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1 Deadline Constrained Packet Scheduling in Wireless Networking Aditya Dua, Nicholas Bambos Department of Electrical Engineering Stanford University {dua,bambos}@stanford.edu Abstract Third generation (3G) of wireless communication networks aim to provide quality-of-service (QoS) sensitive services to downlink users. Amongst these are real-time multimedia services like video- conferencing and live media streaming, where the packets to be delivered to the user have deadlines associated with them. Downlink packet scheduling plays a key role in efficiently allocating base station resources to meet the desired level of QoS for various users. In this paper, we study the design of a downlink wireless packet scheduler that is capable of supporting applications with packet deadlines. We use a dynamic programming (DP) approach to study a parsimonious model of the scheduling problem. We propose a heuristic scheduling policy based on the structural properties of the optimal solution to the DP. We contrast the performance of the proposed scheduler to that of standard benchmark schedulers via extensive link-level simulations. The proposed scheduler offers a significantly better trade-off between user fairness and system performance. Index Terms 3GPP, HSDPA, QoS, Packet scheduling, Dynamic Programming. July 8, 2005 DRAFT
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Deadline Constrained Packet Scheduling in

Wireless Networking

Aditya Dua, Nicholas Bambos

Department of Electrical Engineering

Stanford University

{dua,bambos}@stanford.edu

Abstract

Third generation (3G) of wireless communication networks aim to provide quality-of-service (QoS)

sensitive services to downlink users. Amongst these are real-time multimedia services like video-

conferencing and live media streaming, where the packets tobe delivered to the user have deadlines

associated with them. Downlink packet scheduling plays a key role in efficiently allocating base station

resources to meet the desired level of QoS for various users.In this paper, we study the design of a

downlink wireless packet scheduler that is capable of supporting applications with packet deadlines. We

use a dynamic programming (DP) approach to study a parsimonious model of the scheduling problem.

We propose a heuristic scheduling policy based on the structural properties of the optimal solution to the

DP. We contrast the performance of the proposed scheduler tothat of standard benchmark schedulers via

extensive link-level simulations. The proposed scheduleroffers a significantly better trade-off between

user fairness and system performance.

Index Terms

3GPP, HSDPA, QoS, Packet scheduling, Dynamic Programming.

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I. INTRODUCTION

Third generation (3G) wireless communication systems aim to support quality-of-service

(QoS) based services like interactive multimedia and high-speed data [1]. However, bandwidth

constraints (due to physical and regulatory reasons), power constraints (due to battery limitations)

and the erratic nature of the wireless channel (due to shadowing, fading and mobility) make this

a challenging task for wireless system designers [2].

While traditional digital communication system design approaches (source and channel code

design, constellation design etc.) [3] have been investigated thoroughly, cross-layer design method-

ology has attracted much interest in recent years [4]. This approach tries to exploit the synergy

existing between different layers of the communication protocol stack to achieve more efficient

designs, instead of treating each layer as an individual entity. QoS and channel aware packet

scheduling is an important illustration of the cross-layerdesign approach, which exploits the

interactions between the physical, network and application layers.

Packet scheduling is an important component of the high-speed downlink packet access

(HSDPA) technology, which has been introduced in the 3GPP Release 5 specifications to provide

high data rates and QoS sensitive services to downlink users[5]. In fact, the packet scheduler

has been moved from the radio network controller (RNC) to thebase-station (BS) to support

fast physical layer re-transmissions.

Packet scheduling for wireline networks has been studied extensively over the years. However,

scheduling algorithms designed for wireline networks are not directly applicable to the wireless

scenario, because wireless channels suffer from bursty errors and time and location dependent

capacity. Much work has been done on adapting wireline scheduling algorithms to wireless

networks. See [6] and references therein for a review.

Since the performance of any wireless packet scheduler is affected by random channel fluctu-

ations, it is essential to incorporate channel state information into the scheduling decision. For

instance, such information is available in a 3G cellular system via the channel quality indicator

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(CQI) signal, which is periodically transmitted by the downlink users to the BS. Also, using

other protocol and application related information (backlog states of various queues, link-layer

deadlines on packets, minimum data rate requirements, etc.) can enhance the performance of

the scheduler from a system perspective. From a user perspective, a scheduler must possess an

element offairness, so that each user can get its due share of service. Fair scheduling of packets

without deadlines in wireless networks has been studied in [7]. Also, a scheduler should be able

to provide differentiated QoS to users. For instance, the packets of a user who pays more to

receive service should have higher priority for scheduling. A “good” scheduler should be capable

of achieving a trade-off between system performance and user fairness in a flexible manner and

should have tunable parameters to provide differentiated service to different classes of users and

data.

