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Extended Abstract: Real-Time Implications of Multiple

Transmission Rates in Wireless Networks

Vartika Bhandari

Dept. of Computer Science

University of Illinois at

Urbana-Champaign

vbhandar@uiuc.edu

∗

Vivek Raghunathan

Dept. of Electrical and

Computer Engineering

University of Illinois at

Urbana-Champaign

vivek@control.csl.uiuc.edu

†

Bach Duy Bui

Dept. of Computer Science

University of Illinois at

Urbana-Champaign

bachbui2@uiuc.edu

Marco Caccamo

Dept. of Computer Science

University of Illinois at

Urbana-Champaign

mcaccamo@cs.uiuc.edu

‡

ABSTRACT

Wireless networks are increasingly being used for latency-

sensitive applications that require data delivery to be timely,

efficient and reliable. This trend is primarily driven by the

proliferation of wireless networks of real-time data-gathering

sensor-actuator devices. This has led to a strong need to

bring real-time concerns to the forefront of an integrated

research thrust into wireless real-time systems. In this pa-

per, we introduce and analyze a specific instance of the rich

set of problems in this domain. We consider a wireless net-

work serving real-time flows in which the underlying physi-

cal layer provides multiple transmission rates. Higher rates

have more stringent SINR requirements and thus represent

a trade-off between raw transmission speed and packet error

rate. We adopt a first principles approach to the design of

optimal real-time scheduling algorithms for such a multi-rate

wireless network. We illustrate the inherent complexities of

the problem through examples and obtain provably optimal

structural results. We then characterize the optimal policy

for an approximate model. Our theoretical analysis provides

guidelines for heuristic scheduler design. Our initial work in-

dicates that this is a rich problem domain with the potential

for a unifying theory that integrates real-time requirements

into multi-rate wireless network design.

∗Supported by a Vodafone Graduate Fellowship.

†Supported by a Vodafone Graduate Fellowship.

‡This research was supported in part by the National Science

Foundation through grant CNS-0613665.

Copyright is held by the author/owner(s).

MobiCom’07, September 9–14, 2007, Montréal, Québec, Canada.

ACM 978-1-59593-681-3/07/0009.

Categories and Subject Descriptors

C.2.1 [Computer Communication Networks]: Network

Architecture and Design—Wireless communication; C.3 [Special-

Purpose and Application-Based Systems]: Real-time

and embedded systems

General Terms

Performance, Theory

Keywords

Real-time scheduling, wireless networks, rate adaptation,

dynamic programming.

1. INTRODUCTION

The rapid proliferation of wireless sensor-actuator net-

works has brought to a fore the need to address real-time

concerns in wireless networking. Such real-time applications

do not merely require efficient and reliable data delivery.

Timeliness is of utmost importance, as data communication

is often part of a complex sequence of time-critical actions

performed by the sensor network in response to external

events. Real-time guarantees may also be required for mul-

timedia applications in wireless mesh networks.

Typical wireless systems provide multiple transmission rates,

e.g., 1, 2, 5.5 and 11 Mbps in IEEE 802.11b, 6, 9, 12, 18, 24,

36, 48, 54 Mbps in IEEE 802.11g and 11, 22, 33, 44, 55 Mbps

in IEEE 802.15.3. Higher rates have more stringent SINR

requirements for decoding, and represent a trade-off between

raw transmission speed and packet error rate. In this work,

we analyze the real-time implications of this trade-off using

a first principles approach. We study a canonical problem,

focusing on the case where all flows are over a single wire-

less link. Given the probabilistic nature of wireless losses, we

take a soft-real-time approach and focus on minimizing the

expected deadline-miss-ratio. We use illustrative examples

to demonstrate the complex set of issues introduced by this

seemingly simple problem. We show that intuitive solutions,

e.g., earliest-deadline-first (EDF) scheduling at the rate with

the minimum expected transmission time (min-ETT), are

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not necessarily optimal. We also show that packet schedul-

ing and rate selection are inextricably linked, and even ap-

parently simple scenarios are computationally intractable.

Motivated by the intractability of the problem, we adopt

a two pronged approach toward developing theoretical in-

sight. Firstly, we establish the structure of the opti-

mal policy by showing that it is possible to associate each

packet with a“pseudo-deadline”(a value between its arrival-

time and deadline) such that the optimal scheduler has an

earliest pseudo-deadline first (EPDF) structure. Secondly,

we consider an approximate, albeit more tractable

model. For this model, we show that the “packet arbitra-

tion problem”, i.e., which packet to schedule, is decoupled

from the“rate selection problem”, i.e.., what rate to schedule

the packet at. We further establish that EDF is the optimal

packet arbitration policy, and min-ETT is the optimal rate

selection rule.

