Scheduling in Multi-hop Wireless Networks with Priorities

0.17 MB·English

by  Qiao Li

Scheduling in Multi-hop Wireless Networks with
Priorities
Qiao Li Rohit Negi
[email protected] [email protected]
Department of Electrical and Computer Engineering Department of Electrical and Computer Engineering
Carnegie Mellon University Carnegie Mellon University
5000 Forbes Ave., Pittsburgh, PA 15213 5000 Forbes Ave., Pittsburgh, PA 15213
9
0
Abstract—In this paper we consider prioritized maximal byconsideringspecific maximalschedulers,since the class of
0
2 scheduling in multi-hop wireless networks, where the scheduler maximal schedulers is quite broad. Recently, LQF scheduling
chooses a maximal independent set greedily according to a
has been shown to yield a much larger stability region than
n sequence specified by certain priorities. We show that if the
a probability distributions of the priorities are properly chosen, the worst case maximal scheduling. During scheduling, an
J we can achieve the optimal (maximum) stability region using LQF scheduler produces a maximal independent set greedily

9 an i.i.d random priority assignment process, for any set of following a sequence according to the queue length order of
1 arrival processes that satisfy Law of Large Numbers. The pre- each link, from the longest to the shortest. It is shown that in
computation of the priorities is, in general, NP-hard, but there
certain networks, where the topology satisfies the so called
] exists polynomial time approximation scheme (PTAS) to achieve
T “local-pooling” condition [8], LQF is optimal. For general
any fraction of the optimal stability region. We next focus on
I the simple case of static priority and specify a greedy priority networks,theperformanceofLQFschedulingcanbebounded
.
s assignment algorithm, which can achieve the same fraction of by its “local-pooling factor” [9], which is a function of the
c the optimal stabilityregion as the state of art result for Longest network topology and interference model. In the worst case,
[ QueueFirst (LQF)schedulers.Wealsoshow that thisalgorithm
LQFschedulingcanguaranteeanefficiencyratiobetween1/6
can be easily adapted to satisfy delay constraints in the large
1 and 1/3 in geometric networks with the K-hop interference
deviations regime, and therefore, supports Quality of Service
v
(QoS) for each link. model.
2
2 Keywords: Maximal scheduling, wireless networks, stabi- Promisingasitis, presentlytherearestillsomeopenissues
9 lity, delay, priority, large deviations. for the LQF scheduling. First of all, the stability region of
2 LQF is not well characterized, i.e., it is unclear whether a
. I. INTRODUCTION network is stable or not under an arrival process whose rate
1
0 Efficient scheduling algorithms in wireless networks have is outside the stability region of the the worst case maximal
9 been the subject of intensive research in the past few years. scheduler. Moreover, unlike maximal matching for switch
0 A fundamentalissue is that the optimalscheduling algorithm, scheduling [10], where a centralized controller is available,
:
v which can achieve a superset stability region of any schedu- distributed LQF scheduling in wireless networks is hard to
i ler, involves solving a Maximum Weighted Independent Set implement, and is subject to performance degradation due to
X
(MWIS) problem [1], which is, in general, NP-Hard [2]. This asynchronous queue length information updates. Finally, it is
r
a phenomenonisexaggeratedbytherequirementthattheMWIS hard to adapt LQF scheduling to support QoS in wireless
problem has to be solved in every time slot, which renders multi-hop networks, due to the intractability of its analysis,
it impossible to be implemented in applications due to the whichallowsitsperformanceto beunderstoodonlyin certain
highcomputingresourceconsumptions.Asanalternative,low networks with sufficient symmetries [11].
complexity suboptimal schedulers with guaranteed efficiency Realizing the limits of LQF scheduling, we try to improve
ratio are needed, among which, maximalscheduling has been the performance of maximal scheduling from a different
the focusof recentresearch.Similar to the maximalmatching perspective.Notethatin LQF scheduling,in each time slotn,
in the switch scheduling literature [10], maximal scheduling the scheduler picks the links following a sequence according
is a low complexity algorithm in wireless networks with to queue lengths. We generalize this to prioritized maximal
a constant efficiency ratio which is inversely proportional scheduling, where in each time slot the scheduler chooses a
to the maximum number of independent links in a link’s maximalschedulefollowingasequencespecifiedbyapriority
neighborhood [3], [5]. Not only is it sound theoretically, vector p(n), which is not necessarily the same as the queue
maximal scheduling is also well supported in practice by length orders. The priority process p(n) is, in general, a
efficient distributed algorithms with constant overheads (e.g. random process, and includes the priorities generated by the
see [4]). max-weight scheduling [1] and LQF scheduling as special
Furthermore,ithasbeenobservedthattheperformancegua- cases. Therefore, it can achieve the optimal stability region,
rantee for maximal schedulers can be improved significantly if the priorities are properly chosen. In this paper, we show
R 2
that a simple i.i.d random process p(n) suffices to achieve 1
the optimal stability region. Furthermore, the distribution of R5
p(n) can be computed, or approximated by a PTAS, if the
arrival rate value provided as of each link is available. In T5 1 5
T
applications, the arrival rates can be obtained from either 1 T
R 3 3
online estimation or the parameters by upper layer services, 6
R
such as digital voice or encoded video. Since arrival rates R2 3 4
T
parameters and pre-computation of the priorities are only 4 R
4
needed when the network topology changes, this combined T 6 2
6 T
priority pre-computationand maximal scheduling approach is 2
suitable for slowly changing wireless multi-hop networks.
