A Theory of QoS for Wireless
ABSTRACT Wireless networks are increasingly used to carry applications with QoS constraints. Two problems arise when dealing with traffic with QoS constraints. One is admission control, which consists of determining whether it is possible to fulfill the demands of a set of clients. The other is finding an optimal scheduling policy to meet the demands of all clients. In this paper, we propose a framework for jointly addressing three QoS criteria: delay, delivery ratio, and channel reliability. We analytically prove the necessary and sufficient condition for a set of clients to be feasible with respect to the above three criteria. We then establish an efficient algorithm for admission control to decide whether a set of clients is feasible. We further propose two scheduling policies and prove that they are feasibility optimal in the sense that they can meet the demands of every feasible set of clients. In addition, we show that these policies are easily implementable on the IEEE 802.11 mechanisms. We also present the results of simulation studies that appear to confirm the theoretical studies and suggest that the proposed policies outperform others tested under a variety of settings.

Conference Paper: Fair scheduling with deadline guarantees in singlehop networks
[Show abstract] [Hide abstract]
ABSTRACT: We address the problem of simultaneously ensuring longterm fairness and deterministic delay guarantees for realtime traffic over a singlehop network. Specifically, we propose a network control policy that maximises a concave utility function of the average throughput of each flow, while guaranteeing that each packet is delivered within a deterministic deadline. Although this problem has been addressed in the past, prior work makes restrictive assumptions, by allowing only binary packet arrival and service processes at each link. The present paper allows for any bounded burst size distributions for the arrival and service processes.2014 Sixth International Conference on Communication Systems and Networks (COMSNETS); 01/2014  SourceAvailable from: de.arxiv.org[Show abstract] [Hide abstract]
ABSTRACT: The model is a "generalized switch", serving multiple traffic flows in discrete time. The switch uses MaxWeight algorithm to make a service decision (scheduling choice) at each time step, which determines the probability distribution of the amount of service that will be provided. We are primarily motivated by the following question: in the heavy traffic regime, when the switch load approaches critical level, will the service processes provided to each flow remain "smooth" (i.e., without large gaps in service)? Addressing this question reduces to the analysis of the asymptotic behavior of the unscaled queuedifferential process in heavy traffic. We prove that the stationary regime of this process converges to that of a positive recurrent Markov chain, whose structure we explicitly describe. This in turn implies asymptotic "smoothness" of the service processes.02/2015; 
Conference Paper: Scheduling multicast traffic with deadlines in wireless networks
[Show abstract] [Hide abstract]
ABSTRACT: We consider the problem of transmitting multicast flows with hard deadlines over unreliable wireless channels. Every user in the network subscribes to several multicast flows, and requires a minimum throughput for each subscribed flow to meet the QoS constraints. The network controller schedules the transmissions of multicast traffic based on the instant feedback from the users. We characterize the multicast throughput region by analyzing its boundary points, each of which is the solution to a finitehorizon dynamic programming problem over an exponentially large state space. Using backward induction and interchange arguments, we show that the dynamic programming problems are solved by greedy policies that maximize the immediate weighted sum throughput in every slot. Furthermore, we develop a dynamic throughputoptimal policy that achieves any feasible throughput vector by tracking the running performance received by the users.IEEE INFOCOM 2014  IEEE Conference on Computer Communications; 04/2014
Page 1
A Theory of QoS for Wireless
IHong Hou
Department of Computer Science
University of Illinois
Urbana, IL 61801, USA
ihou2@illinois.edu
Vivek Borkar
Tata Institute of Fundamental Research
Mumbai 400 005, India
borkar@tifr.res.in
P. R. Kumar
CSL and Department of ECE
University of Illinois
Urbana, IL 61801, USA
prkumar@illinois.edu
Abstract—Wireless networks are increasingly used to
carry applications with QoS constraints. Two problems arise
when dealing with traffic with QoS constraints. One is
admission control, which consists of determining whether it
is possible to fulfill the demands of a set of clients. The other
is finding an optimal scheduling policy to meet the demands
of all clients. In this paper, we propose a framework for
jointly addressing three QoS criteria: delay, delivery ratio,
and channel reliability.
