Realtime implications of multiple transmission rates in wireless networks.
ABSTRACT Wireless networks are increasingly being used for latencysensitive applications that require data delivery to be timely, efficient and reliable. This trend is primarily driven by the proliferation of wireless networks of realtime datagathering sensoractuator devices. This has led to a strong need to bring realtime concerns to the forefront of an integrated research thrust into wireless realtime systems. In this paper, we introduce and analyze a specific instance of the rich set of problems in this domain. We consider a wireless network serving realtime flows in which the underlying physical layer provides multiple transmission rates. Higher rates have more stringent SINR requirements and thus represent a tradeoff between raw transmission speed and packet error rate. We adopt a first principles approach to the design of optimal realtime scheduling algorithms for such a multirate 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 indicates that this is a rich problem domain with the potential for a unifying theory that integrates realtime requirements into multirate wireless network design.

Article: Extended Abstract: RealTime Implications of Multiple Transmission Rates in Wireless Networks
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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 realtime datagathering sensoractuator devices. This has led to a strong need to bring realtime concerns to the forefront of an integrated research thrust into wireless realtime 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 realtime flows in which the underlying physi cal layer provides multiple transmission rates. Higher rates have more stringent SINR requirements and thus represent a tradeoff between raw transmission speed and packet error rate. We adopt a first principles approach to the design of optimal realtime scheduling algorithms for such a multirate 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 realtime requirements into multirate wireless network design. 
Article: A note on preemptive scheduling of multiclass jobs with geometric service times and hard deadlines
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ABSTRACT: We consider the scheduling of multiclass jobs with deadlines to the completion of their service. Deadlines are deterministic and job arrivals in each class occur at the times of deadline expirations in the respective class. Assuming geometric service times with class dependent means, we derive structural properties of preemptive server allocation policies that maximize the expected number of job completions. Our work extends results that have appeared in the realtime wireless scheduling literature.Journal of Scheduling 08/2013; 16(4). · 0.94 Impact Factor
Page 1
Extended Abstract: RealTime Implications of Multiple
Transmission Rates in Wireless Networks
Vartika Bhandari
Dept. of Computer Science
University of Illinois at
UrbanaChampaign
vbhandar@uiuc.edu
∗
Vivek Raghunathan
Dept. of Electrical and
Computer Engineering
University of Illinois at
UrbanaChampaign
vivek@control.csl.uiuc.edu
†
Bach Duy Bui
Dept. of Computer Science
University of Illinois at
UrbanaChampaign
bachbui2@uiuc.edu
Marco Caccamo
Dept. of Computer Science
University of Illinois at
UrbanaChampaign
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 realtime datagathering
sensoractuator devices. This has led to a strong need to
bring realtime concerns to the forefront of an integrated
research thrust into wireless realtime 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 realtime flows in which the underlying physi
cal layer provides multiple transmission rates. Higher rates
have more stringent SINR requirements and thus represent
a tradeoff between raw transmission speed and packet error
rate. We adopt a first principles approach to the design of
optimal realtime scheduling algorithms for such a multirate
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 realtime requirements
into multirate 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 CNS0613665.
Copyright is held by the author/owner(s).
MobiCom’07, September 9–14, 2007, Montréal, Québec, Canada.
ACM 9781595936813/07/0009.
Categories and Subject Descriptors
C.2.1 [Computer Communication Networks]: Network
Architecture and Design—Wireless communication; C.3 [Special
Purpose and ApplicationBased Systems]: Realtime
and embedded systems
General Terms
Performance, Theory
Keywords
Realtime scheduling, wireless networks, rate adaptation,
dynamic programming.
1. INTRODUCTION
The rapid proliferation of wireless sensoractuator net
works has brought to a fore the need to address realtime
concerns in wireless networking. Such realtime 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 timecritical actions
performed by the sensor network in response to external
events. Realtime 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 tradeoff between
raw transmission speed and packet error rate. In this work,
we analyze the realtime implications of this tradeoff 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 softrealtime approach and focus on minimizing the
expected deadlinemissratio. We use illustrative examples
to demonstrate the complex set of issues introduced by this
seemingly simple problem. We show that intuitive solutions,
e.g., earliestdeadlinefirst (EDF) scheduling at the rate with
the minimum expected transmission time (minETT), are
Page 2
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“pseudodeadline”(a value between its arrival
time and deadline) such that the optimal scheduler has an
earliest pseudodeadline 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 minETT is the optimal rate
selection rule.
