large number of orthogonalsub-carriers,which delay rate-function optimality (or near-optimality). Although
1 can be used to carry data for different users. In this system, the results hold for general arrivals (e.g., time-correlated
:
v theBaseStation(BS)maintainsaseparatequeuetostoredata arrivalsareallowed),thechannelsareassumedtobei.i.d.over
i packets requested by each user. When the sub-carrier seen time. In practice, the current channel condition could depend
X
by a user is in good channel condition, the sub-carrier can on past channel conditions. Therefore, the following impor-
r
a successfully transmit a packet to the user from its designated tant question remains: How do we design a low-complexity
queue.We willfocuson thesetting ofa single-hopmulti-user schedulingpolicythatachievesprovablygoodthroughputand
multi-channelsystem and study the delay performanceof this delay performance in the OFDM downlink system with time-
system from a large-deviationsperspective. correlated channels?
In wireless networks, a key problem that has been exten- While it is relatively straightforward to develop throughput
sively studied is the design of high-performance scheduling optimalpoliciesevenfortime-correlatedchannels,developing
policies. It is well known from the seminal work [1] that policies that are delay-optimal or delay-efficient for time-
the MaxWeightpolicyisthroughput-optimal,in the sense that correlated channels remains an open problem.
it can stabilize the system under any feasible arrival rates. To that end, we are motivated to consider the following
However,it has been shown in [2] that the MaxWeightpolicy question: Can we find index scheduling policies that are
sacrifices the delay performance (and may lead to very large delay-optimal even when the channels are time-correlated?
queue lengths) for better throughput. This fact has motivated In this paper, we provide a positive answer in some cases.
researchers to look for policies that can improve the delay Specifically,we analyzethedelayrate-functionofthe classof
performancemeasuredbyaqueue-length-basedmetric.In[3], oldestpacketsfirst(OPF)policieswhichgiveahigherpriority
the authors showed that the maximum-throughput and load- topacketswithalargedelayandpresenttwoconditionsunder
balancing (MTLB) policy can achieve delay optimality for which delay rate-function optimality can be achieved by any
OPF policy. Q S
1 1
The key contributions of this paper are summarized as
follows. We use an alternating renewal process to model a Q S
2 2
general ON/OFF time-correlated channel. We first prove an
upper bound on the delay rate-function for any scheduling
policy. Then, we analyze the delay rate-function of the class
ofOPFpolicies,whichgiveahigherprioritytoolderpackets. Q S
n n
We present two conditions and show that if both conditions
aresatisfied,delayrate-functionoptimalitycanbeachievedby Fig.1. Amulti-queue multi-server systemwithstochastic connectivity. The
any OPF policy. The first condition requires that the channel connectivity betweenqueueQi andserverSj is“ON”iftheyareconnected
byasolidline,and“OFF”otherwise (connected byadashedline).
conditionisnon-negativelycorrelated overtime.Thisis often
observed in practical time-correlated channels. The second
condition requires that the “OFF” period distribution has the denotethecumulativepacketarrivalstothesystemfromtime-
memoryless property, whereas the “ON” period distribution slot t to time-slot t .
1 2
could be arbitrary. Next, we will introduce several assumptions on the arrival
The rest of the paper is organizedas follows. In Section II, process for purpose of rate-function delay analysis.
we describe the system modeland the performancemetric. In Assumption 1: The number of arrivals are bounded, i.e.,
Section III, we derive an upper bound on the rate-function there exists a finite number L such that Ai(t) ≤ L for any i
for any possible policy, and in Section IV, we obtain an and t. Also, we assume P(A(s,s+t−1)=Lnt)>0 forany
achievable rate-functionof the class of OPF policies. Then in s,t and n.
Section V, we propose two conditions that imply delay rate- Assumption 2:The arrivalprocessarei.i.d acrossall users,
function optimality of the class of OPF policies. We conduct andthemeanarrivalrateisp(weassumep<1,otherwisethe
simulations to validate our theoretical results in Section VI system could not be stable under any scheduling policy) for
and make concluding remarks in Section VII. every user. Given any ǫ>0 and δ >0, there exists a positive
function I (ǫ,δ) independent of n and t such that
B
II. SYSTEMMODEL Σt 1
P τ=1 {|A(τ)−pn|>ǫn} >δ <exp(−ntI (ǫ,δ)). (1)
We use a time-slotted multi-queue multi-server system to t B
model the downlink phase of a single cell OFDM system. In for a(cid:16)ll t≥T (ǫ,δ) and n≥N (cid:17)(ǫ,δ).
B B
particular,we assume that there are n serverswhich stand for Assumptions 1 and 2 are mild. Packet arrivals per time-
frequency sub-carriers. Furthermore, we assume the number slot are typically boundedin practice. In addition, it has been
of users is equal to the number of channels for ease of shown in [7] that Assumption 2 is a general result of the
presentation [10]. The Base Station maintains a queue/buffer statisticalmultiplexingeffectofalargenumberofsourcesand
to store packets requested by each user, hence there are also holdsfor both i.i.d. arrivalsand Markovchain drivenarrivals.
n queuesin the queueingsystem. (We use terms “server”and
“channel”,“queue”and“user”interchangeablythroughoutthis
paper.) Next, we present several notations that will be used
D U
later in this paper. We use Q to denote the queue associated
i
ON OFF ON OFF
to the i-th user, and use S to denote the j-th server for
j time
1 ≤ i,j ≤ n. We use Q (t) to denote the queue length of
i
State Change
queue Q at the beginning of time-slot t immediately after
i
new packet arrivals. All queues are assumed to have infinite
buffer size. Further, we use W (t) to denote the head-of-line Fig.2. Time-correlated channel model
i
(HOL) delay of queue Q at the beginning of time-slot t and
i
use W(t) = max1≤i≤nWi(t) to denote the largest packet B. Stochastic Connectivity
delayinthesystematthebeginningoftime-slott.Finally,we
We assumethateachchannelhasunitcapacityandchanges
use 1 to denotethe indicatorfunctionthatindicateswhether
A between “ON” state and “OFF” state from time to time. We
event A occurs or not.