We analytically prove the necessary and sufficient con
dition for a set of clients to be feasible with respect to
the above three criteria. We then establish an efficient
algorithm for admission control to decide whether a set
of clients is feasible. We further propose two scheduling
policies and prove that they are feasibility optimal in the
sense that they can meet the demands of every feasible
set of clients. In addition, we show that these policies are
easily implementable on the IEEE 802.11 mechanisms. We
also present the results of simulation studies that appear
to confirm the theoretical studies and suggest that the
proposed policies outperform others tested under a variety
of settings.
I. INTRODUCTION
Wireless networks have been widely deployed for a
variety of purposes. Among the many applications that
benefit from wireless networks, those with quality of
service (QoS) constraints are increasingly of interest.
They include video streaming, VoIP, realtime monitoring,
networked control, etc. One common characteristic of
these applications is that they have some requirements
on throughput, delay, and delivery ratio. Hence, most
current network mechanisms, which only provide “best
effort” service, are not adequate for these applications.
While there has been much research interest in provid
ing QoS, there is a dearth of analytical studies and theo
retical guarantees on the service that can be provided. A
fundamental difficulty is that it is important to specifically
take into account a most important feature of wireless
network, that is, the lossy channel. As more and more
devices, such as cordless phones, Bluetooth and Zigbee
This material is based upon work partially supported by USARO under
Contract Nos. W911NF0810238 and W911NF0710287, NSF under
Contract Nos. ECCS0701604, CNS0721992, CNS0626584, and CNS
0519535, and a grant from General Motors India Lab. Any opinions,
findings, and conclusions or recommendations expressed in this publi
cation are those of the authors and do not necessarily reflect the views
of the above agencies.
devices, are accessing the same unlicensed channel as
wireless networks, packet loss can no longer be neglected.
We provide an analytical framework for addressing QoS
constraints in wireless networks that allows the incorpora
tion of three criteria with each flow: delay, delivery ratio,
and channel reliability. We first identify a necessary con
dition for a set of flows to be feasible with respect to the
above three QoS criteria. Next, two dynamic scheduling
policies for these applications are proposed. We prove that
the proposed policies can meet the demand of every set
of flows that satisfies the identified necessary condition.
Thus, we not only show that the necessary condition is
indeed sufficient, but we also prove our proposed policies
are optimal. Finally, while the necessary and sufficient
condition involves exponentially many inequalities to be
checked, we show that those criteria can be simplified into
linearly many tests. We thus obtain an efficient admission
control algorithm for flows with QoS.
Our contribution is therefore threefold. First we pro
pose a mathematical framework for QoS for handling
deadlines, delivery ratios and channel unreliability. Sec
ond, the linear time algorithm makes admission control
computationally efficient. Third, the simple nature of
the policies proved to be feasibility optimal shows that
scheduling for QoS is tractable and feasible at run time.
In addition to the theoretical results, we also evaluate
the proposed policies by simulation. We implement the
two policies by widelyused IEEE 802.11 mechanisms. We
compare the two policies against the IEEE 802.11 Dis
tributed Coordination Function (DCF) and a servercentric
scheduler that gives equal transmission opportunities to
all clients. Simulation results suggest that the proposed
policies offer much better service than the two compared
mechanisms.
The rest of the paper is organized as follows: Section
II summarizes some existing work on providing QoS.
Section III formally models the wireless channels and
formulates a framework for addressing QoS constraints.
Based on the formulation, Section IV develops some
preliminary results that provide insights into designing
scheduling policies and employing admission control. In
Section V, we propose two scheduling policies. We prove
they are both feasibility optimal in Section VI. In addition,
in Section VII, we propose an efficient admission control
algorithm. In Section VIII, we show how to implement the
Page 2
2
proposed scheduling policies on IEEE 802.11. Simulation
results are described in Section IX. Section X concludes
the paper.