Our work indicates the potential for a unifying theory

that integrates real-time requirements into multi-rate wire-

less network design.

2.MODEL/NOTATION/TERMINOLOGY

We study a slotted-time, single-link wireless system, where

all flows have the same source-destination pair. We model

a multi-rate wireless system, with each rate corresponding

to a different modulation, e.g., BPSK, QPSK, 16-QAM and

64-QAM in IEEE 802.11b. Faster rates have higher SINR re-

quirements for successful decoding, and thus exhibit higher

packet error rates. There are k rates, labeled as r1,...,rk,

where each ri is specified by a two-tuple (li,pi). A packet

transmission at rate ri takes li time slots, and is successful

with probability 1−pi. All li’s are assumed integral. Thus,

we implicitly assume that all packets have the same size.

We denote absolute deadlines by d, and relative deadlines

by D. We consider two packet arrival models:

1. One shot model: n flows, comprising one packet each,

arrive at t = 0 with deadlines d1 < d2 < ... < dN. No

arrivals occur after t = 0.

2. Periodic model: For flow i, packets arrive periodically

with period Ti with initial relative-deadline Di = Ti. The

hyper-period is the least common multiple(LCM) of all Ti’s.

Traditional real-time schedulability analysis focuses on find-

ing conditions on arrival rates so that all flows are guaran-

teed to meet their deadlines. In a system with probabilistic

losses, this is an impossible undertaking, and we focus on

minimizing the deadline miss ratio.

3. A FIRST GLANCE AT THE PROBLEM

We begin with some observations demonstrating that many

apparently intuitive properties do not hold for the optimal

scheduling policy.

Example 1. Consider a single one-shot flow with d = 4

over a link with two rates r1 = (2,p1) and r2 = (3,p2), with

p2

1> p2, e.g., p1 = 0.5, p2 = 0.2. Then, the optimal rate

sequence is r2, which yields a deadline miss probability of p2,

but leaves one slot unused, even though the packet’s deadline

is yet to expire.

This yields:

Observation 1. Unless there is a rate ri with li = 1, the

optimal scheduler may not be work-conserving.

Similarly, EDF scheduling may seem intuitively optimal.

This is not always true, as shown by the following counter-

example:

Example 2. One-shot arrivals: Consider two flows τ1

and τ2 with deadlines d1 = 1 and d2 = 2 respectively. There

are two rates r1 = (1,0.6),r2 = (2,0.1). Using EDF, we

are forced to use r1 in both slots, and incur 0.6 + 0.6 = 1.2

expected deadline misses. If instead, we play only τ2 at rate

r2, the expected deadline misses would be 1 + 0.1 = 1.1.

Periodic arrivals: Consider two flows τ1 and τ2 with peri-

ods T1 = 2 and T2 = 4 respectively. The hyper-period is

thus of duration 4. There are two rates r1 = (1,0.99),r2 =

(4,0.01). Using any EDF ordering, the expected deadline

misses per hyper-period is 2.96000001. Instead, by playing

only τ2 at rate r2, we can reduce the expected deadline misses

to 2 + 0.01 = 2.01.

Thus we obtain:

Observation 2. EDF is not necessarily optimal for the

multi-rate real-time scheduling problem.

4.REALISTIC MODEL: CONSTANT SIZE

PACKETS

We now build a theoretical understanding of real-time

multi-rate scheduling for fixed size packets. Due to lack of

space, results are stated without proof. For details, please

see [2].

4.1Single-Packet Rate Adaptation

Consider the apparently simple RATE-ADAPT problem,

where a single packet arriving at t = 0 with deadline d,

must be transmitted over a lossy multi-rate link. The link

has a set of k possible transmission rates ri = (li,pi). We

seek to find the sequence of transmission rates to be used, to

minimize the probability of missing the deadline. An opti-

mization instance of RATE-ADAPT can be represented as a

3-tuple (d,Pe,L), with |Pe| = |L| = k, Pe = {p1,p2,...,pk},

and L = {l1,l2,...,lk}.

Theorem 1. RATE-ADAPT is NP-hard.