Wireless schedulers that have received a lot of attention incorporate one or more of the

attributes discussed above. Theround-robinscheduler, while being fair, is insensitive to wireless

channel fluctuations and incapable of providing differentiated QoS to users. Themaximum SNR

scheduler utilizes the available bandwidth efficiently by transmitting to the user with the best

quality channel. However, it is unfair to users with poor channels and can lead to large fluctuations

in QoS experienced by downlink users. The proportional-fair scheduler [8] achieves a trade-off

between system throughput and long-term user fairness. None of these schedulers are suited to

real-time applications with packet deadlines.

The QoS requirements for real-time applications are very different from those of non-real-time

applications. For instance, video-conferencing, which isa real-time variable-bit-rate (rt-VBR)

application has a delay bound of 40-90 ms and an acceptable loss rate of10−3. In contrast, non-

real-time applications like web browsing and file transfer have a stringent acceptable loss rate

requirement of10−8, but can withstand large delays. These numbers clearly indicate the need for

designing packet scheduling algorithms tailored to the requirements of real-time applications.

The earliest-deadline-first(EDF) scheduling policy, also known as theearliest-due-dateor

shortest time-to-extinction(STE) has been shown to be optimal for deadline constrained schedul-

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ing over wireline (error-free) channels under various modeling assumptions. A variant of EDD

applicable to wireless networks, namelyfeasible earliest deadline due(FEDD) is studied in

[9] and shown to perform well for two-state Markovian (ON-OFF) wireless channels. The

EDF scheduling policy and its variants are easily implementable, but inflexible and incapable

of providing differentiated QoS to users. In addition, EDF does not account for variations in

channel conditions and is expected to be unfair to users withpoor channel conditions. In [10],

the authors study downlink scheduling of a mixture of real-time and non-real-time traffic and

show that theexponential rule(EXP-rule) performs well for both types of traffic. In [11], the

authors study deadline constrained scheduling for a revenue based model and show that a simple

greedy algorithm achieves a competitive ratio of12

under certain assumptions on the revenue

function. In [12], the authors consider delay sensitive scheduling based on minimizing a heuristic

cost function that accounts for both channel and queue states.

Our goal in this paper is to design a flexible and easily implementable downlink packet

scheduling algorithm for a scenario in which packets in eachqueue have associated deadlines,

as is the case in multimedia and mobile computing applications. Such constraints can also be

artificially imposed to achieve objectives like load balancing across streams in mobile computing

and networking applications.

We define the scheduling problem and an associated parsimonious model in Section II. In

Section III, we capture the scheduling trade-offs within a dynamic program (DP) framework

[13], [14] and state some key results regarding the structural properties of its optimal solution.

A DP based approach lends itself naturally to the problem of packet scheduling. For instance,

downlink scheduling for non-real time data using a DP formulation is studied in [15]. We study

the asymptotic behavior of the optimal solution in Section IV and construct an approximation to

the optimal solution in Section V. Based on the latter, we propose a heuristic scheduling policy

and evaluate its performance via extensive link level simulations in Section VII. We demonstrate

that under several different scenarios, the proposed scheduler offers a significantly better trade-

off between user fairness and system performance compared to other schedulers. We conclude

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in Section VIII.

II. M ODEL CONSTRUCTION ANDREDUCTION

We consider a time-division multiplexed (TDM) wireless cellular communication system. Time

is slotted into fixed size transmission time intervals (TTI)or slots. There is a queue corresponding

to each downlink user at the base station (BS) which is servedaccording to a first-come first-

served (FCFS) discipline. In each slot, the BS schedules thehead-of-line (HOL) packet of a

non-empty queue for transmission, and selects a modulationand coding scheme (MCS) and a

transmission power for the scheduled packet. The channel for each user exhibits temporal and

spatial random fluctuations. Packets in each queue have associated deadlines, that is, if they

are not (successfully) transmitted before the expiration of the deadline they get dropped. The

scheduled user is notified of the scheduling decision of the BS prior to transmission. We assume

that the notification and transmission occur in the same slot. An instantaneous and error free

feedback indicating a successful/failed transmission is available on the uplink to the BS. Also,

we assume that the scheduler has perfect and most recent knowledge of the queue backlog state

and channel conditions for each user.

Despite the simplifying assumptions, the scheduling problem posed in the aforementioned

form is hard to solve. We can cast it into a DP framework and solve it numerically. However,

such a solution will neither provide any structural insightinto the scheduling problem, nor will

its complexity be desirable from an implementation perspective. In what follows, we construct

a parsimonious reduction of the original problem and derivea heuristic scheduling policy based

on the optimal solution to the reduced problem.