II. RELATED WORK
Scheduling policies for QoS support on errorprone
wireless channels have been increasingly of interest in
recent years. Tassiulas and Ephremides [17] have pro
posed a max weight scheduling policy and proved that it
is throughput optimal. Neely [10] have further evaluated
this policy and have shown that the policy achieves a
constant average delay. Shakkottai and Stolyar [14] have
evaluated various scheduling policies to support a mixture
of realtime and nonrealtime traffic. Johnsson and Cox
[7] have proposed a policy that aims to achieve both
small packet delay and high user throughput. Dua and
Bambos [4] have focused on the tradeoff between user
fairness and system performance and designed a policy
for this purpose. However, all these works lack a thorough
theoretical study with provable performance guarantees.
Raghunathan et al [12] and Shakkottai and Srikant [13]
have developed analytical results on scheduling. However,
the goal of their works is to minimize the total number of
expired packets among all users, which will inevitably be
unfair to clients with poor channel qualities. Stolyar and
Ramanan [16] aim at offering QoS guarantees on a per
client basis. Their approach offers asymptotic optimality
only when the period is large. Kawata et al [8] have
focused more on implementation issues and enhancing
QoS for the IEEE 802.11 mechanisms. Their simulations
have been conducted in a controlled environment where
packet losses are infrequent. Other works [1] [2] have
considered different performance metrics and modeling
assumptions.
Compared to scheduling policies, there are fewer an
alytical studies on admission control. Xiao et al [18]
and Pong el al [11] have proposed admission control
algorithms to guarantee a certain amount of bandwidth
for each user but do not take into account any latency
constraints. Garg et al [5], Zhai et al [19], and Shin and
Schulzrinne [15] have used various metrics to predict
system performance statistically but lack a theoretical
study.
III. A MODEL FOR QOS
We consider a system with N wireless clients and one
access point (AP). Each client wishes to transmit packets
to the AP with some QoS constraints. It is assumed that
time is slotted. At the beginning of a time slot, the AP
broadcasts a control message, indicating which client can
transmit in the time slot. The assigned client then sends
out a packet if it has a packet waiting to be transmitted.
The size of a time slot is the time required for the AP
to send the control packet plus the time for a client to
transmit a data packet. While this model appears to as
sume that there is no traffic from the server to clients, we
show in Section VIIIB that all the results are applicable
to the case where clients require QoSconstrained traffic
from the server.
The QoS constraints for a client are described as
follows: At the beginning of every period of length τ,
where the length is measured in timeslots, each client
n ∈ {1,2,··· ,N} generates one data packet. The packets
of all clients are to be delivered to the AP within the
next τ time slots before the end of the period. If a
packet is not delivered by the end of the period, it is
marked as expired and removed from the system. Thus,
we can guarantee that the delay of each delivered packet
is less than τ. Further, client n requires a delivery ratio
of at least qn. That is, the proportion of expired packets
cannot exceed 1 − qn. Finally, reflecting the nature of
the unreliable wireless channels, client n has a channel
reliability of pn; that is, the proportion of transmissions
of client n that are successfully delivered to the AP is pn.
This channel reliability reflects qualities of both uplink
and downlink since a successful transmission includes the
delivery of both the control message by the server and
the data packet by the designated client. The value of pn
can be obtained by probing messages before the client
is admitted into the system and updated as long as the
client stays in the system. The different values of pnfor
different clients also reflect the fact that wireless links are
not homogeneous and vary in quality from user to user.
The decision on which client is chosen to transmit on a
slot is specified by a scheduling policy which makes the
decision causally based on the entire past history of events
up to that time slot.
We wish to provide a service for clients with QoS
constraints as described above.
Definition 1: A set of clients with the above QoS con
straints is said to be fulfilled by a particular scheduling
policy η if the time averaged delivery ratio of each client
n is at least qnwith probability 1.1
Due to the limited wireless resource, the requested QoS
demands of the set of clients may exceed the capacity of
the wireless network. In this case, a service that aims to
fulfill all clients may end up providing poor performance.
Therefore, a desired service must incorporate some admis
sion control mechanism. To perform admission control, it
is vital to verify whether a set of clients is feasible:
Definition 2: A set of clients is feasible if there exists
some scheduling policy that fulfills it.