Greedy Heuristic for RATE-ADAPT. Consider the fol-

lowing greedy heuristic: sort the rates in non-decreasing or-

der of mi = p1/li

i

, and let the order be ri1,ri2,...,rik. We

greedily include as many transmission attempts at ri1as

possible, then move on to ri2, and so on.

Lemma 1. If the packet transmission durations liare mu-

tually harmonic, i.e., for all 1 ≤ i,j ≤ k, one of li,lj is a

multiple of the other, then the greedy heuristic is optimal.

Lemma 2. Let pgreedy be the probability of deadline miss

when using the heuristic. Let popt be the optimal deadline

miss probability. Then pgreedy ≤√popt.

4.2Multi-Rate Scheduling Problem

The multi-rate real-time scheduling problem MRATE-SCHED

models the single-link scheduling situation described in Sec-

tion 2. Given a flow-set τ, we seek to determine the schedul-

ing algorithm that minimizes the expected number of dead-

line misses. Note that the problem is a joint packet and

rate selection problem.

Lemma 3. MRATE-SCHED is NP-hard.

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Structure of Optimal MRATE-SCHED Algorithm

Theorem 2. There exists an optimal MRATE-SCHED

scheduling algorithm that has earliest pseudo-deadline first

(EPDF) structure. If at some time t, this optimal scheduler

bypasses the packet with the earliest deadline (call it i), and

schedules another packet j, then i will never be scheduled

again by the optimal scheduler. This property holds for both

one-shot and periodic arrival models. Additionally, the dif-

ference between a packet’s deadline and pseudo-deadline is

less than the maximum packet transmission time, over all

possible rates.

This structural result can be intuitively explained by ob-

serving that, since all packets experience the same channel

conditions in the single-link problem, one can benefit from

bypassing the packet with earliest deadline only if the packet

we schedule instead is played at a slower and better rate that

was not usable for the earliest deadline packet due to dead-

line constraints. When different packets see different chan-

nel conditions (as in a multi-link scenario), this intuition no

longer holds, and it can be shown via counter-example that

EPDF is not necessarily optimal.

4.3Rate-Selection

In light of Theorem 2, one might wonder whether within

a packet’s pseudo-deadline, the optimal scheduler always se-

lects the rate that is optimal for that individual packet. We

show via counter-example that this is not the case.

Example 3. Consider 2k periodic flows, each having pe-

riod Ti = T = 2k (this example may also be interpreted as

an instance of the one-shot model). There are two avail-

able rates: r1 = (1,0.4),r2 = (k + 1,0.0). The hyper-period

is T, and it suffices to consider this duration. From Theo-

rem 2, there exists an optimal EPDF scheduler that bypasses

a packet only if another packet can be scheduled at a slower

(but better) rate. This implies that the pseudo-deadlines are

the same as the deadlines for this example. Hence at the be-

ginning of slot 1, one can schedule the first packet at the

greedy best rate r2 without exceeding its pseudo-deadline,

leading to expected deadline misses of at least k. Instead,

consider the following naive policy: try sending packet i ex-

actly once at rate r1. This yields an expected loss of 0.4(2k) =

0.8k < k.

This yields the following observation:

Observation 3. Rate-selection within a packet’s pseudo-

deadline cannot always be optimally performed independent

of other unfinished packets in the system.

Additionally, performance also depends on the the order

in which rates are attempted, as shown by the following ex-

ample:

Example 4. There are two one-shot flows τ1and τ2, with

deadlines d1 = 3 and d2 = 5 respectively. There are three

rates r1 = (1,p1),r2 = (2,p2),r3 = (4,p3). We assume that

the greedy ordering of rates is r3,r2,r1, i.e., p1 ≥ p

Note that τ1 cannot use rate r3. The possible rate-set in this

case is r2,r1, and the greedy order of rates for τ1 is r2,r1.

Suppose we use EDF, and τ1 is played in this order upto its

deadline, followed by τ2. The expected number of deadline

misses is X = p2p1+(p2·p2+(1−p2)p2p1). Now consider the

1

2

2≥ p

1

4

3.

alternate order r1,r2 for scheduling τ1. The expected number

of deadline misses is now Y = p1p2+ (p1·p2+ (1 − p1)p3)).