(a) (b)
Next we consider a special class of prioritized maximal
schedulers, where p(n) is constant over time. In principle,
Fig.1. (a)isanarbitrarynetworkof6links,and(b)isitsinterferencegraph.
this is simpler than the LQF scheduler, where dynamic prio-
Theblacknodesformamaximumindependentset,whilethegraynodesform
rities are used, and its performance is not influenced by amaximalindependent set.
asynchronous queue length updates. Interestingly, we show
that if the priorities are properly chosen, we can achieve the
of communication links. A link i = (u,v) ∈ E only if node
sameefficiencyratioasLQFscheduling.Moreover,we givea
v is in the transmission range of node u. The interference is
specific lower bound characterization of the stability region
modeled by an undirected interference graph G = (V ,E ),
of prioritized maximal scheduling with a fixed p. Further, c c c
where V is the set of links, and E is the set of pairwise
we provide an algorithm that can successfully compute a c c
stabilizing priority p as long as the arrival rate vector is conflicts. Two links (i,j) ∈ Ec if and only if they are not
allowed to transmit together. For example, in the primary
inside the lower bound region of any fixed priority scheduler.
interferencemodel,the onlyconstraintis thata user node can
Therefore, it is easy to check the stability of the prioritized
not transmit and receive simultaneously. Therefore, two links
maximal scheduler for any arrival process that is within a
(i,j)∈E ifandonlyiftheyshareacommonnode,inwhich
certain guaranteed fraction of the optimal stability region. c
casetheschedulingproblemisreducedtoamatchingproblem
Finally we consider the delay constrained scheduling pro-
in G. This modelarises naturally in switch scheduling,and is
blem in wireless networks. With a major class of services
suitable for wireless networks using Bluetooth or FH-CDMA
being real time services, which are sensitive to congestion in
physical layers [3]. If secondary interference is considered,
thenetworkcausingbufferoverflows,theissueofdetermining
onemodelistheK-hopinterferencemodel[2],whichrequires
and guaranteeing delay in wireless networks is of equal, if
that two links within K hops can not transmit at the same
not more, importance compared to the issue of stability. It is
time (Note that the 802.11 DCF (Distributed Coordination
desirable to implement a scheduling algorithm such that the
Function)correspondstoK =2).SeeFig.1foranillustration
queue overflow probability for each link is below a certain
of the network topology and interference graph for a sample
small threshold. In this paper we try to solve this problem
network consisting of 6 links.
by using maximal scheduling with constant priority. We first
For each link i, define its neighbor set as N = {j :
analyzedelayguaranteesoftheworstcasemaximalscheduler, i
(i,j) ∈ E }. Thus, the transmission of link i is successful
as well as those of the prioritized maximal scheduler in the c
if and only if no link in N is transmitting. Denote the
large deviations regime, and formulate an upper bound on i
interference degree of link i as ∆ , which is the cardinality
the queue overflow probability. By exploring the similarity i
of the maximum independent set in the subgraph formed by
between the stability constrained and the delay constrained
{i}∪N . It has been shown that ∆=max ∆ is related
scheduling problems, we then propose a greedy priority i i∈Vc i
to the efficiency ratio of maximal scheduling [5]. To analyze
assignment algorithm, which is adapted from the priority
the performance ratio of the prioritized maximal scheduling,
assignment algorithm designed for the stability case, with the
we need the following definitions. Denote a node removal
guarantee that it will generate a satisfying priority p as long
sequence as r = (i ,i ,...,i ), which is a permutation
as the delay constraint can be achieved by some priority. 1 2 |Vc|
of (1,2,...,|V |), and consider removing the nodes from G
Thepaperisorganizedasfollows.SectionII introducesthe c c
according to the sequence r. Thus, i denotes the index of
system model. Section III describes the prioritized maximal k
the kth node that is removed. If a certain node i is removed
scheduler with random priorities. We consider the constant
in some step k (i.e., i = i), define δ(r) as the interference
priority case in Section IV, and adapt it to support delay k i
degree of link i when it gets removed in the subgraph at that
constraints in Section V. Section VI concludes this paper.
step (i.e., without nodes i ,i ,...,i ). Define
1 2 k−1
II. SYSTEMMODEL δ =minmaxδ(r),
i
A. Network Model r∈Π i∈Vc
where Π is the set of n! removal (permutation) sequences.
We model the topology of the network as a directed graph It has been shown that 1/δ is a lower bound on the local
G=(V,E),whereV isthesetofusernodes,andE istheset pooling factor for a network [9], and hence, the lower bound
3
on the efficiency ratio of the LQF scheduling. We will show paper). We begin our study of prioritized maximal schedulers
later that δ is also a lower bound on the efficiency ratio of by considering the stability region of prioritized maximal
maximal scheduling with constant priority. schedulers with random priorities, in the next section.
B. Traffic Model III. SCHEDULING WITH RANDOMPRIORITIES
We assume the network is time synchronized, and that the Inthissection,weanalyzethestabilityregionofprioritized
transmitterofeach linki isassociated with anexternalarrival maximal schedulers when the priority vectors p(n) form
process Ai(n), which is the cumulative number of packet a random process. First of all, as a subclass(cid:0)of m(cid:1)aximal
arrivals during the first n time slots. We further assume that schedulers,any prioritizedmaximalschedulercan supportthe
the arrivals happen at the end of each time slot, and in each following region [5]:
timeslot,thepacketarrivalsareboundedabovebyaconstant.