In addition to an efficient admission control mecha
nism,we also aim to design a feasibility optimal scheduling
policy:
Definition 3: A scheduling policy is said to be feasibility
optimal if it fulfills every feasible set of clients.2
1More formally, liminfK→∞
ered successfully in period k) ≥ qn, with probability one, for each client
n = 1,2,··· ,N, where 1(·) is the indicator function of the event.
2This is analogous to the notion of “throughput optimality” in queuing
systems.
1
K
?K
k=11(A packet of client n is deliv
Page 3
3
IV. NECESSARY CONDITION FOR FEASIBILITY OF QOS
It is quite evident that the more time slots we allocate
to a particular client, the more likely it is that we can
meet the demand of that client. We further observe that
whether the demand of a client is met is only related
to the proportion of time slots in which the client is
transmitting.
Lemma 1: The delivery ratio of client n is at least
qn with probability 1 if and only if the longterm time
average of the proportion of time slots that client n is
transmitting is at least wn=
Proof: Define:
?
and
?
Let Ft be the σalgebra generated by {(un(k),dn(k −
1)),for 1 ≤ k ≤ t and 1 ≤ n ≤ N}. (We set dn(0) = 0 for
all n.)
Then E[dn(t)Ft] = pnun(t). Hence, by the martingale
stability theorem of Loeve [9],
qn
pnτ.
un(t) =
1,
0,
if client n makes a transmission at time t,
otherwise,
dn(t) =
1,
0,
if client n delivers a packet at time t,
otherwise.
lim
T→∞
1
T
T
?
t=1
[dn(t) − pnun(t)] = 0, a.s.(1)
Since the delivery ratio of client n must be at least qn,
liminf
T→∞
1
T
T
?
?T
t=1
dn(t) ≥qn
τ, a.s.
Hence, liminfT→∞
1
T
t=1un(t) ≥
qn
pnτfrom (1).
We will hereafter refer to wnas the implied work load
for client n. Determining whether client n is fulfilled is
therefore equivalent to deciding whether the share of time
that client n gets is at least as large as its implied work
load. This helps to partially decouple the clients.
We next study whether it is possible to fulfill a set of
clients. Since there is at most one client that can transmit
in any time slot, we immediately obtain the following
necessary condition:
Lemma 2: A set of N clients can be feasible only if
?N
Since each client only generates one packet in each
period, it might be the case that at some slot in a period,
all packets in the system are delivered and the system
is forced to stay idle. (Recall that expired packets are
deleted from the system at the end of a period, and so only
new packets are available in a system at the beginning of
each period). While the amount of time that the system
is idling may be influenced by the scheduling policy, we
show it is the same for some particular policies:
n=1wn≤ 1.
This necessary condition, however, is not sufficient.
Definition 4: A scheduling policy is work conserving if
it never idles whenever there is an undelivered packet in
the system.
Lemma 3: The probability distribution of the amount
of time that the system is idling in any period is identical
for every work conserving policy.
Proof: Let the random variable γndenote the number
of slots a packet of client n is transmitted before it is
delivered. The distribution of γnis Prob(γn= t) = pn(1−
pn)t−1. Under any work conserving policy, the number of
idle time slots in a period, L, is the number of time slots
left after all packets in the system are delivered:
?
Hence its probability distribution is the same under
every work conserving policy.
The following observation allows us to focus only on
work conserving policies when designing a feasibility
optimal policy:
Lemma 4: Let η be a scheduling policy that can meet
the demands of a particular set of clients. Then there
exists a work conserving policy η?that can also meet the
requirements of the same set of clients.
Proof: Policy η can be extended to be a work
conserving policy η?by simply filling slots that η would
leave idle by transmitting any undelivered packets in arbi
trary order. This cannot reduce the number of undelivered
packets in any period.
Note that for at least E[L] number of slots of each
period, on average, the server must be idle. The fraction of
idle time is therefore at least
the necessary condition in Lemma 2 to
L =
τ −?N
n=1γn,
if
otherwise.
?N
n=1γn< τ,
0,
E[L]
τ. Hence we can improve
N
?
n=1
wn≤ 1 −E[L]
τ
.
(2)
However, we can go even further by considering subsets
of the set of all clients {1,2,··· ,N}. For any subset S ⊆
{1,2,··· ,N}, let
IS:=E[max{0,τ −?