Our work indicates the potential for a unifying theory
that integrates realtime requirements into multirate wire
less network design.
2.MODEL/NOTATION/TERMINOLOGY
We study a slottedtime, singlelink wireless system, where
all flows have the same sourcedestination pair. We model
a multirate wireless system, with each rate corresponding
to a different modulation, e.g., BPSK, QPSK, 16QAM and
64QAM 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 twotuple (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 relativedeadline Di = Ti. The
hyperperiod is the least common multiple(LCM) of all Ti’s.
Traditional realtime 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 oneshot 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 workconserving.
Similarly, EDF scheduling may seem intuitively optimal.
This is not always true, as shown by the following counter
example:
Example 2. Oneshot 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 hyperperiod 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 hyperperiod 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
multirate realtime scheduling problem.
4.REALISTIC MODEL: CONSTANT SIZE
PACKETS
We now build a theoretical understanding of realtime
multirate scheduling for fixed size packets. Due to lack of
space, results are stated without proof. For details, please
see [2].
4.1SinglePacket Rate Adaptation
Consider the apparently simple RATEADAPT problem,
where a single packet arriving at t = 0 with deadline d,
must be transmitted over a lossy multirate 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 RATEADAPT can be represented as a
3tuple (d,Pe,L), with Pe = L = k, Pe = {p1,p2,...,pk},
and L = {l1,l2,...,lk}.
Theorem 1. RATEADAPT is NPhard.
Greedy Heuristic for RATEADAPT. Consider the fol
lowing greedy heuristic: sort the rates in nondecreasing 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.
4.2MultiRate Scheduling Problem
The multirate realtime scheduling problem MRATESCHED
models the singlelink scheduling situation described in Sec
tion 2. Given a flowset 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. MRATESCHED is NPhard.
Page 3
Structure of Optimal MRATESCHED Algorithm
Theorem 2. There exists an optimal MRATESCHED
scheduling algorithm that has earliest pseudodeadline 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
oneshot and periodic arrival models. Additionally, the dif
ference between a packet’s deadline and pseudodeadline 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 singlelink 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 multilink scenario), this intuition no
longer holds, and it can be shown via counterexample that
EPDF is not necessarily optimal.
4.3RateSelection
In light of Theorem 2, one might wonder whether within
a packet’s pseudodeadline, the optimal scheduler always se
lects the rate that is optimal for that individual packet. We
show via counterexample 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 oneshot model). There are two avail
able rates: r1 = (1,0.4),r2 = (k + 1,0.0). The hyperperiod
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 pseudodeadlines 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 pseudodeadline,
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. Rateselection 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 oneshot 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 rateset 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 nongreedy rate order r1,r2 than
the greedy sequence r2,r1.
One might ask whether the ordersensitivity is only an
artefact of the potentially suboptimal use of EDF in the
above example. This is easily shown not to be the case. From
Theorem 2, τ1’s pseudodeadline 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 pseudodeadline 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 nongreedy 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 pseudodeadline.
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 subschedule 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
Page 4
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 oneshot model, the optimal policy
π∗(¯D) that minimizes the total number of deadline misses
has the following structure:
1. It schedules packets in earliestdeadlinefirst (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 earliestdeadlinefirst 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 multirate realtime 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, NPhard 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 rateselection 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 rateandpacket 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 realtime
scheduling established the optimality of the earliestdeadline
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 realtime schedul
ing in wireless environments with multiuser diversity, and
use stochastic dynamic programming to show a “virtual
deadlinefirst” structure for the optimal policy.
developed a similar dynamic programming approach for an
interpacket 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 multirate 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. Realtime implications of multiple
transmission rates in wireless networks. Technical
report, University of Illinois, UrbanaChampaign, 2007.
[3] J. Bicket. Bitrate 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. Channelaware
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 hardrealtime environment.
Journal of the ACM, 20(1), 1973.
[7] V. Raghunathan, M. Cao, and P. R. Kumar. Realtime
scheduling for wireless systems with losses. Technical
report, University of Illinois, Urbana Champaign, 2007.
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