A ={a:a + a <1,1≤i≤|V |}. (4)
min i j c
The arrival processes are also subject to the Strong Law of
Large Numbers (SLLN), i.e., jX∈Ni
In order to obtain a better bound, we need to explore the
lim A (n)/n=a for all i∈V , (1) impactof the priorityvectors. In fact, the stability regionof a
i i c
n→∞
priorityscheduleris closely related to the choice of priorities.
with probability1 (w.p.1),where the constanta is the arrival
i For example, consider a star shaped interference graph, with
rate for link i. Note that these assumptions on the arrival
center link 1 and n−1 outer links. If we assign the priorities
process are quite mild, as the arrivals can be dependent over
suchthatthe centerlink1 alwayshasthelowestpriority,then
time slots and also among different links. As a consequence,
for any arrival rate a + n a >1, we can find an arrival
in this paper we only focus on single hop traffic, since it 1 j=2 j
process which makes linkP1 unstable. Therefore, the stability
is straightforward to generalize these results to the multi-
regioncoincideswithA .Asanotherexample,considerthe
hop case (since departures from one queue may very well min
followingchoice of priorities:in each time slot, we first solve
be arrivals to another queue, in our mild assumption).
the MWIS problem
C. Scheduling Model m⋆(n)=argmax Q(n)Tm. (5)
m∈M
In eachtime slot, a schedulerπ choosesan independentset Then,weassignalllinkswithm⋆(n)=1thehighestpriorities
i
for transmission. In this paper, we are interested in maximal (the order does not matter) while the links with m⋆(n) = 0
i
scheduling,whereascheduledsetoflinkshasthepropertythat are assigned the remaining (lowest) priorities (the order does
no other link can be added without violating the interference notmatter).Sincem⋆(n)isanindependentset,alllinksinthe
constraints.Denotethefamilyofmaximalindependentsetsas independentsethavethelocalhighestpriority,i.e.,thehighest
M. We will also treat a maximal independent set m ∈ M priority in that specific neighborhood. After scheduling, the
as a column vector such that mi = 1 if i ∈ m, and mi = scheduled links maximize the weighted sum (5), and thus
0 otherwise, as long as there is no confusion. The queueing A , in this example, can be achieved [1]. However, the
max
dynamics at each link can be expressed as computation of priorities is prohibitive, since (5) requires
Q (n)=Q (0)+A (n)−D (n) (2) solving an NP-hard problem in every time slot. In fact, a
i i i i
much simpler prioritized scheduler can also achieve A ,
max
where Q (n) is the queue length at link i at the end of time
i by utilizing only an i.i.d random process p(n) .
slot n and D (n) is the cumulative departures during the first
i We need the following lemma. (cid:0) (cid:1)
n time slots. Clearly, D (n) must ensure that Q (n)≥0.
i i Lemma 1: For any arrivalrate vector a and an i.i.d random
Forprioritizedmaximalschedulers,duringschedulinginti-
process p(n) of priority vectors, if there exists a S =
meslotn,anindependentsetisproduced,asdescribedabove,
guided by a priority vector p(n), where p (n) represents the (S1,S2,.(cid:0)..,Sn(cid:1)) where set Si ⊆Ni, such that
i
priority of link i. We assume that link i has higher priority a + a <Pr(p <p ,∀j ∈S ), (6)
i j i j i
than link j if and only if pi(n)<pj(n) and that all priorities j∈XNi/Si
are distinct. Thus, the scheduler choosesdeparturesfollowing
a is stable underthe maximalschedulerwith process p(n) .
a sequencespecifiedbyp, fromthe highesttothe lowest,and
schedules link i for departure if it has a nonempty queue and Proof: See in Appendix. (cid:0) (cid:4)(cid:1)
Thus, this lemma provide a lower bound on the stability
no higher priority neighbor has already been scheduled.
region achievable by p(n) . The following is the main
The throughput of a scheduler π is represented by its
stability region A , which is the set of stable arrival rate theorem of this section(cid:0). It us(cid:1)es Lemma 1 to show that any
π
a∈A can be stabilized by a properly chosen p(n) .
vectors under π. We define stability to be rate stability, i.e., max
Theorem 1: In an arbitrary network with n nod(cid:0)es, for(cid:1)any
lim A (n)/n= lim D (n)/n=a w.p.1 (3)
i i i a∈A , there exists a stabilizing prioritizedmaximalsche-
n→∞ n→∞ max
foralli∈V .Ithasbeenshownthatthemax-weightscheduler duler with an i.i.d process p(n) . Furthermore, the support
c
in[1]canachievethemaximumstabilityregion,Amax,which of p(n) consists of at mo(cid:0)st n+(cid:1)1 elements.
is the interior of the convex hull of the maximal independent P(cid:0)roof:(cid:1)SinceaisinanopensetAmax,thereexistsanǫ>0
sets in G (we do not consider the boundary points in this suchthata+ǫe∈A ,wheree=(1,1,...,1)T.According
c max
4
to the Carathe´odory theorem, a+ǫe can be represented as a Ithasbeenshownthatanǫ-approximatesolutionfor(9)can
convex combination of at most n+1 maximal independent be obtained by solving Θ(ǫ)-approximate solutions for (11)
sets, i.e., O(|V |(ǫ−2+log|V |)) times, see [16] for an approximation
n+1 c c
a+ǫe= θ m(k), m(k) ∈M (7) algorithmusinglogarithmicpotentialreduction.Theonlything
k
Xk=1 remains is to check that (11) is a MWIS problem. Note
where θ (cid:23) 0 and eTθ = 1. We associate a priority vector that (11) is a linear program, with a simplex constraint, and
p(k) with each m(k) such that if m(k) = 1, i has the local therefore, the optimal value is attained at a vertex, i.e.,
i
highest priority, and other priorities are arbitrary. During the f(λ)= maxw(λ)Tm (12)
scheduling in time slot n, the scheduler chooses p(k) with m∈M
probability θk. Therefore, we have wthheetrheeworie(mλ)h=oldatsiλ+.iǫ, showingthatf(λ) findsa MWIS.Thu(cid:4)s,
Pr(p <p ,∀j ∈N ) ≥ θ
i j i k
For most network and interference models, there exists a
k:mXi(k)=1 PTAS for the MWIS problem, see [7] for an example using
= θ m(k) thegraphpartitioningtechniqueingeometricgraphs.Notethat
k i
Xk one can also achieve similar performance by using a PTAS
= ai+ǫ for the max-weight scheduler during the scheduling [12].
