# 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.

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**ABSTRACT:**We consider the joint design of transmission schedules and controllers for networked control loops that use WirelessHART communication for sensor and actuator data. By parameterizing the design problem in terms of the sampling rate of the control loop, the co-design problem separates into two well-defined subproblems which admit optimal solutions: transmission scheduling should be done to maximize the delay-constrained reliability while the control design should optimize closed-loop performance under packet loss. We illustrate how these problems can be solved and demonstrate our co-design framework for the case of linear-quadratic control.01/2011; - SourceAvailable from: Mikael Johansson[Show abstract] [Hide abstract]

**ABSTRACT:**We consider the joint design of forwarding policies and controllers for wireless control loops where sensor measurements are sent over an unreliable multi-hop wireless network to the controller node. For fixed sampling rate of the control loop, the co-design problem separates into two well-defined and independent subproblems: transmission scheduling for maximizing the delay-constrained reliability and optimal control under packet loss. We develop optimal solutions for these subproblems and show that the optimally co-designed system can be found by a one-dimensional search over the sampling interval. Several numerical examples illustrate the power of our approach.04/2012; - SourceAvailable from: arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**Wireless sensor networks have been increasingly used for real-time surveillance over large areas. In such applications, it is important to support end-to-end delay constraints for packet deliveries even when the corresponding flows require multi-hop transmissions. In addition to delay constraints, each flow of real-time surveillance may require some guarantees on throughput of packets that meet the delay constraints. Further, as wireless sensor networks are usually deployed in challenging environments, it is important to specifically consider the effects of unreliable wireless transmissions. In this paper, we study the problem of providing end-to-end delay guarantees for multi-hop wireless networks. We propose a model that jointly considers the end-to-end delay constraints and throughput requirements of flows, the need for multi-hop transmissions, and the unreliable nature of wireless transmissions. We develop a framework for designing feasibility-optimal policies. We then demonstrate the utility of this framework by considering two types of systems: one where sensors are equipped with full-duplex radios, and the other where sensors are equipped with half-duplex radios. When sensors are equipped with full-duplex radios, we propose an online distributed scheduling policy and proves the policy is feasibility-optimal. We also provide a heuristic for systems where sensors are equipped with half-duplex radios. We show that this heuristic is still feasibility-optimal for some topologies.04/2012;

Page 1

A Theory of QoS for Wireless

I-Hong 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, real-time 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. W911NF-08-1-0238 and W-911-NF-0710287, NSF under

Contract Nos. ECCS-0701604, CNS-07-21992, CNS-0626584, and CNS-

05-19535, 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 widely-used IEEE 802.11 mechanisms. We

compare the two policies against the IEEE 802.11 Dis-

tributed Coordination Function (DCF) and a server-centric

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

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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 error-prone

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 real-time and non-real-time 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 trade-off 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 VIII-B that all the results are applicable

to the case where clients require QoS-constrained 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 time-slots, 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 long-term time

average of the proportion of time slots that client n is

transmitting is at least wn=

Proof: Define:

?

and

?

Let Ft be the σ-algebra 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 time-based 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 time-based 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 weighted-delivery 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 single-player repeated game. The payoff,

instead of being a single number, is an N-dimensional

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 long-term average payoff, limj→∞

study the problem, Blackwell introduced the concept of

approachability:

Definition 7: Let A ⊆ RNbe any set in N-dimensional

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 long-term 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 time-based debt first policy is

feasibility optimal.

Proof: We first translate the largest time-based 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 time-based 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 time-based 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 time-based 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 time-based debt first policy is

therefore feasibility optimal.

Theorem 3: The largest weighted-delivery 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 weighted-delivery debt of each

client. In other words, the payoff is an N-dimensional

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

weighted-delivery 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.