useC (t)to indicatetheconnectivitybetweenqueueQ and
i,j i
serverS intime-slott:C (t)=1whenthechannelis“ON”
A. Arrival Process j i,j
and C (t)=0 when the channel is “OFF.” We define “ON”
i,j
Thearrivalprocesstoeachqueueisassumedtobestationary periodtobethenumberoftime-slotsbetweenthelasttimethe
and ergodic. We also assume the arrivals are i.i.d. across all channelwas “OFF” untilthe nexttime-slotitbecomes“OFF”
users, but could be correlated over time. Let A (t) denote again. “OFF” period is defined in the similar way. From time
i
the number of packet arrivals to queue Q in time-slot t. Let to time, the channel state alternates between “ON” periods
i
A(t) = n A (t) denote the total packet arrivals coming and “OFF” periods. We use an alternating renewal process to
i=1 i
intothesystemintime-slott,andletA(t ,t )= t2 A(τ) modelthe stochastic connectivity.In other words, the channel
P 1 2 τ=t1
P
is initially “ON” for a time period U1 and then “OFF” for a III. AN UPPERBOUND ON THE RATE-FUNCTION
time period D , followed by another “ON” period U and so
1 2 In this section, we derive an upper bound on the best
on. In particular, the sequences of “ON” times {U : n ≥1}
n achievable delay rate-function. Later, we will use this upper
and “OFF” times {D :n≥1} are independentsequencesof
n bound as a baseline to evaluate the delay performance of the
i.i.d.positiverandomvariables.LetU beageneric“ON”time
OPF policies.
and D be a generic “OFF” time. We use F (·) and F (·) to
U D First, as in [6], [7], we define quantity I (t,x) for any
A
denote the CDF of random variable U and D, respectively.
integer t>0 and any real number x≥0:
Assumption3:Thesumof“ON”and“OFF”periodsU+D
is aperiodic with E[U]<∞ and E[D]<∞. I (t,x),sup[θ(t+x)−λ (θ)]. (4)
A Ai(−t+1,0)
It is well known (e.g. [9]) that under Assumption 3, we θ>0
have: where λAi(−t+1,0)(θ) = logE[eθAi(−t+1,0)] is the cumulant-
tl→im∞P(Ci,j(t)=0)= E[UE]+[DE][D] =π0. (2) genFeroramtinCgrafmunecrt’isoTnhoeforAeim(−,ItA+(t1,x,0))is=eqPua0τl=t−ott+h1eAasiy(mτ)p.totic
decay-rateoftheprobabilitythatinanyintervalofttime-slots,
for any i,j. the totalnumberof packetarrivalsto the system is no smaller
Remark 1: If U +D is periodic with period d, the above than n(t+x) as n tends to infinity, i.e.,
result is true if t is an integral multiple of d. For simplicity,
−1
we only focus on the aperiodic case. lim logP(A(−t+1,0)≥n(t+x))=IA(t,x). (5)
n→∞ n
Note that this is a general model that can capture the
We define t for L>1 and non-negativeinteger x:
time-correlation of a channel. If U and D are geometri- x
cally distributed with parameters 1 − q and q, respectively, t , x . (6)
it degenerates to a static i.i.d. channel model with channel x L−1
“ON” probability q. Similarly, if U and D have a geometric
Then we define an integer set Ψ , {c ∈
distribution with parameter p and p , respectively, it be- b
10 01 {1,2,··· ,b}|t ∈Z+}. For any integer b≥0, let
comes the Markovian channel model with transition matrix b−c
1−p p
01 01 1−F (τ)
T = . I∗(b),min log max D −logπ ,
" p10 1−p10 # U τ∈{0,1,···}1−FD(τ +b) 0
In each time-slot, a scheduling policy allocates servers to min innf I (cid:0)(t,b), min { inf I (t,b−(cid:1) c)−logπ
A A 0
serve packets from user queues. We further assume that a t>tb 1≤c≤b t>tb−c
server can only serve one queue in a time-slot, however, a (cid:8) 1−FD(τ)
+log max ,
queuecangetservicefrommultipleserverssimultaneouslyin τ∈{0,1,···}1−FD(τ +c−1)
one time-slot. In addition, one packet from queue Qi can be min{I(cid:0)A(tb−c,b−c)−logπ0+ (cid:1)(cid:9)
served if an “ON” channel is allocated to queue Qi. c∈Ψb
1−F (τ)
D
log max } . (7)
C. Problem Formulation τ∈{0,1,···}1−FD(τ +c)
o
(cid:0) (cid:1)
In this paper, the metric we use to measure the delay In addition, we define:
performanceis the largedeviationrate-functionofthe steady-
−logπ +log max 1−FD(τ) , L=1
state probability that the largest packet delay exceeds a given I (b)= 0 τ∈{0,1,···} 1−FD(τ+b)
U
threshold b. Assume the system starts at minus infinity, then (I∗(b), (cid:16) (cid:17) L>1
U
W(0) is the largest packet delay over all the queues in the (8)
steady-state.Wedefinetherate-functionI(b)astheasymptotic The following theorem shows that for any integer b ≥ 0,
decay-rate of the probability that W(0) > b for a given IU(b) is an upper bound on the delay rate-function for any
threshold b: feasible scheduling policy.
Theorem 1: For any integer threshold b ≥ 0 and any
−1
I(b), lim logP(W(0)>b). (3) scheduling policy, we have:
n→∞ n
−1
Notethatbythedefinitionofrate-functionI(b),wecanesti- limsup logP(W(0)>b)≤IU(b). (9)
n
mate the order of delay violation probability (i.e., P(W(0)> n→∞
b) by exp(−nI(b)). It is obvious that a larger rate-function Proof: We consider two cases L>1 and L=1. For the
impliesasmallerdelayviolationprobabilityandabetterdelay caseL>1,wewillconsiderthreetypesofevents:χ1,χc2 and
performance. In this paper, our objective is to maximize the χc3, which are subsets of the delay-violation event {W(0) >
rate-function I(b). 1 b}. Note that bursty arrivals and sluggish services both cause
largepacketdelay in the system. In particular,χ is the event
1
1We mainly focus on the delay analysis, since the results for throughput with sluggishserviceswhile χc2 and χc3 are eventswith bursty
performance canbeeasilygeneralized from[10]. arrivals and sluggish services.