and the stability follows from Lemma 1 by setting S = N Comparedtotheirapproach,ourmethodhastheadvantagethat
i i
for all 1≤i≤n. (cid:4) theschedulingphasehaslowcomplexity,whichisindependent
Intuitively,Theorem1showsthatthetraditionalcoloringap- of approximation ratio, since the approximation algorithm is
proach,whichisusedforschedulinginnetworkswithconstant only executed during the priorities pre-computation phase.
traffics, is still applicable in the presence of stochastic packet
dynamics.Furthermore,thehardnessremainsalmostthesame, IV. SCHEDULING WITHCONSTANT PRIORITY
since (7) is equivalentto a coloringproblem.Note (7) is only In this section we consider a special class of prioritized
executed during the pre-computation phase. Thus, compared schedulers, where p(n) = p is a constant vector, i.e., the
to the max-weightscheduler, which essentially solves an NP- priority for each link is fixed. This is certainly easier to
hard problem in every time slot, the burden on the prioritized implement, since it does not require a global coordination
maximalschedulerduringtheschedulingphaseissignificantly variable, specifying the priority vector applying in each time
relieved. Finally, we show that there exists PTAS for the pre- slot. We will focus on both of the priority pre-computation
computation of the priorities. and the performance guarantees.
Theorem2:Givenanetworkandinterferencemodel,ifthere
existsaPTASfortheMWISproblem,thenthereexistsaPTAS A. Priority Assignment
for computing the the stabilizing priorities. In order to search for the optimal priority, we need to
Proof: Suppose we have a+ǫ′e ∈ (1−ǫ)Amax for some analyzethe associated stability regionfor an arbitrarypriority
ǫ′ >0.Theprioritiescanbeobtainedbysolvingthefollowing vector p. By setting S = {j ∈ N : p < p } for each link
i i i j
optimization: i∈V and apply Lemma 1, we get a lower bound
c
min eTx
x ai+ aj1(pi−pj)<1 ∀i∈Vc (13)
subject to Mx(cid:23)a+ǫ′e,x(cid:23)0 (8) jX∈Ni
where 1(·) is the indicator function
where M is the matrixwhose columnscorrespondto all m∈
1 if x>0
M.Thisproblemcanbesolvedbybinarysearchovert,where 1(x)= (14)
in each step we assume eTx=t and solve the following (cid:26) 0 otherwise
Denote this stability region as A and the stability region
max s p
x,s achieved by any constant priority as A = ∪ A . We are
p p
subject to (Mx)i ≥s 1≤i≤|V | interested in computing a stabilizing priority for a given
a +ǫ′ c a ∈ A. We will first propose the algorithm, and then show
i
eTx=t,x(cid:23)0. (9) its optimality. See also in [6]. Note that the proof below is
new, compared to [6], and so is the condition that A⊆A′.
By strong duality, this can be solved by the dual problem
min f(λ)
Algorithm 1 Stable-Priority (G ,a)
λ c
subject to λ(cid:23)0,eTλ=1 (10) 1: for k =1 to |Vc| do
where f(λ) is the optimal value of the following problem 2: s←argmini∈Vc{ai+ j∈Niaj};
3: ps ←n+1−k; P
max n λ (Mx)i 1≤i≤|V | 4: Remove s from Vc and its incident edges from Ec;
x Xi=1 iai+ǫ′ c 5: end for
6: return p
subject to eTx=t,x(cid:23)0. (11)
In each step, the algorithmchoosesnodes, the center node d(r) is the degree of node i when it gets removed. Suppose5
k k
of a neighborhood,and assign it the next lowest priority. that the above maximum is achieved at node i . When i
k⋆ k⋆
The following lemma is the key to the proof. gets removed,its neighborhoodneeds at least ⌈d(r)+1⌉ colors
δ(r)
Lemma 2: (Alternate definition of A) For the A defined (in any optimal coloring), and at most d(r) + 1 colors in
above, we have A = A′ where A′ is any set of rate vectors the sequential coloring. Therefore the efficiency ratio for this
in Rn that satisfies the following conditions: choice of sequence is lower bounded by 1/δ(r). In the worst
+
1)(Coordinate-convex)Ifa∈A′ anda′ (cid:22)a, thena′ ∈A′. case, we have the efficiency ratio of 1/∆ = 1/max δ(r),
r∈Π
2) (Worst-case stable) For any nonzeroa∈A′, there exists which corresponds to the efficiency ratio of the worst case
1≤i≤n such that a >0 and a + a <1. maximal scheduler [5]. On the other hand, if we choose the
i i j∈Ni j
Proof:Supposea∈A,thenthereexPistspsuchthata∈Ap. sequence properly, we can achieve 1/δ = 1/minr∈Πδ(r). In
Thus, property (1) holds from the fact that A is a polytope, thefollowingweshowthattheefficiencyratio1/δcan,indeed,
p
and thus contains any a′ (cid:22)a. For this priority p, denote i as be achieved by A.
thelowestprioritylinkwithpositivearrivalrate.Then,S =φ, Theorem 4: 1A ⊆A.