It is a lower bound on the fraction of time spent idling,
if S were the set of all clients. Clearly if {1,2,··· ,N} is
feasible, then every subset S must be feasible. Hence we
can tighten the condition (2):
Lemma 5: A set of clients is feasible only if?
The reason why the condition for a subset S is not
subsumed by the condition for all clients is that?
creasing in S. Surprisingly, we will show that the above
necessary condition is actually sufficient in Section VI.
n∈Sγn}]
τ
.
n∈Swn≤
1 − ISholds for every subset S.
n∈Swn
is monotone increasing in S, while IS is monotone de
V. SCHEDULING POLICIES
In this section, we propose two scheduling policies on
providing QoS. Both policies are what we call largest debt
Page 4
4
first policies. The idea of a largest debt first policy is to
compute the debt owed to each client at the beginning of
every period. The server then determines the priorities of
clients according to their debts, a clients with larger debt
getting a higher priority. Ties are broken by lexicographic
order. In each time slot of the period, the client with the
highest priority among those who have not yet succeeded
in a transmission is scheduled to transmit. The only
difference between the two policies lies in the definitions
of debts.
The first policy, which we call the largest timebased debt
first policy, tries to make every client get a share of time at
least as large as its implied work load. To see how much
a client lags behind its implied work load, we define debt
as follows:
Definition 5: The timebased debt of client n at time t is
defined as t × wnminus the actual number of time slots
that client n has transmitted by time slot t. The policy
which assigns priorities accordingly is the largest time
based debt first policy.
The next policy, which we call the largest weighted
delivery debt first policy, approaches the QoS requirements
more directly. It seeks to make every client have a success
rate higher than the desired delivery ratio, that is, qn.
Definition 6: Let cn(t) be the number of successful
transmissions of client n up to time t. The weighted
delivery debt of client n at time t is defined as (t
cn(t))/pn. The policy which assigns priorities accordingly
is called the largest weighteddelivery debt first policy.
τ× qn−
VI. PROOFS OF OPTIMALITY
We prove that the two largest debt first policies are fea
sibility optimal policies. The proof is based on Blackwell’s
approachability theorem [3]. We first describe the content
of this theorem.
Consider a singleplayer repeated game. The payoff,
instead of being a single number, is an Ndimensional
vector, v ∈ RN. In each stage i, the player picks an action,
a(i), according to some history dependent policy, based
on past actions a(1),··· ,a(i − 1), and past payoffs in
the previous stages, v(1),··· ,v(i − 1). The payoff of this
stage, v(i), is a random vector whose distribution is a
given function, which we call the payoff function, of the
action a(i) taken. We are interested in the distribution
of the longterm average payoff, limj→∞
study the problem, Blackwell introduced the concept of
approachability:
Definition 7: Let A ⊆ RNbe any set in Ndimensional
space. We shall say A is approachable under policy η, if
for every δ > 0 and ε > 0 there is a j0such that,
?j
i=1v(i)/j. To
Prob{ρ(j) ≥ δ for some j ≥ j0} ≤ ε,
where ρ(j) denotes the distance of the point?j
payoff will approach A with probability 1.
Let R(a) be the expected payoff of action a, i.e., the ex
pected value corresponding to the probability distribution
i=1v(i)/j
from A. In other words, this means the longterm average
of the payoff function. Blackwell’s sufficient condition for
approachability is the following:
Theorem 1: Let A ⊆ RNbe any closed set. If for
every x / ∈ A there is an action a (= a(x)) such that
the hyperplane through y, the closest point in A to x,
perpendicular to the line segment xy, separates x from
R(a), then A is approachable with the policy η which uses
action a(xj) if xj= (?j
Since every feasible set of clients has to satisfy the
necessary condition stated in Lemma 5, it suffices to show
that both policies fulfill every set of clients that satisfies
the necessary condition.
Theorem 2: The largest timebased debt first policy is
feasibility optimal.
Proof: We first translate the largest timebased debt
first policy into a policy for the single player game. A
stage in the game corresponds to a period in our model.