Event χ : Suppose there is a packet that arrives to the and thus
1
network at the beginning of time-slot −b − 1. Since the
−1
arrival process is independent across all queues, without loss limsup logP(W(0)>b)
n
of generality, we assume that this packet arrives to Q . n→∞
1
1−F (τ)
Furthermore, Q is assumed to be disconnected from all n D
1 ≤−logπ +log max .
0
servers from time-slot −b − 1 to −1. As a result, at the τ∈{0,1,···}1−FD(τ +b)
beginning of time-slot 0, this packet is still in the network (cid:16) (cid:17)(13)
and has a delay of b+1, which violates the delay threshold
Event χc: Consider any fixed c ∈ {0,1,2,··· ,b} and any
b. Therefore, χ1 ⊆{W(0)>b}. t > t 2. Recall that t = b−c. Then, for all t > t ,
FirstwewanttocalculatetheprobabilitythatQ1 isdiscon- b−c b−c L−1 b−c
nected from an arbitrary server Sj from time-slot −b−1 to we have b−c < (L−1)t, and thus IA(t,b−c)=IA+(t,b−
−1, i.e., C1,j(t) = 0 for all −b−1 ≤ t ≤ −1. In this case, c) from Lemma 11 (Right continuity of function IA(t,x)) in
time-slot −b − 1 must fall in some “OFF” period and this [10]. Hence, for any fixed ǫ > 0, there exists a δ > 0 such
“OFF” period must cover the time from time-slot −b−1 to that IA(t,b−c+δ) ≤ IA+(t,b−c)+ǫ = IA(t,b−c)+ǫ.
time-slot−1.LetTˆ(−b,C )bethelengthofthetimeperiod Supposethatfromtime-slot−t−bto−b−1,thetotalnumber
i,j
from the beginning of channel C ’s last state-change (”ON” of packet arrivals to the system is greater than or equal to
i,j
to ”OFF”) time-slot before time-slot −b to the beginning of nt+n(b−c+δ), and let p(b−c+δ) denote the probabilitythat
time-slot −b−1. Thus, we have: this event occurs. Then, from Cramer’s Theorem, we have
lim −1logp =I (t,b−c+δ)≤I (t,b−c)+ǫ.
P(C (t)=0 for all −b−1≤t≤−1) n→∞ n (b−c+δ) A A
1,j Clearly,thetotalnumberofpacketsthatareservedinanytime-
∞ slot is no greater than n. For any fixed δ, we have nδ ≥ 1
= P(C1,j(−b−1)=0,Tˆ(−b,C1,j)=τ,D ≥τ +b+1) for large enough n (when n ≥ 1). Hence, if the above event
δ
Xτ=0 occurs, at the end of time-slot −c−1, the system contains at
∞
= P(C (−b−1)=0)·P(Tˆ(−b,C )=τ, least one packet that arrived before time-slot −b.
1,j 1,j
Without loss of generality, we assume that this packet is in
τ=0
X
D ≥τ +b+1|C1,j(−b−1)=0) Q1. Now, assume that Q1 is disconnected from all n servers
from time-slot −c to −1, i.e., C (t)=0 for all 1≤j ≤n,
∞ 1,j
(=a)π P(Tˆ(−b,C )=τ, −c ≤ t ≤ −1. Then, at the beginning of time-slot 0, there is
0 1,j
still a packet that arrived before time-slot −b. Thus, we have
τ=0
X D ≥τ +b+1|C1,j(−b−1)=0) W(0)>b in this case. This implies χc2 ⊆{W(0)>b}.
∞ Case 1: c=0
=π0 P(D ≥τ +b+1|Tˆ(−b,C1,j)=τ,C1,j(−b−1)=0) In this case, the probability that event χc2 occurs can be
τ=0 computed by:
X
·P(Tˆ(−b,C )=τ|C (−b−1)=0)
1,j 1,j
∞ P(χ02)=p(b+δ). (14)
=π P(D ≥τ +b+1|D ≥τ +1)·
0
τ=0 And since χ0 ⊆{W(0)>b}, we have:
X 2
P(Tˆ(−b,C )=τ|C (−b−1)=0)
1,j 1,j
−1
∞ P(D ≥τ +b+1) limsup logP(W(0)>b)≤IA(t,b)+ǫ. (15)
≥π0 P(D ≥τ +1) · n→∞ n
τ=0
X
P(Tˆ(−b,C )=τ|C (−b−1)=0) Case 2: c≥1
1,j 1,j
Applyingthe same method we used to boundthe eventχ ,
1−F (τ +b) 1
D
≥π min . (10) we have:
0
τ∈{0,1,···} 1−FD(τ)
where (a) is from (2) since the system starts from −∞ and is P(C (t)=0 for all 1≤j ≤n,−c≤t≤−1)
1,j
now in the steady-state.
1−F (τ +c−1) n
D
Since all channelsare independentfromeach other,we can ≥ π0· min . (16)
easily obtain the probability for event χ1: (cid:16) τ∈{0,1,···} 1−FD(τ) (cid:17)
1−F (τ +b) n Note that the channel condition is independent from the
P(χ )≥ π · min D . (11)
1 0 τ∈{0,1,···} 1−FD(τ) arrivalprocess, and the probabilityχc2 occurscan be bounded
(cid:16) (cid:17) as:
Hence, we have:
1−F (τ +b) n 1−F (τ +c−1) n
P(W(0)>b)≥P(χ )≥ π · min D . P(χc)≥p π · min D .