i δ max
so that (13) is the same as property (2). Therefore we have Proof: Denote the removal sequence which achieves δ as
A⊆A′. r = (i ,i ,...,i ). For any a ∈ A, define a sequence of
1 2 |Vc|
Suppose a(1) ∈A′. From property (2) we have arrival rate vectors as follows:
a(1)+ a(1) <1 a(1) = a
i1 j
for some i . Assign p j=∈XN|iV1 |, and form a new rate vector a(k) = a(k−1)−ai(kk−1)eik 2≤k ≤|Vc|
a(2) from a1(1) by settiin1g a(1)c= 0 (which induces a reduced where eik is the vector that is all-zero except for an 1 in the
graph).Fromproperty(1),ai(12) isalsoinA,andfromproperty ikth entry. In other words, a(k) is obtained from a(k−1) by
setting the i th entry to zero. We have for any k
(2) there exists i 6=i such that a(2) >0 and k
2 1 i2 a(k)+ a(k) <δ (15)
a(2)+ a(2) <1. ik j
i2 j∈XNi2 j j∈XNik
duetothefactthatatmostδlinks’sini ’sneighborhoodwhen
We then set p = |V |−1 and repeat the similar procedure k
i2 c it gets removed can transmit in each time slot. Now for the
until the highest priority,i.e., priority (1) is assigned. Thus, a
priority assignment such that p =n+1−k for 1≤k ≤n,
p vector is obtained. We claim that a ∈ Ap, due to the fact ik
we have
that
a + a 1(p −p )=a(k)+ a(k) <δ (16)
aik +j∈XNikaj1(pi−pj)=a(ikk)+j∈XNika(jk) <1 for ailkl i .j∈XTNhiuks wje haikve 1aj∈A ikand jth∈XeNicklaijm holds. (cid:4)
for all 1 ≤ k ≤ |V |. Therefore we have A′ ⊆ A, and the k δ p
c Interestingly, the efficiency ratio of 1/δ has been shown to
lemma holds. (cid:4)
also be a lower bound on the local pooling factor [9] for an
From the above proof, it is straightforward to prove the
interference graph, which is the efficiency ratio promised by
following theorem:
LQF scheduling. This is no coincidence, since in the special
Theorem 3: If a ∈ A, Stable-Priority will return a p such
case of constant fluid arrivals with equal rate, both belong
that a∈A .
p to the family of greedy coloring algorithms, and hence have
similar performance guarantees.
B. Performance Guarantee
We next apply this result to certain networks and interfe-
Wenextanalyzetheperformanceofthemaximalscheduling
rence models to obtain examples of worst case guarantees.
with the constantpriorityp generatedbyStable-Priority.This
1) K-hop Interference:
can bemeasuredbythe efficiencyratioγ, whichis definedas
We first consider the K-hop interference model, which
can be used for a large class of networks. For instance, the
γ =sup{σ :σA ∈A}.
max
ubiquitous IEEE 802.11 DCF is usually modeled as a K-
To gain insight, we first consider the efficiency ratio of hop interference model with K = 2, due to the RTS-CTS
greedy graph vertex coloring, where a sequence of vertices message exchanges. In the K-hop interference model, two
are removed first and the colors are assigned in the reverse links (i,j) ∈ E if and only if the distance between one
c
order. In fact, if the arrival processes have equal rate, and node in link i (transmitter or receiver) and one node in link
can be approximated by constant fluids, then the scheduling j (transmitter of receiver) is less than a threshold Kr, where
problem is reduced to a graph coloring problem, where the r is the transmission range of a node. Summarizingthe result
rate is inversely proportional to the number of colors used. in [9], we have the following corollary.
Denotetheremovalsequenceinagreedycoloringalgorithmas Corollary 1: In a geometric graph with K-hop interference
r=(i ,i ,...,i ).Whenthenodesarecoloredinthereverse model,the prioritizedmaximalschedulingwith constantprio-
1 2 n
order,weneedatmostd(r)+1=. max d(r)+1colors,where rity can achieve an efficiency ratio between 1/6 and 1/3.
k k
6
2) PHY-Graph: For each link i, the log moment generation function
We next consider the PHY-Graph [17], which is a more
1
realistic interferencemodelexplicitly incorporatingthe physi- Λi(θ)= lim logE(eθAi(n)) (18)
n→∞n
cal layer parameters, i.e., the signal-to-interference-plus-noise
exists for all θ, from which we can get the rate function via
ratio (SINR). In the PHY-Graph model, two links (i,j)∈E
c
the Legendre transform
if and only if either of the following is true: 1) the distance
between the transmitter of link i and the receiver of link j Λ∗(µ)=sup(θµ−Λ (θ)). (19)
i i
is less than clj, or 2) the distance between the transmitter θ
of link j and the receiver of link i is less than cl , where
i A2: (The sample path LDP applies)
c=(SNRt)κ1 isafunctionoftheSINRthresholdSNRt and For all s ∈ N,ǫ ,ǫ > 0 and for every scalars
1 2
path loss exponent κ, and l ,l are the link lengths of i,j,
i j b ,b ,...,b , there exists N > 0, such that for all n > N
0 1 s−1
respectively. From their coloring bound, which is essentially
and all 1=k ≤k ≤...≤k =n,
0 1 s
an upper bound on δ, we have the following corollary.
Corollary 2: In a geometric graph with PHY-Graph inter- Pi ≥e−nǫ2+Psj=−01(kj+1−kj)Λ∗(bj) (20)
ference model, for fixed κ, the efficiency ratio of prioritized
where P is the probability of the following event:
maximal scheduling with constant priority is a nondecreasing i
functionofSNR .Particularly,whentheSNR issufficiently
t t {|A (k )−A (k )−(k −k )b |≤ǫ n,0≤j ≤s}
i j+1 i j j+1 j j 1
high, the efficiency ratio is bounded above by 1/7.