The action that the player, which is the server in our
system, can take, is to decide the priorities of the clients,
with the interpretation that a client with a certain priority
is transmitted only after the client with the immediately
higher priority is successful. The payoff a player gets in
the stage is the net change of the timebased debt of each
client. To be more precise, the payoff is a vector whose
nthelement equals τwnminus the actual number of time
slots that client n is transmitting in this period.
Lemma 1 shows that a group of clients is fulfilled if
every of them gets an average share of time at least wn.
In terms of approachability, this means a group of clients
is fulfilled if the set A := {z = [z1,z2,··· ,zN]zn≤ 0,∀n}
is approachable.
Now we apply Theorem 1. Suppose at the beginning
of some period the average payoff x = [x1,x2,··· ,xN]
does not lie in A. We can reorder the clients so that x1≥
x2≥ ··· ≥ xm> 0 ≥ xm+1≥ ··· ≥ xN. The closest point
in A to x is y = [0,0,··· ,0,xm+1,xm+2,··· ,xN]. The
hyperplane H through y and perpendicular to the line
segment xy is comprised of points z satisfying h(z) :=
?m
the largest timebased debt first policy. Also, let ¯ wnbe the
expected portion of time slots that client n is transmitting
during this period. We can now express
¯ x = [τw1− τ ¯ w1,τw2− τ ¯ w2,···]
= τ[w1− ¯ w1,w2− ¯ w2,···].
Since h(x) =
separates x and ¯ x, it suffices to show h(¯ x) ≤ 0. We have
h(¯ x) = τxn(wn− ¯ wn)
i=1v(i)/j) / ∈ A, and an arbitrary
action otherwise.
n=1xnzn= 0.
Let ¯ x be the expected payoff in this round according to
?m
m
?
m−1
?
n=1x2
n> 0, in order to show that H
n=1
= τ
n=1
[(xn− xn+1)(
n
?
k=1
wk−
n
?
k=1
¯ wk)]
+ τxm(
m
?
k=1
wk−
m
?
k=1
¯ wk).
Page 5
5
By the largest timebased debt first policy, the server
will give priorities according to the ordering 1,2,··· ,N.
Hence,
the system spends working if only clients 1 through n
are present in the system. In other words,?n
stated in Lemma 5, we have?n
h(¯ x) ≤ 0 and the largest timebased debt first policy is
therefore feasibility optimal.
Theorem 3: The largest weighteddelivery debt first
policy is feasibility optimal.
Proof: Again, we translate the largest weighted
delivery debt first policy into one for the single player
game. As in the previous proof, a stage in the game
corresponds to a period in the system, and the action
of a player is to decide the priorities of the clients at
each stage. However, in this proof, the payoff corresponds
to the net change of the weighteddelivery debt of each
client. In other words, the payoff is an Ndimensional
vector, whose nthelement is (qn− 1)/pn if the packet
of client n is delivered in the period, and qn/pn if not.
A set of clients is fulfilled if each client has a negative
weighteddelivery debt. Hence, we need to show that the
set A := {z = [z1,z2,··· ,zN]zn≤ 0,∀n} is approachable.
Suppose at the beginning of some period, the average
payoff x = [x1,x2,··· ,xN] does not lie in A. We can
reorder the clients so that x1 ≥ x2 ≥ ··· ≥ xm >
0 ≥ xm+1 ≥ ··· ≥ xN. The closest point in A to x is
y = [0,0,··· ,0,xm+1,xm+2,··· ,xN]. The hyperplane H
through y and perpendicular to the line segment xy is
comprised of points z with h(z) :=?m
in the period. The expected payoff of this period is ¯ x =
[(q1−π1)/p1,(q2−π2)/p2,··· ,(qN−πN)/pN]. Since h(x) =
?m
h(¯ x) =
pn
?n
k=1¯ wk will be the average portion of time
k=1¯ wk =
1 − I{1,2,···,n}. Now, according to the necessary condition
?n
k=1wk≤ 1 − I{1,2,···,n}=
k=1¯ wk. Further, x1 ≥ x2 ≥ ··· ≥ xm > 0. Hence,
n=1xnzn= 0.
Let πnbe the probability that client n delivers its packet
n=1x2
approachability of A. We have
?