1 0 τ∈{0,1,···} 1−FD(τ) 2 (b−c+δ) 0 τ∈{0,1,···} 1−FD(τ)
(cid:16) ((cid:17)12) (cid:16) (cid:17)(17)
Hence, we have: Combining events χ , χc, χc, the delay rate-function is
1 2 3
upper bounded by I∗(b):
U
−1
limsup n logP(W(0)>b) limsup−1logP(W(0)>b)
n→∞ n
n→∞
≤I (t,b−c)+ǫ−logπ
A 0 1−F (τ)
D
≤min log max −logπ ,
1−F (τ) 0
+log max D . (18) τ∈{0,1,···}1−FD(τ +b)
τ∈{0,1,···}1−FD(τ +c−1) min innf I (cid:0)(t,b), min { inf I (t,b−(cid:1) c)−logπ
(cid:16) (cid:17) t>tb A 1≤c≤b t>tb−c A 0
Since inequality (15) or (18) holds for any c ∈ (cid:8) 1−FD(τ)
+log max ,min{I (t ,b−c)
A b−c
{0,1,2,··· ,b}, for any t > tb−c, and any ǫ > 0, applying τ∈{0,1,···}1−FD(τ +c−1) c∈Ψb
theresultswe havein(15)and(18)andbylettingǫtendto0, (cid:0) 1−F ((cid:1)τ(cid:9))
D
−logπ +log max }
takingtheinfimumoverallt>tb−c,andtakingtheminimum 0 τ∈{0,1,···}1−FD(τ +c)
over all c∈{0,1,2,··· ,b}, we have: ,I∗(b). (cid:0) (cid:1) o (22)
U
−1 Next, we consider the case where L = 1. In this case, we
limsup logP(W(0)>b)
n→∞ n only need to consider event χ1, combining the case L = 1
and L > 1, we have shown that I (b) is an upper bound on
U
≤min inf I (t,b), min { inf I (t,b−c)
A A the delay rate-function.
t>tb c∈{1,2,···,b} t>tb−c
n
−logπ +log max 1−FD(τ) . (19) IV. ACHIEVABLE RATE-FUNCTION OFOPF POLICIES
0
τ∈{0,1,···}1−FD(τ +c−1) In this section, we aim to derive a non-trivial achievable
(cid:16) (cid:17)o
delayrate-functionoftheclassofOPFpolicies.First,westate
Event χc: Consider any fixed c ∈ Ψ , {c ∈ the definition of the class of OPF policies.
3 b
{1,2,··· ,b}|t ∈ Z+}. Suppose that from time-slot Definition 1: A scheduling policy P is said to be an OPF
b−c
−t −b to −b−1, the total number of packet arrivals to (oldest packets first) policy if in any time-slot, policy P can
b−c
the system is equal to nt +n(b−c) = nLt , and let servethekoldestpacketsinthesystemforthelargestpossible
b−c b−c
p′ denote the probability that this event occurs. Note that value of k ∈{1,2,··· ,n}.
(b−c)
the total number of packet arrivals to the system from time- We want to show that the achievable rate-function of any
slot −tb−c − b to −b−1 can never exceed nLtb−c. Then, OPF policy P is no smaller than I0(b), defined as:
from Cramer’s Theorem, we have lim −1logp′ =
n→∞ n (b−c) −logπ +b·log 1 , L=1
IA(tb−c,b−c). Clearly, the total number of packets that can I0(b)= 0 1−qˆ (23)
be served duringthe interval[−t −b,−c−1]is no greater (I∗(b), L>1
b−c 0
than n(t + b − c) = nLt . Suppose that there exists
b−c b−c where the parameter qˆis defined to be
one queue, say Q that is disconnected from all the servers
1
in time-slot −c−1. Then, at the end of time-slot −c−1, the qˆ,min{ min P(D =k+1), min 1−FU(k+1)}.
system contains at least one packet that arrived before time- k∈{0,1,···} 1−FD(k) k∈{0,1,···} 1−FU(k)
slot −b. Further, if queue Q is disconnected from all the n (24)
1
servers from time-slot −c−1 to −1. Then, at the beginning and
of time-slot 0, we have a packet in the system that arrived
1
before time-slot −b. Thus, in this case we have W(0) > b, I0∗(b),min b·log1−qˆ−logπ0,
and χc3 ⊆ {W(0) > b}. Note that the probability that event min{inf I (nt,b), min { inf I (t,b−c)
χc3 occurs can be bounded as t>tb A 1≤c≤b t>tb−c A
1
−logπ +(c−1)·log }}
P(χc)≥p′ π · min 1−FD(τ +c) n. (20) 0 1−qˆ
3 (b−c) 0 τ∈{0,1,···} 1−FD(τ) min{I (t ,b−c)−logπ +c·log 1 } . (25)
(cid:16) (cid:17) A b−c 0
c∈Ψb 1−qˆ
o
Since the above inequality holds for any c∈Ψb, by taking The analysis of delay rate-function follows a similar line
the minimum over all c∈Ψb, we have, for of argument as in the case of i.i.d. channels. Specifically,
we analyze the rate-function of the Frame Based Scheduling
limsup−1logP(W(0)>b) (FBS) policy and the perfect-matching policy and exploit
n→∞ n the dominance property of the OPF policies over both of
≤ min{I (t ,b−c)−logπ them. However, in the case of time-correlated channels, it
A b−c 0
c∈Ψb becomes more challenging to derive a good lower bound on
1−F (τ)
+log max D }. (21) the achievable rate-function. Since the channel has different
τ∈{0,1,···}1−FD(τ +c) behaviors(distributions)forstate-changeandstate-keeping.To
(cid:16) (cid:17)
addressthiskey challenge,we provetwo importantproperties Proof: We provide the proof in APPENDIX A.
of the FBS policy and the perfect-matching policy (Section Lemma 1 shows that given the past channel state informa-
IV.A), which will play a key role in the proof. We start by tion,a framecanbesuccessfullyservedwithhighprobability.