Intuitively,thisisthe eventthatthearrivalprocessisconstrai-
V. DELAY-AWARESCHEDULING ned to lie within a tube around s linear segments of slopes
In this section we try to adapt the prioritized maximal b ,b ,...,b , respectively.
0 1 s−1
schedulingtosupportQoSconstraints,intheformofanupper A3: (Convex dual analog of the sample path LDP)
boundon queue-overflowprobability.Specifically, we assume For all s∈N, there exists N >0 and a function g(·) with
that the system constraint on the queue over flow probability 0 ≤ g(x) < ∞ when x > 0, such that for all n ≥ N and all
for link i is 1=k ≤k ≤...≤k =n,
0 1 s
s
Pr(Q (0)>B )≤ǫ
i i E(exp(θTZ))≤exp{ (k −k )Λ(θ )+g(θ )}
j j−1 j j
where Qi(0) is the stationary queue length and Bi is buffer Xj=1
capacity.ForlargeB , we canassume thatthebuffercapacity
i where θ = (θ ,θ ,...,θ ) and Z = (A (k ),A (k ) −
1 2 s i 0 i 2
if infinite and approximatethe queueoverflowproblemas the
A (k ),...,A (k )−A (k )).
following: calculate θ∗, where i 1 i s i s−1
i It should be noted that the processes satisfying the above
−1 three properties form a broad class, which includes most
θ∗ =liminf logPr(Q (0)>B )≥ǫ′. (17)
i Bi→∞ Bi i i common models for bursty traffic in realistic networks, such
as renewal and Markov-modulated processes. For detailed
Here ǫ=exp(−B ǫ′) while θ∗ represents the delay exponent
i i discussions, see [13].
of Q in the large deviations regime. In order to measure the
i
delay performance of a scheduler π in the large deviations
B. Worst Case Maximal Scheduling
regime,usinga similarnotationto thestability regionA ,we
π
We first consider the delay region of an arbitrary maximal
define the delay region
scheduler.Inthefollowing,weanalyzethedelayexponentfor
Θ ={θ ∈Rn :θ (cid:22)θ∗} afixedlinki.Denotetheoriginalnetworkwithachosenarbi-
π +
trary maximal scheduler as system S. Since it is analytically
where θ∗ is defined as Eqn. (17), i.e., the set of guaranteed
intractabletogiveanexactcharacterizationofthedelayregion,
delay exponents under π. In contrast to the stability region,
wewanttocreateasystemS′suchthat1)itisrelativelyeasier
where any the arrival process satisfying SLLN applies, the
to analyze the delay region in S′ and that 2) the delay region
delay region is quite sensitive to the arrival process model,
of i in S′ is a lower bound of that in S. Note that due to
especiallytheburstinessoftheprocess.Therefore,we haveto
maximal scheduling, when link i has a nonempty queue, we
first re-define the arrival process model before analyzing the
canguaranteethat, ineachsuchtime slot,thereisatleastone
delay region of maximal schedulers.
departure in i’s neighborhood (which includes i). Therefore
A. Assumptions an obvious dominant system for S is a server with service
capacity 1, which is fed by one queue of queue length equal
We need to slightly modify the arrival process model, so
to the sum of the queues in {i}∪N . We can lower bound
as to apply the Large Deviations Principle (LDP). We now i
the delay exponent of link i in S using the following bound
assume that the arrival processes are independent among the
based on S′:
links. For each link i, we will follow the model in [14].
Specifically, we have the following assumptions: Pr(Q (0)≥x)≤Pr(Q′(0)+ Q′(0)≥x),
i i j
A1: (The Ga¨tner-Ellis theorem applies) jX∈Ni
7
wheretheRHScorrespondstothesinglequeueinS′ (equalto when S does not. Thus from s >s′, we have D (s′ −1)=
1 1 i 1
the sum of queuesin S) and can be calculated using standard D′(s′ −1)=0 and
i 1
large deviations techniques. However, empirical results show
D (s′ −1)≥ D′(s′ −1),
that this bound is quite loose, especially in the cases where j 1 j 1
some neighbors in N are likely to grow large, whereas Q jX∈Ni jX∈Ni
i i following Eqn. (22). Therefore, we have
is not. To get a better lower bound, we need to analyze the
queuelengthdynamicsoflinkialone.Sinceweareanalyzing 0(=a) Q′(s′ −1) = A′(s′ −1)− D′(s′ −1)
j 1 j 1 j 1
the worst case maximal scheduler, we assume a prioritized jX∈Ni jX∈Ni jX∈Ni
maximalscheduler where link i has the globallowest priority ≥ A (s′ −1)− D (s′ −1)
and the other links in N have the global highest priority, so j 1 j 1
that i can not transmit ifiany link in N has nonemptyqueue. jX∈Ni jX∈Ni
i (b)
Based on this, we create a system S′ consisting of a clique = Q (s′ −1) > 0. (23)
j 1
formedby the nodes in {i}∪Ni, and i is assigned the lowest jX∈Ni
priority.The prioritiesof the otherlinksare arbitrary.We will Equality(a)isexplainedasfollows.SinceitransmitsinS′ at
show that S′ is a dominant system over S in the following time s′, no other neighbor is contending for that slot (neigh-
1
lemma.Intuitively,weareassumingtheworstcasecorrelations bors have higher priority). It must be that all the neighbors
amongtheneighborsinNi,suchthattheavailabletimeslotfor have empty queues in the previous slot. The inequality (b) is
link i is minimized. This, together with the scheduler which explained as follows. Since i does not transmit in time slot
assigns link i the lowest priority, can be shown to dominate s′ in S (we know it transmits at s > s′), that must be
1 1 1
the original system S with any scheduler. We will prove this due to some transmitting neighbor occupying that slot. Since
domination in the following lemma. arrivalsoccur at the end of each slot, that neighbormusthave
Lemma 3: For any x≥0 and n≥0, if Qi(0)=Q′i(0), we a nonempty queue in the previous slot s′1−1. Since this is a
have contradiction, we have proved the case for k =1.
