Power Optimal Scheduling with Maximum Delay Constraints
Mohammad Ali Khojastepour∗ and Ashutosh Sabharwal∗
Department of ECE, Rice University, 6100 Main St., MS-366, Houston, TX 77005
e-mail: {amir,ashu}@rice.edu
Most multimedia sources are bursty in nature, a property which can be used to trade
queuing delay with the resulting average transmission power [2, 3, 4]. In this paper, we
study the relation between average transmission power and strict delay constraints.
Our main contributions are two-fold. First, we establish necessary and sufficient
conditions on the service rates of the wireless transmitter, to meet the delay deadline
of every packet in the queue. Second, the conditions are used to show that a scheduler
which meets a delay guarantee Dmax for each of the packet over Gaussian channels is
a time-varying low-pass filter of order no more than Dmax . This confirms the intuitive
explanation for power reduction due to additional queuing delay provided in [3]. Using the
relation between delay bounded scheduling and linear filtering, we construct schedulers
without the knowledge of source statistics. This marks a significant departure from most
information theoretic work on power efficient scheduling [2, 3]. We construct the optimal
time-invariant scheduler, which does not require the knowledge of the source statistics.
The work of this paper is motivated by several factors. In most situations, source and
channel probability distribution is seldom known completely. Thus, optimal schedulers
built using the probability distributions can possibly incur significant loss in practical
systems. Thus seeking robust schedulers, we investigate the achievable performance with
no prior assumptions about source probability distribution. Though our results only
focus on Gaussian channels with bursty sources, we foresee using source-channel duality
at packet time-scales to arrive at scheduler design for transmission of bursty sources over
fading channels with no assumptions on their probability distributions.
Consider the system in Figure 1, where the number of input packets to the queue
at time t is given by a random process Xt . The arriving packets are queued in a buffer
whose backlog before time t is denoted by Bt . We assume that all packets have the same
size. At any time t, a causal scheduler transmits Rt packets out of the queue by looking
at the queue size Bt , and having the knowledge of all the previous input arrivals {Xi }ti=1 .
On the other hand, a non-causal scheduler uses all arrivals {Xi }∞
i=1 .
In this paper, we will consider the case when each packet is required to be delivered
within the Dmax time-slots of its arrival. The scheduler will be designed to adapt the
transmission rate Rt such the average transmit power E [Pt ] is minimized while meeting
the maximum queuing delay constraint for each packet. We consider a Gaussian channel
between transmitter and receiver. Motivated by outage analysis in [1], we assume that
all transmissions occur in packets and the length of the packet is long enough to allow
reliable communication close to the
of the channel. Therefore, the
h mutual information
P N Rt i
1
is minimized.
scheduler is designed such that E limN →∞ N t=1 2
∗
This work was supported in part by Nokia Corporation and NSF Grant CCR-0311398.
Input arrival
Output service rate
Xt
Rt
Bt
Queue backlog
Figure 1: Conceptual depiction of a queuing system.
It is shown that the necessary and sufficient conditions for a scheduler to guarantee
maximum delay of Dmax are given by
Xt + Xt+1 + . . . + Xt+k ≤ Rt + Rt+1 + . . . + Rt+k+Dmax −1
(1)
Rt + Rt+1 + . . . + Rt+k ≤ Bt + Xt + Xt+1 + . . . + Xt+k
(2)
Using (1), we show that structure of any scheduler which guarantees the maximum
delay of Dmax is equivalent to a linear time-varying filter of size no more than Dmax given
by
t
Xt−Dmax +1 ,
(3)
Rt = α0t Xt + α1t Xt−1 + . . . + αD
max −1
P max −1 t+i
where the filter coefficients satisfy D
αi = 1 and 0 ≤ αit ≤ 1 for all t and i.
i=0
This turns the design of the guaranteed maximum delay scheduler into the problem
of filter design with a ‘linear’ structure. Thus, it is possible that the vast literature on
linear filtering theory could have bearing in the design of power-efficient schedulers. For
example it is easy to show that every feasible time-invariant scheduler is a low-pass filter,
which confirms the intuition behind smoothening of the input arrivals via queuing [3].
Also, the optimal time invariant scheduler is given by
Xt + Xt−1 + . . . + Xt−Dmax +1
.
(4)
Dmax
Interestingly, the optimal scheduler is simply a moving average low-pass filter of length
Dmax , whose filter bandwidth decreases as Dmax increases. Note that even though the
objective function (expected power) depends on the source distribution, the optimal timeinvariant scheduler is same for all i.i.d. sources and hence robust to source distribution.
Rt =
References
[1] L.H. Ozarow, S. Shamai, and A.D. Wyner, “Information theoritic considerations for
cellular mobile radio,” IEEE Tran. on Veh. Tech., Vol 43, No 2, May 1994.
[2] R. Berry and R. Gallager, “Communication Over Fading Channels with Delay Constraints,” IEEE Transactions on Information Theory, vol. 48, no. 5, May 2002.
[3] D. Rajan, A. Sabharwal, and B. Aazhang, “ Delay-bounded Scheduling of Bursty
Sources over Wireless Channels,” accepted for IEEE Trans. on IT, Dec. 2002.
[4] E. Uysal-Biyikoglu, A. El Gamal, and B. Prabhakar, “Adaptive transmission of
variable-rate data over a fading channel for energy efficiency,” GLOBECOM 2002.