briefly describing the operations of the FBS policy and the Weareinterestedinfindinganupperboundontheprobability
perfect-matching policy. thatduringthetime interval[−t−b,−1],exactlyt+aframes
Underthe FBS policy,packetsare servedin unitof frames. can be successfully served by the FBS scheduling policy. We
Each frame is constructed according to a given operating have the following lemma:
parameter h, such that: 1) the difference of the arrival times Lemma 2: For all a≤b−1, we have:
of any two packets within a frame must be no greater than −1
h; and 2) the total number of packets in each frame is no P( X (τ)=t+a)
F
greater than n0 = n − Lh. In each time-slot, the packets τ=X−t−b
arrived at the beginning of this time-slot are filled into the ≤2t+b n 7H n 7bHe−n{−logπ0+(b−a−1)log1−1qˆ}.
last frame until any of the above two conditions are violated, π 1−qˆ
0
in which case a new frame will be opened. In each time-slot, (cid:0) (cid:1) (cid:0) (cid:1) (28)
the HOL frame can be served only if there exists a matching Proof: We provide the proof in APPENDIX B.
thatcanserveallthepacketsintheHOLframe.Otherwise,no Likewise, we define X as:
PM
packet will be served.In any time-slot, the FBS policy serves
1 if G has a perfect matching at time-slot t,
the HOL frame that contains the oldest (up to n ) packets
0 X (t)=
PM
withhighprobabilityforalargen.Undertheperfect-matching (0 otherwise.
policy, if a perfect matching can be found, i.e., every queue (29)
can be matched with a different server that is connected to
Similarly, we have the following lemma:
this queue, the HOL packet of every queue will be served
Lemma 3: Consider an n × n bipartite graph G, where
by the respective server determined by the perfect matching.
the time-varying connectivity has general time-correlation
Otherwise, none of the packets will be served. It has been
property. There exists an N > 0, for all n ≥ N the
shown in [10] that any OPF policy dominates the FBS policy PM PM
probability that G has no perfect matching can be bounded
and the perfect-matching policy, i.e., given the same packet
as:
arrivals and channel realization, any OPF policy will serve
every packetthat the FBS policy has served up to time t; and P(X (t)=0|S(t ),··· ,S(t ),S(t−1))
PM 1 d
the same for the perfect-matching policy. Therefore, the FBS ≤3ne−nlog1−1qˆ. (30)
and perfect-matchingpolicywill providelowerboundson the
delay rate-function that any OPF policy can achieve. Proof: We omit the proof here, as the same technique
used in the proof of Lemma 1 can be applied.
A. Properties of FBS and Perfect Matching Policy Similarly, it can be shown that for all a≤b−1:
In this subsection, we derive the following properties of
−1
FBS and perfectmatchingpolicy,which willlater be usedfor P( X (τ)=t+a)
PM
the rate-function analysis. For ease of presentation, we define
τ=−t−b
X
function XF(t) as: ≤2t+3bnbe−n{−logπ0+(b−a−1)log1−1qˆ} (31)
1 if a frame can be served in time-slot t Theaboveinequalityholdsforsufficientlylargen≥N .
PM
XF(t)= under FBS policy, (26) Note that the R. H. S. of inequalities (28) and (31) are both
0 otherwise. monotonically increasing with respect to a.
B. Achievable Rate-function
We havethefollowinglemmathatgivesa lowerboundonthe WefirstconsiderthecasewhereL>1.We needtopickan
probability that X (t)=1.
F appropriatechoiceforthevalueofparameterhforFBSbased
Lemma 1: Consider an n × n bipartite graph G, where
on the statistics of the arrival process. We fix δ < 2 and ǫ<
the time-varyingconnectivity has the generaltime-correlation 3
p/2.Then,fromAssumption2,thereexistsapositivefunction
propertydescribedinSectionII.Then,thereexistsanN >0,
F I (ǫ,δ) such that for all n≥ N (ǫ,δ) and t ≥T (ǫ,δ), we
such that for all n ≥ N the conditional probability that B B B
F have
X (t)=1 is bounded by:
F
l+t 1
P(XF(t)=1|S(t1),··· ,S(td),S(t−1)) P τ=l+1 {|At(τ)−pn|>ǫn} >δ <exp(−ntIB(ǫ,δ)).
≥1− n 7He−nlog1−1qˆ. (27) (cid:16)P (cid:17) (32)
1−qˆ where l is any arbitrary integer. Choose parameter h to be:
(cid:0) (cid:1)
for any positive integer d, t1 < t2 < ··· < td < t − 1, h=max T (ǫ,δ), 1 , 2I0(b) +1.
and any S(t1),··· ,S(td) and all n>NF, where S(·) is the B (p−ǫ)(1− 3δ) IB(ǫ,δ)
connectivity in the corresponding time-slot. n l 2 m l mo (33)
and define H =Lh. By combining part 1 and part 2, there exists a finite N ,
The reason for choosing this value of h will later become max{N ,N }, such that for all n≥N,
1 2
clearer. Note that in Assumption 2, the maximum number of
arrivals in a time-slot is L.
Let L(−b) be the last time before time-slot −b, when the P(W(0)>b)≤ C1n7(b+1)H +4 e−nI0(b). (44)
backlog is empty, i.e., all the queues have a queue-length of (cid:16) (cid:17)
zero.Also,letE bethesetofsamplepathssuchthatL(−b)=
t
−t−b−1 and W(0)>b under policy P. Then, we have If we take logarithm and limit as n goes to infinity, we
obtainliminf −1logP(W(0)>b)≥I (b), whichis the
∞ n→∞ n 0
P(W(0)>b)= P(E ). (34) desired result.
t
t=1
X
LetEF andEPM bethesetofsamplepathssuchthatgiven
t t
L(−b) = −t − b − 1, the event W(0) > b occurs under C. Part 1
the FBS policy and the perfect-matching policy, respectively.