Pr(Qi(n)≥x)≤Pr(Q′i(n)≥x) (21) Now suppose it is true for up to k−1 packets. For the kth
packet,anytimeslotbetweent′+1tos′−1isoccupiedbythe
where Q (n), and Q′(n) are the queue lengths of link i at k k
i i neighborsinS′.Ifs >s′,wehaves ≤s′ <s′ <s ,
time slot n in S and S′, respectively. k k k−1 k−1 k k
and
Proof: We will show that in an arbitrary sample path ω
D (s′ −1)=D′(s′ −1)=k−1 (24)
with the same arrival processes and initial queue lengths for i k i k
both systems, we have Q (n,ω) ≤ Q′(n,ω) for all n ≥ 0.
i i Similar to (23), we get
We assume both systems use the first-in-first-out policy, and
denote the arrival time and departure time of the kth packet 0= Q′j(s′k−1) = A′j(s′k−1)−Dj′(s′k−1)
at link i in S as tk and sk, respectively. Similarly for S′ we jX∈Ni jX∈Ni
use t′ and s′. From the queueing equations in both systems, ≥ A (s′ −1)−D (s′ −1)
k k j k j k
we have jX∈Ni
∞ ∞ = Q (s′ −1)>0,
Q (n) = Q (0)+ 1 (t )− 1 (s ) j k
i i Xk=1 [0,n] k kX=1 [0,n] k i.e., a contradiction. Thus thejX∈leNmima holds. (cid:4)
∞ ∞
Q′(n) = Q′(0)+ 1 (t′)− 1 (s′), Having shown the the dominance of S′, we next analyze
i i [0,n] k [0,n] k
Xk=1 kX=1 the delay performanceof link i in S′ in the followinglemma.
Lemma 4: For link i in system S′, we have
where1 (x)istheindicatorfunctionthatx∈S.Notethatfor
S
−1
danuye stoamthpeleapsasuthmωpt,ioQnis(.0T,ωhu)s=toQs′ih(0ow,ω)thaantdQtk((nω))≤=Qt′k′((ωn)) xl→im∞ x logPr(Q′i(t)≥x)=θi∗ (25)
i i
w.p.1foralln≥0,itissufficienttoshowthatsk(ω)≤s′k(ω) where θi∗ is the largest root of the following equation
for all k ≥1 and ω. For simplicity we will drop the index ω
Λ (θ)+ inf [ Λ (u)−u]=0 (26)
in the following and proof by contradiction. i j
Wewilluseinductiononk.Supposethatthisisnottruefor Proof:Weconstruct0≤aus≤ysθtejXm∈NSi′′equivalenttoS′,consisting
thefirstpacket,i.e.,s >s′.InS′,duetomaximalscheduling,
1 1 of two nodes, where node 1 corresponds to link i and node
eachtimeslotfromt′+1tos′−1isoccupiedbytheneighbors
1 1 2 corresponds to the links in Ni. Thus, the queue of node 2
of i (which is why the packet in i has not been served yet).
is the sum of the queues of link i’s neighbors. We assume
Note that we always have
that the arrival process to node 2 is equal to the sum arrivals
Di(n)+ Dj(n)≥Di′(n)+ Dj′(n), (22) in Ni, and that there is an edge between 1 and 2. During
jX∈Ni jX∈Ni the scheduling, we assign higher priority to node 2. Note that
foranyn≥1sincebothsystemsarework-conservingandthe from link i’s perspective, S′ and S′′ yield the same queueing
linksinS′haveedgesbetweenthem(theyformaclique),even dynamics.
8
For a 2-node system, accordingto [14], the delay exponent lowest priority link i with θ > 0 according to p. We have
i
θ∗ for queue 1 is given by the largest root of the equation θ <θ∗ where θ∗ is the largest rootof the followingequation
i i i
Λ1(θ)+ inf [Λ2(u)−u]=0. (27) Λi(θi)+ inf [ Λj(u)1(pi−pj)−u]=0. (30)
0≤u≤θ 0≤u≤θi jX∈Ni
Thusthe lemmaholdsfromthe factthatthearrivalprocessof Thusproperty(2) holdsfrom the fact that 1(p −p )=1(θ )
i j j
node 2 is A2(n)= j∈NiAj(n). (cid:4) for all j ∈Ni. Hence we have Θ⊆Θ′.
FromtheabovediPscussions,wehavethefollowingtheorem: Nowsupposeθ(1) ∈Θ′.Wefirstassignthelowestpriorities
Theorem5:Ina networkwherethearrivalprocessessatisfy to any link i such that θ(1) = 0. According to property (2),
i
wAh1e-Are3,Θwmeinh=av{eθΘ∈mRin+n⊆: θΘπ≺fθo∗r}aannydmθai∗xiims tahlesclahregdeuslterroπot, wdeesccraibnefidnidnEliqnnk.i(129s)u.cWhethaastsi0gn<i1θi(t11h)e<cuθri(r11e)n∗tlwowheersetpθi(r11io)∗ritiys
of Eqn. (26). availableandsetθ(1) =0togetanewvectorθ(2) ∈Θ.Repeat
This theorem provides a lower bound on the delay region the above processi1until all the links are assigned priorities.
of any maximal scheduler. Denotingthe resulting priorityvectoras p, it is easy to check
that θ(1) ∈ Θ , and hence Θ′ ⊆ Θ. Therefore the lemma
p
C. Delay-Aware Maximal Scheduling holds. (cid:4)
Based on the above lemma, we can construct a priority
Having obtained the asymptotic limit of the worst case
assignment algorithm, such that whenever θ ∈ Θ, the al-
maximal scheduler, we next consider improving the guaran-
gorithm will output a satisfying priority. This is the delay-
tees through proper priority assignment. In the following we
equivalentofStable-Priorityalgorithm,whichwasconstructed
assumethattrafficparameterΛ (θ)foreachlinkiisavailable.
i
using Lemma 2.