X−Y = (p2

To illustrate, consider the following values of the pi’s: p1 =

0.75,p2 = 0.4,p3 = 0.1. These respect the greedy ordering,

and provide a valid instantiation of this example. In this

case, X = 0.3+0.16+0.18 = 0.64, Y = 0.3+0.3+0.025 =

0.625, and X−Y = (p2

Thus it is better to play the non-greedy rate order r1,r2 than

the greedy sequence r2,r1.

One might ask whether the order-sensitivity is only an

artefact of the potentially sub-optimal use of EDF in the

above example. This is easily shown not to be the case. From

Theorem 2, τ1’s pseudo-deadline can only be either of 1, 2

or 3. The last case is already covered in the EDF cases de-

scribed above. Consider the first two cases. When pseudo-

deadline is 1, τ1 can only be played once. Then the best ex-

pected loss is Z = p1+p3 = 0.75+0.1 = 0.85. This is greater

than X,Y above. When the pseudo-deadline is 2, using rate

r2 for τ1 yields expected loss W = p2+ p2p1 = 0.7, and us-

ing r1,r1 yields expected loss p2

0.5625+0.225+0.025 = 0.8125. Thus the optimal scheduler

has loss Y and it uses a non-greedy ordering.

2−p3)(1−p1) ≥ 0 by our assumed greedy ordering.

2−p3)(1−p1) = (0.16−0.1)(0.25) > 0.

1+ p1(p2p1) + (1 − p1)p3 =

This yields the following observation:

Observation 4. The performance of the scheduling pol-

icy is sensitive to the order in which rates are attempted

within the packet’s pseudo-deadline.

5.APPROXIMATION MODEL

Motivated by the intractability results, we have considered

an approximate approach. We provide a brief summary of

results here. For details, please see [2]. Consider the follow-

ing channel model: suppose at time t, a flow is scheduled

for transmission at rate ri, and consider a small interval

(t,t + li∆t). Then, with probability (1 − pi)∆t, the packet

successfully completes in this time interval. This model en-

sures that when rate ri is used, the time between successful

transmissions is exponentially distributed with parameter

li(1 − pi), and is the continuous analogue of the geometric

distribution between successful slots for the constant packet

model, with the means of both distributions being equal.

We can formulate the problem of finding the optimal pol-

icy as a stochastic control problem using the theory of dy-

namic programming. For the one shot model, this is a finite

time horizon problem. It is clear that the optimal policy

never schedules a flow whose relative deadline is 0. Suppose

then, that the relative deadlines are denoted by the vector

¯D := (D1,...,DN), with 0 < D1 < ... < DN. A policy πt(¯D)

is characterized by a two tuple (flowt(¯D),rt(¯D)) describing

which flow and rate to schedule at time t. Let Vπ(¯D) be the

total number of deadline misses incurred by π starting with

initial relative deadlines¯D. We wish to find the stationary

π∗that minimizes Vπ(¯D). Observe that the optimal V∗(¯D)

satisfies the optimal substructure property of dynamic pro-

gramming, i.e., any sub-schedule of the optimal schedule is

also optimal.

For the periodic model, we focus on the expected dead-

line miss ratio per slot over the infinite time horizon (0,∞).

Assuming for simplicity that there are two flows, we de-

note the relative deadlines as D1,D2. Define x and y as

0 − 1 variables indicating whether the packet correspond-

ing to the first and second flows has left the system (1) or

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not (0). Then, a policy πt(D1,D2,x,y) is characterized by

the two tuple (flowt(D1,D2,x,y),rt(D1,D2,x,y)) describ-

ing what flow and transmission rate to use at time t. Define

Vπ(D1,D2,x,y) as the expected average number of dead-

line misses starting from (D1,D2,x,y). We wish to find the

stationary π∗(D1,D2,x,y) that minimizes the average num-

ber of deadline misses. Again, the optimal V∗(D1,D2,x,y)

satisfies an optimal substructure property, i.e., every sub-

schedule of the optimal schedule is also optimal.

5.1One shot arrivals

1

Theorem 3. For the one-shot model, the optimal policy

π∗(¯D) that minimizes the total number of deadline misses

has the following structure:

1. It schedules packets in earliest-deadline-first (EDF) or-

der, i.e, flow∗(¯D) = argminiDi.

2. The optimal rate selection rule is independent of packet

scheduling rule, i.e., r∗(¯D) = r∗.

3. It selects the rate with the minimum expected transmis-

sion time, i.e., r∗= argmink

lk

1−pk.