Recall that policy P dominates both the FBS policy and the In this section, we want to show that there exists a finite
perfect-matching policy. Since each packet not served by the value N , such that for all n≥N , we have:
1 1
OPF policy is also not served by the FBS policy or perfect
matching policy, then for any t>0 we have
P ≤C n7(b+1)He−nI0(b). (45)
1 1
E ⊆EF ∩EPM. (35)
t t t
Recall that p is the mean arrival rate to a queue. Now, we
First we consider the case when t < t∗, let Eα denote the
choose any fixed real number pˆ∈(p,1), and fix a finite time t
set of sample paths in which there are at least n arrivals seen
t∗ as
byeveryh−1 time-slotsin the time interval[−t−b,−b−1].
t∗ ,max T1, II0B(Xb) ,max{tb−c|c∈Ψb} , (36) Le−t 1Etβ beXthe(τse)t>of0s.aAmcpcloerdpiantghstosu[c7h],tdhuate Ato(−tht−enbc0,−hob−ic1e) o−f
n l m o τ=−t−b F
where T is defined as: parameter h, we have
1
P
1−pˆ 6
T ,max T (pˆ−p, ), . (37)
1 B 6(L+2) 1−pˆ EF ⊆(Eα)c∪Eβ (46)
n l mo t t t
and
I ,min (1−pˆ)log1−1qˆ,I (pˆ−p, 1−pˆ ) . (38) and also there exists N3 > 0 and C2 > 0 such that for all
BX B n≥N ,
9 6(L+2) 3
n o
Hence, if we let
P(Eα)>1−C te−nI0(b). (47)
t 2
t∗
P , P(EF ∩EPM), (39)
1 t t
t=1 FortheprobabilityofEβ foreacht,wecanderiveanupper
X t
and boundon the probabilityof a large burst of arrivalsduring an
∞ interval of t time-slots [10].
P , P(EF ∩EPM). (40)
2 t t
t=t∗
X
From the relation in (35), we can bound P(E ) as: P(A(−t+1,0)>n0(t+x)) (48)
t
=P(A(−t+1,0)≥(n−H)(t+x)+1)
P(Et)≤P1+P2. (41) ≤e−nIA(t,x)e(H(t+x)−1)θ∗. (49)
Hence, we can divide the rate-function analysis into two
parts. In part 1, we show that there exists a finite N > 0
such that for all n≥N , we have 1 forallx∈[0,(L−1)t]andθ∗ ,max{θ1,θ2,··· ,θt∗}where
1 θ ,argmax [θ(t+x)−λ (θ)].Recallthatt = x .
t θ Ai(−t+1,0) x L−1
P1 ≤C1n7(b+1)He−nI0(b). (42) We first consider any t ∈ {1,2,··· ,t∗}\{tb−c′|c′ ∈ Ψb}.
Let c ∈ Z be the smallest integer such that t < t (i.e.,
Then, in part 2, we show that there exists a finite N2 > 0 t b−ct
t < t < t or c = 0 if t > t ). Then for z ∈
such that for all n≥N2, b−ct b−ct+1 t b
{c ,c +1,··· ,b}, we have t <t, thust+b−z <Lt and
t t b−z
P2 ≤4e−nI0(b). (43) forallz′ <ctwehavetb−z′ >tandthust+b−z′ >Lt.From
the properties we have derived for FBS and perfect matching Next, we need to deal with any tb−c ∈{tb−c′|c′ ∈Ψb}. In
policy in section IV-A, we have for all n≥N : this case, we can use the dominance property over FBS and
F
perfect matching policy, i.e., E ⊆ EF and E ⊆ EPM. Note
P(Eβ) t t t t
t that we have t = b−c >0, thus
A(−t−b,−b−1) b−c L−1
=P −X (−t−b,−1)>0
F
n
0
t+(cid:16)b−ct −1 (cid:17) tb−c+b−c=Ltb−c (53)
= P X (τ)=a P(A(−t−b,−b−1)>an )
F 0
Xa=0 (cid:16)τ=X−t−b (cid:17) If we define
−1
≤(t+b+1) max {P X (τ)=a
F
0≤a≤t+b−ct (cid:16)τ=X−t−b (cid:17) K1 ,P(EtFb−c ∩EtPb−Mc,A(−tb−c−b,−b−1)<(tb−c+b−c)n0)
×P(A(−t−b,−b−1)>an0)} K2 ,P(EtFb−c ∩EtPb−Mc,A(−tb−c−b,−b−1)≥(tb−c+b−c)n0)
−1 (54)
≤(t+b+1)max max P X (τ)=a ,
F
0≤a≤t−1
n (cid:8) (cid:0)τ=X−t−b (cid:1)(cid:9) According to union bound, we have:
P(A(−t−b,−b−1)>(t+b)n ),
0
−1
max P( XF(τ)=t+a) P(EtFb−c ∩EtPb−Mc)≤K1+K2 (55)
0≤a≤b−ct
τ=−t−b
(cid:8) X
×P(A(−t−b,−b−1)>(t+a)n ) (50) In particular,
0
Applying the results in section IV-A here,(cid:9)aond we know that
these values are monotonic increasing with respect to a. K1 ≤P(EtFb−c,A(−tb−c−b,−b−1)<(tb−c+b−c)n0)
P(Etβ) ≤1−P(Etαb−c)+K1′, (56)
≤(t+b+1)max 2t+b π0n(1(b−+1q)ˆ)b 7He−n{−logπ0+blog1−1qˆ},whereK1′ ,P(Etβb−c,A(−tb−c−b,−b−1)<(tb−c+b−c)n0).