Λ (0) = a , so that Λ (θ) is more detailed than a . First we
i i i i
generalize Theorem 5 to prioritized maximal schedulers. Algorithm 2 Delay-Priority (G ,θ)
c
Lemma 5: The delay region Θp = {θ ∈ Rn+ : θ ≺ θ∗} can 1: for k =1 to n do
be achieved by the prioritized maximal scheduler πp, where 2: for i∈Vc do
θi∗ is the largest root of the following equation 3: Compute the largest root θi∗ of the equation:
Λ (θ)+inf [ Λ (u)1(θ )−u]=0
Λi(θ)+ inf [ Λj(u)1(pi−pj)−u]=0 (28) i 0≤u≤θ j∈Ni j j
0≤u≤θ jX∈Ni 54:: esn←d fmorin{i∈Vc :θiP<θi∗}
Proof: Note that the neighbors in Ni with lower priorities 6: ps ←n+1−k
than link i are invisible to link i. Therefore, we create a 7: θs ←0
dominating system S′ consisting of a clique formed by link 8: end for
i and its neighbors that have higher priorities than link i. 9: return p
Using a similar argument,one can show that Lemma 3 holds.
Therefore, Lemma 5 holds, following Lemma 4. (cid:4) We conclude with the following theorem, which is the
delay-analogue of Theorem 3.
Similar to A, we denote Θ = ∪ Θ , i.e., the delay
p p
Theorem6:Ifθ ∈Θ,Delay-Prioritywillgenerateapriority
region guaranteed by any constant priority scheduler. We are
p such that the θ ∈Θ .
interestedincomputingaproperpriorityp,whengivenaQoS p
constraint in the form of the vector θ. We will show this can VI. CONCLUSION
be solvedbyan algorithmadaptedfromStable-Priority,under
This paper considered the prioritized maximal scheduling
the assumption that θ ∈Θ, by showing that Θ has a structure
problem in multi-hop networks for arbitrary correlated arrival
similar as A.
processes. We first considered the random priority case and
Lemma 6: Θ = Θ′, where Θ′ is any delay region in Rn
+ showed that one can achieve the optimal stability region
that satisfies the following conditions:
with i.i.d priorities. Then we focused on the constant priority
1) Θ′ is coordinate-convex; case and proposed a priority assignment algorithm, which,
2) For any nonzero θ ∈ Θ′, there exists 1 ≤ i ≤ n such combinedwith maximalscheduling,canachievean efficiency
that θi > 0 and θi is less than the largest root of the ratio, which is the same as that achieved by the state of art
following equation LQF scheduling. We also analyzed the delay performance of
maximal schedulers in the large deviations regime, assuming
Λ (θ)+ inf [ Λ (u)1(θ )−u]=0 (29)
i j j
0≤u≤θ jX∈Ni ipnrdioerpietinzdedenmt aaxrrimivaall spcrhoecdeuslsiensg, aalngdoriptrhomp.osed a delay-aware
where 1(·) is the indicator function in (14).
Proof:Supposeθ ∈Θ.Then,θ ∈Θ forsomep.Therefore APPENDIX
p
property (1) holds following the fact that Θ is coordinate- Proof of Lemma 1: Due to space limit, we only give a
p
convex.We can assume thatp issuch thatthe linkswith zero outlineoftheproof,foradetaileddiscussionaboutfluidlimits
delay exponentshave the lowest priorities. Next, consider the please refer to [5] and [15].
[D (r t)−D (r t′)] 9
1) The existence of fluid limits: Note that the support of lim j∈{i}∪(Ni/Si) j nk j nk
functions Ai(n),Di(n) and Qi(n) is N, we extend it to k→∞P rnk
R using linear interpolation, which results in continuous ≥ (t−t′)Pr(queue i has local highest priority)
+
functions(e.g., Ai(t),t∈R+). For any sample path ω, define = Pr(p <p ,∀j ∈S )(t−t′).
i j i
a family of functions as
f(rt,ω) Note that if,
fr(t,ω)= (31) [A (r t)−A (r t′)]
r lim j∈{i}∪(Ni/Si) j nk j nk
where f(·) could be Ai(·),Di(·) or Qi(·). Since both the k→∞P rnk
arrivals and the departuresin each time slot are bounded,and = (a + a )(t−t′) (SLLN)
i j
by linearity in (2), the functions defined in (31) are Lipschitz
X
j∈Ni/Si
continuous with the same Lipschitz constant (irrespective of
< Pr(p <p ,∀j ∈S )(t−t′)
r), and hence equi-continuouson any compactinterval. Since i j i
then we have
theyarealsouniformlyboundedon[0,t],accordingtoArzela-
aAnsdcoclionthtienouroeums, finunacntiyon[0s,At¯],i(t·h)e,rD¯eie(·x)isatsnda sQ¯uib(s·e)q,useuncche trhnakt whichL¯ic(ot)nt−raLd¯iic(tts′)E=qnk.l→im(∞34L).irnTkh(et)re−foLreirntkh(et′s)ta<bi0lity of the
(uniform convergence) network is proved and the lemma holds. (cid:4)
lim sup |frnk(τ,ω)−f¯(τ,ω)|=0
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