5.2 Periodic arrivals

We use a discounted cost infinite horizon total cost formu-

lation of the periodic model problem, where a deadline miss

incurred t units in the future is discounted by e−γtwith re-

spect to a deadline miss incurred in the present. This ensures

that the infinite horizon total cost is finite. An optimality re-

sult for the infinite horizon discounted cost formulation can

be converted into an optimality result for the average cost

formulation, which is the deadline miss ratio formulation we

are interested in, by taking the limit as γ → 0 [1].

We have proved the following theorem for the periodic

model with two flows2:

Theorem 4. The optimal policy π∗(D1,D2,x,y) that min-

imizes the average number of deadline misses has the follow-

ing structure:

1. When packets of both flows are in the system, it sched-

ules packets in earliest-deadline-first order, i.e.,

flow∗(D1,D2) = argminiDi. When packets of only one

of the flows is in the system, it schedules that flow’s

packet.

2. The optimal rate selection rule is independent of the

packet scheduling rule, i.e., r∗(D1,D2) = r∗.

3. The optimal transmission rate selection rule selects the

rate with the minimum expected transmission time, i.e.,

r∗= argmini

li

(1−pi).

Observation 5. The multi-rate real-time scheduling prob-

lem in the approximate model has a closed form solution, and

rate selection is decoupled from packet arbitration. This is

in contrast to the realistic model with constant size packets,

where the rate selection problem is, in general, NP-hard and

cannot be decoupled from packet scheduling. This is true for

both one shot and periodic arrivals.

1There is a technical difference between the two formula-

tions; the one shot model is a finite horizon problem, while

the periodic model is an infinite horizon problem.

2We can generalize the periodic model proof to the case with

N periodic flows; the calculations are more involved in this

case

6.HEURISTIC APPROACHES

The theoretical insights derived in Sections 4 and 5 can be

used to design heuristic policies. We have investigated the

following rate-selection heuristics suggested by our theory:

1. Select the rate that minimizes p1/li

i

2. Select the rate with the minimum expected transmission

time (ETT)

1−pi

li

3. Select the rate that minimizes a hybrid metric:

αp1/li

i

+ (1 − α)

maxiETTi

In terms of joint rate-and-packet selection policies, our

theory suggests heuristics such as instantaneous greedy choice

policies, and partial lookahead policies.

ETTi

7.RELATED WORK

The seminal work of Liu and Layland [6] in real-time

scheduling established the optimality of the earliest-deadline-

first (EDF) scheduling policy.

Many rate adaptation algorithms have focused on boost-

ing throughput performance. Of these, SampleRate [3] has

most relevance to our work because it addresses the di-

chotomy between transmission rates and packet error rate,

albeit from a throughput maximization perspective.

[7] describe a first principles approach to real-time schedul-

ing in wireless environments with multi-user diversity, and

use stochastic dynamic programming to show a “virtual-

deadline-first” structure for the optimal policy.

developed a similar dynamic programming approach for an

inter-packet deadline (IPD) arrival model, and shown a switch-

ing curve structure for the optimal policy. A simulation-

based approach is described in [5] for scheduling packets

with deadlines in multi-rate wireless networks under a“con-

tinuum of rates” assumption.

[4] have

8.

[1] D. Bertsekas. Dynamic Programming and Optimal

Control: Vol I and II. Athena Scientific, 2005.

[2] V. Bhandari, V. Raghunathan, B. D. Bui, and

M. Caccamo. Real-time implications of multiple

transmission rates in wireless networks. Technical

report, University of Illinois, Urbana-Champaign, 2007.

[3] J. Bicket. Bit-rate selection in wireless networks.

Master’s thesis, MIT, 2005.

[4] A. Dua and N. Bambos. Deadline constrained packet

scheduling in wireless networks. In Proc. of IEEE VTC

2005.

[5] K. M. Elsayed and A. K. Khattab. Channel-aware

earliest deadline due fair scheduling for wireless

multimedia networks. Wirel. Pers. Commun.,

38(2):233–252, 2006.

[6] C. L. Liu and J. W. Layland. Scheduling algorithms for

multiprogramming in a hard-real-time environment.

Journal of the ACM, 20(1), 1973.

[7] V. Raghunathan, M. Cao, and P. R. Kumar. Real-time

scheduling for wireless systems with losses. Technical

report, University of Illinois, Urbana Champaign, 2007.

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