max Pn(A(−t(cid:0)−b,−b−1(cid:1))>(t+a)n0), We have the similar result for K1′, n≥NF:
a∈{0,···,b−ct,ct=0}
−1
a∈{0,···m,ba−xct,ct6=0}{P(τ=−t−bXF(τ)=t+a) K1′ =P(A(−tb−c−nb,−b−1) − −1 XF(τ)>0,
X 0
×P(A(−t−b,−b−1)>(t+a)n )} τ=−Xtb−c−b
0 A(−t −b,−b−1)<(t +b−c)n )
b−c b−c 0
≤(t+b+1)2t+b n(b+1) 7He(H(t+b)−o1)θ∗ tb−c+b−c−1 −1
π0(1−qˆ)b ≤ P XF(τ)=a
×max{e−n{−logπ(cid:0)0+blog1−1qˆ},(cid:1)e−nIA(t,b), Xa=0 (cid:16) (cid:0)τ=−Xtb−c−b (cid:1)
max {e−n{IA(t,a)−logπ0+(b−a−1)log1−1qˆ}}} ×P A(−tb−c−b,−b−1)>an0
a∈{0,···,b−ct,ct6=0} (cid:0) −1 (cid:1)(cid:17)
≤C3n7(b+1)Hexpn−nmin{−logπ0+blog1−1 qˆ1,cmt=in0IA(t,b) ≤(tb−c+b−c)0≤a≤tmb−ac+xb−c−1P(cid:0)τ=−Xtb−c−bXF(τ)=a(cid:1)
min {IA(t,b−z)−logπ0+(z−1)log }} ×P A(−tb−c−b,−b−1)>an0
z∈{ct,ct+1,···,b} 1−qˆ 1
(51o) ≤C n7(b+1)H(cid:0)exp −nmin{−logπ +bl(cid:1)og ,
3 0
1−qˆ
awnhderweeCl3et=z(t=∗ +b−b+a1i)n2tt∗h+ebl(cid:16)asπt0(s11t−epqˆ).b(cid:17)H7eHncee(,Hf(ot∗r+abl)l−n1)θ≥∗, cm≤zi≤nb{IA(t,b−nz)−logπ0+(z−1)log1−1 qˆ}(}5o7)
N max{N ,N }, we have
4 3 F
P(EF ∩EPM)≤P(EF)
t t t For K , we have
2
≤1−P(Eα)+P(Eβ)
t t
≤C4n7(b+1)He−nI0(b) (52) K2 ,P(EtFb−c ∩EtPb−Mc,A(−tb−c−b,−b−1)≥(tb−c+b−c)n0)
w{1h,e2r,e··· ,Ct4∗}\{tb−,c′|c′ ∈mΨabx}{.C2t∗,C3} and t ∈ ≤P(EtPb−Mc,A(−tb−c−b,−b−1)≥(tb−c+b−c)n0)(58)
For all n≥N originalarrivalprocessA(·).TheresultingarrivalprocessAˆ(·)
PM
is simple, and has the following property:
K ≤P(EPM|A(−t −b,−b−1)≥(t +b−c)n )
2 tb−c b−c b−c 0 pˆn if A(τ)≤pˆn
×P(A(−tb−c−b,−b−1)≥(tb−c+b−c)n0) Aˆ(τ)= (64)
≤P(EPM|A(−t −b,−b−1)=Ln t ) (Ln if A(τ)>pˆn
tb−c b−c 0 b−c
×e−nIA(tb−c,b−c)e(H(tb−c+b−c)−1)θ∗ Since Aˆ(τ) ≥ A(τ), if we can find an upper bound on
Aˆ (−t−b,−b−1),thenitisalsoanupperboundonA (−t−
−1 F F
≤P( X (τ)<t +b−c) b,−b−1).
PM b−c
Consider any t ≥ t∗. Let B = {b ,b ,··· ,b } be the
τ=−Xtb−c−b 1 2 |B|
×e−nIA(tb−c,b−c)e(H(tb−c+b−c)−1)θ∗ set of time-slots in the interval from −t−b to −b−1 when
Aˆ(τ)=Ln. Given L(−b)=−t−b−1, from Corollary 2 of
tb−c+b−c−1 −1 [6] and the proof of Theorem 2 in [10], we have
≤ P( X (τ)=a)
PM n
Xa=0 τ=−Xtb−c−b AˆF(−t−b,−b−1)≤ n (pˆt+(L+2)|B|+1) (65)
×e−nIA(tb−c,b−c)e(H(tb−c+b−c)−1)θ∗ 0
From Assumption 2, we know that |B| could be arbitrary
≤(tb−c+b−c) small for large enough t and n. Suppose for n ≥ (2+pˆ)H,
1−pˆ
−1 t> 6 and |B|< 1−pˆ t
× max P( X (τ)=a) 1−p 6(L+2)
PM
a∈{0,···,tb−c+b−c−1} τ=−Xtb−c−b AF(−t−b,−b−1)≤AˆF(−t−b,−b−1)
×e−nIA(tb−c,b−c)e(H(tb−c+b−c)−1)θ∗ n
≤ (pˆt+(L+2)|B|+1)
≤(tb−c+b−c)2t+3be(H(tb−c+b−c)−1)θ∗nb n0
2+pˆ 1−pˆ
×e−n(IA(tb−c,b−c)−logπ0+clog1−1qˆ) < pˆt+ t
1+2pˆ 3
≤C5nbe−n(IA(tb−c,b−c)−logπ0+clog1−1qˆ) (59) ≤(2+pˆ)(cid:16)t. (cid:17) (66)
3
where C ,(t∗+b)2t∗+3be(H(t∗+b)−1)θ∗.
5 In fact, for n ≥ N , max{N (pˆ−p, 1−pˆ ), (2+pˆ)H}
{tCom|cb∈inΨing}reasnudltnsi≥n(N47),anmda(5x5{)N,w,eNha,vNefora}n,ytb−c ∈ and t≥T1. 6 B 6(L+2) 1−pˆ
b−c b 5 3 F PM
P(A (−t−b,−b−1)
F
P(EF ∩EPM)≤C n7(b+1)He−nI0(b), (60)
tb−c tb−c 6 2+pˆ
≥( )t,L(−b)=−t−b−1)
where C ,max{C t∗,C ,C }. 3
6 2 3 5 2+pˆ
Summing over t=1 to t=t∗, we have =1−P(AF(−t−b,−b−1)<( )t,
3
t∗ L(−b)=−t−b−1)
P = P(L(−b)=−t −b−1,E ) (61) 1−pˆ
1 b−c t ≤1−P(|B|≤ t) (67)
t=1 6(L+2)
X
≤C1n7(b+1)He−nI0(b), (62) ≤e−ntIB(pˆ−p,6(1L−+pˆ2)), (68)
for all n≥N1 ,max{N4,N5}, where C1 ,C6t∗. where the last inequality is from Assumption 2.