=
[(xn− xn+1)(
n> 0, we only need to show h(¯ x) ≤ 0 to establish
m
n=1
m−1
?
+ xm(
xnqn− πn
n=1
n
?
πk
pk)
k=1
qk
pk
−
n
?
k=1
πk
pk)]
m
?
[(xn− xn+1)(
k=1
qk
pk
−
m
?
k=1
= τ
m−1
?
n=1
n
?
πk
τpk) (since wn=
k=1
wk−
n
?
k=1
πk
τpk)]
+ τxm(
m
?
k=1
wk−
m
?
k=1
qn
τpn).
Since x1 ≥ x2 ≥ ··· ≥ xm > 0, it suffices to show
?n
I{1,2,···,n}. The proof is therefore complete if we can show
?n
k=1wk ≤?n
k=1
πk
τpk, for every n. Notice now that the
necessary condition in Lemma 5 requires?n
k=1
k=1wk≤ 1−
πk
τpk= 1−I{1,2,···,n}. This is done in Lemma 6 below.
Lemma 6: Under
?n
case where n = 1. Since client 1 has the highest priority,
it fails to deliver its packet only when every transmission
during this period fails. Thus, π1= 1−(1−p1)τ. Next we
compute the value of I{1}. The probability that client 1
delivers its packet on the γthtransmission is p1(1−p1)γ−1,
which will result in τ − γ idle time slots. Hence, I{1}=
1
τ
τ−τI{1}= [1−(1−p1)τ]/p1=π1
Assume
k=1
pk
We wish to show
k=1
pk
∆p :=?m+1
Since client m + 1 is the one with least priority among
clients 1 through m + 1, it can transmit only after all
packets from clients 1 through m are delivered. Suppose
there are σ time slots left when client m+1 can transmit,
with σ a random variable. Let πm+1(t) be the probability
that the packet of client m + 1 is delivered, and let
ζm+1(t) be the number of time slots left when client m+1
succeeds, given that σ = t. As in the case of n = 1, we
have
pm+1
as
?
=
(t − ζm+1(t))Prob(σ = t)
?
= τI{1,···,m}− τI{1,···,m+1}= ∆I.
Finally, since ∆p= ∆I,?m+1
Since both policies fulfill every set of clients that is
consistent with the necessary condition in Lemma 5, the
necessary condition is also sufficient.
Theorem 4: A set of clients is feasible if and only if
?
work conserving policy.
thepriority order
{1,2,··· ,N},
k=1
πk
pk= τ(1 − I{1,2,···,n}), for n = 1,2,··· ,N.
Proof: We prove this by induction. First consider the
?τ
γ=1(τ − γ)p1(1 − p1)γ−1= 1 −1−(1−p1)τ
p1τ
. This yields
p1, for every period. (3)
= τ(1 − I{1,···,n}) for all n ≤ m.
?m+1
k=1
?n
πk
πk
= τ(1 − I{1,···,m+1}). Let
pk=
k=1
πk
pk−?m
πk
πm+1
pm+1and ∆I := τ(1 −
I{1,···,m+1})−τ(1−I{1,···,m}) = τI{1,···,m}−τI{1,···,m+1}.
πm+1(t)
= t − ζm+1(t). ∆pand ∆I can be obtained
∆p=
t
πm+1(t)
pm+1
Prob(σ = t)
?
t
=
t
tProb(σ = t) −
?
t
ζm+1(t)Prob(σ = t)
k=1
πk
pk= τ(1 − I{1,···,m+1}).
By induction,?n
k=1
πk
pk= τ(1−I{1,2,···,n}) holds for all n.
n∈Swn≤ 1−ISfor every subset S of the clients, where
ISis the expected proportion of idle time for S under any
VII. A EFFICIENT ALGORITHM FOR ADMISSION CONTROL
Performing admission control essentially consists of
deciding whether a set of clients is feasible. While The
orem 4 states a necessary and sufficient condition for
feasibility, it requires testing every subset of N clients,
and thus results in exponentially many tests in N. In
this section, we show that we only need to evaluate a
total number of N conditions to determine feasibility.
The following theorem therefore makes admission control
computationally efficient, and feasible to implement.