Next, let’s restate Lemma 1 in [7],
D. Part 2 Lemma 4: Let X , i = 1,2,··· be a sequence of binary
i
random variables such that for all i,
Inthissection,wewanttoshowthatthereexistsanN >0,
2
such that for n≥N2, P Xi =0|Xi′,i′ <i ≤c(n)e−nb (69)
P2 ≤4e−nI0(b) (63) where c(n) is (cid:16)a polynomial in n o(cid:17)f finite degree. Let N9 be
suchthatc(n)<en2b foralln>N9.Then,forany0<a<1,
Let R be the empty space in the end-of-line frame at the
0
oenfdnoewf timfraem-selost ct1re.aTtehdenf,rolemt AtiRFm0e(t-1sl,ott2)tdetnootet t,heinncluumdibnegr P t Xi <(1−a)t <e−tn3ab (70)
1 2
any partially-filled frame in time-slot t , but excluding the (cid:16)Xi=1 (cid:17)
2
partially-filled frame in time-slot t1. Also, let AF(t1,t2) = for all n>N10 :=max{a12b,N9}.
ARF0(t1,t2), if R0 = 0. As in the proof for Theorem 2 of Proof: Due to the choiceof N9, we have forall n>N9,
[a7rr]i,vafolrpraoncyesfisxAeˆd(·)r,eablynaudmdbinegr pˆex∈tra(d0u,1m)m, ywearcriovnaslisdetor tthhee P Xi =0|Xi′,i′ <i ≤e−n2b. (71)
(cid:16) (cid:17)
From Markov’s inequality we have for any r >0: Summing over all t ≥ t∗, we have for n ≥ N ,
2
max{N , log2 },
8 IBX
t
P Xi ≤(1−a)t =P e−rPti=1Xi ≥e−r(1−a)t P = ∞(cid:6) (EF(cid:7)∩EPM)
2 t t
(cid:16)Xi=1 (cid:17) (cid:16) (cid:17) t=t∗
E[e−rPti=1Xi] X∞
≤ (72)
e−r(1−a)t ≤ P(EtF)
t=t∗
X
∞
From the definition of conditional expectation, we have: ≤ P A (−t−b,−b−1)−X (−t−b,−1)>0,
F F
tX=t∗ (cid:16)
E[e−rPti=1Xi]=E[E[e−rPti=1Xi|X1,X2,··· ,Xt−1]] L(−b)=−t−b−1
=E[e−rPti=−11XiE[e−rXt|X1,X2,··· ,Xt−1]] ∞ (cid:17)
≤E[e−rPti=−11Xi] (1−e−n2b)+e−r−n2b ≤t=t∗2e−ntIBX
(cid:16) (cid:17)(73) ≤4Xe−nt∗IBX
≤4e−nI0(b) (78)
The last inequality comes from the condition (69), contin-
uing for t−1,t−2,··· down to 1, we have: where the last two inequalities are from the choice of N2 and
I .
BX
Combining part 1 and part 2, it is easy to verify
t
E[e−rPti=1Xi]≤ (1−e−n2b)+e−r−n2b (74) liminf −1logP(W(0)>b)≥I (b).
n→∞ n 0
(cid:16) (cid:17)
E. Delay Rate-Function Analysis for L=1
Applying the proof technique used in [12], we can get:
In this case, we want to show that for any fixed integer
b > 0, the rate-function achieved by OPF policy is greater
t
P Xi ≤(1−a)t <e−tD(a||e−n2b) (75) than blog1−1qˆ−logπ0.
(cid:16)Xi=1 (cid:17) Similarly, we can choose t′ as:
blog 1 −logπ
From the proofof Lemma 1 in [7], we know thatthe result t′ ,max T , 1−qˆ 0 (79)
1
I
holds for sufficiently large n>N . BX
10 n l mo
From Lemma 4, there exists N , max{N ,N }, such If we define probabilities
7 10 F
that for all n≥N ,
7 t′
P′ , P(EF ∩EPM) (80)
2+pˆ 1 t t
P(XF(−t−b,−b−1)<( 3 )t, Xt∞=1
L(−b)=−t−b−1) P′ , P(EF ∩EPM) (81)
2 t t
2+pˆ t=t′
≤P(X (−t−b,−b−1)<( )(t+b), X
F
3 Due to the dominance property over FBS policy and the
L(−b)=−t−b−1)
perfect-matching policy, we can spilt the delay violation
≤e−n(t+b)(1−9pˆlog1−1qˆ) probability P(W(0)>b) as:
≤e−nt(1−pˆ)l9og1−1qˆ (76) P(W(0)>b)≤P1′ +P2′. (82)
Likewise, we still dividethe proofinto two parts. In part1,
From(68)(76),foralln≥N ,max{N ,N }andt≥T we need to show that for all n≥N′
8 6 7 1 1
P A (−t−b,−b−1)−X (−t−b,−1)>0, P1′ ≤C1′nbe−n b1−1qˆ−logπ0 (83)
F F
(cid:0) (cid:1)
(cid:16) Consider t≤t′, the total packet arrivalsduring the interval
L(−b)=−t−b−1
of [−t−b,−b−1] cannot exceed nt when L = 1, hence if
≤1−(1−e−ntIB(pˆ−p,6(1L−+pˆ2))(cid:17))(1−e−nt(1−pˆ)l9og1−1qˆ) −τ=1−t−bXPM(τ)≥t then all packets that arrive before −b
havebeenservedattime-slot0,eventEPM doesnotoccur.We
≤2e−ntIBX (77) Phave EPM ⊆ L(−b)=−t−b−1, t−1 X (τ)<t
t τ=−t−b PM
(cid:8) P (cid:9)