A. Model Reduction

If we were to solve the scheduling problem as posed above using a DP approach, the state

of the system would comprise the deadlines associated with each packet in each queue, in

addition to all the channel states. This would clearly lead to a huge state-space that grows

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exponentially with the number of users and render the problem intractable. However, for delay

sensitive applications like multimedia streaming, packets in a queue exhibit strong temporal

correlation in that they are typically ordered in increasing order of their deadlines. It is important

for packets to meet differential deadlines relative to the packets ahead of them in the queue, rather

than meeting absolute deadlines. For instance, in a lossless multimedia transmission scenario (no

packet dropping), if the current video frame gets delayed, then the deadlines on subsequent frames

in the sequence are also incremented appropriately to accommodate this delay. We refer to these

differential deadlines asinter-packet deadlines(IPD). We leverage the temporal correlation to

achieve a tremendous reduction in the size of the state spaceby using the IPD of the HOL

packet (relative to the preceding HOL packet) as the state ofeach queue.

B. Reduced Problem

With the above reduction in state-space, let us consider a discrete-time system where two

queues,Q1 andQ2 are competing for a single server (transmitter)S to transmit their packets

to receiversR1 and R2 respectively. We assume that each queue hasexactly one packet. In

each time slot,S selects eitherQ1 or Q2 according to some scheduling policy and transmits its

HOL packet. The HOL packet ofQi has an associated deadline, that is, if it is not transmitted

successfully before the deadline expires, it is dropped. Inthe context of the original problem,

this deadline reflects the the inter-packet deadline or IPD (relative to the preceding HOL packet)

associated with the HOL packet ofQi1. We model the channels from the transmitter to each

receiver as independent and identically distributed (i.i.d) random processes, independent of each

other. A simple truncated ARQ mechanism is employed for eachqueue, that is, a packet is

transmitted till it is received successfully or its associated deadline expires, and the success

probability of a re-transmission is unaffected by previoustransmissions. Once the HOL packet

of a queue is successfully transmitted it stays empty and is not a potential candidate for scheduling

1When there is only packet in the queue, the inter-packet deadline (IPD) is same as the (absolute) deadline. Hence, in the

context of the reduced problem we use deadline and IPD interchangeably.

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in subsequent time-slots. Each packet incurs a buffering cost for every time slot it spends in the

queue, and a dropping cost if it is not transmitted successfully prior to the expiry of its deadline.

We re-emphasize that adopting thislocal perspectiveon delay constrained scheduling based

on inter-packet deadlines yields a tremendous reduction inthe size of the state-space.

We now state some definitions that will be used extensively inthe rest of the paper.

Definition: Let Di be the deadline associated with the HOL packet ofQi.

Definition: Let si be the probability of successful transmission of the HOL packet of Qi,

if it is scheduled.

Definition: Let ci be the buffering cost per slot incurred by the HOL packet ofQi.

Definition: Let λi be the dropping cost incurred by the HOL packet ofQi, if it is dropped

due to expiration of its deadline before successful transmission.

Definition: Let ni be the time-to-expiration (TTE) of the original deadlineDi on the HOL

packet ofQi.

Note that by definition,0 ≤ ni ≤ Di.

Definition: Let thestate of the system be the two-tuple(n1, n2).

Let P denote the set of all non-idling packet scheduling policiesthat schedule the HOL packet

of exactly one non-empty queue in each time slot. With the above defined state-space and cost

structure, our goal is to design the optimal packet scheduling policyP⋆ ∈ P that minimizes the

total expected buffering and dropping costs.

The states(n1, 0) ∀n1 ∈ N and (0, n2) ∀n2 ∈ N respectively correspond toQ2 andQ1 being

empty 2. Thus, there is no scheduling decision to be made in these states andQ1 andQ2 are

respectively scheduled. The system dynamics under a candidate scheduling policyP ∈ P, if it

schedulesQ1 in state(n1, n2) ∀n1, n2 ∈ N are described below:

• n1 > 1 andn2 > 1: The transmission is successful with probabilitys1 and the new state

is (0, n2 − 1). The transmission fails with probability(1 − s1), and the new state is(n1 −

2Throughout this paper, we denoteZ0+ = {0, 1, 2, . . .}, N = {1, 2, 3, . . .} andN\{1} = {2, 3, 4, . . .}.

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1, n2 − 1).

• n1 = 1 andn2 > 1: The new state is always(0, n2 − 1). However, the transmitted packet

gets dropped with probability(1 − s1), and a dropping cost ofλ1 is incurred.

• n1 > 1 andn2 = 1: The transmission is successful with probabilitys1 and the new state is

(0, 0), and the transmission fails with probability(1−s1), and the new state is(n1−1, 0). The

HOL packet ofQ2 is dropped in both cases, and a dropping cost ofλ2 is incurred. Note that

the scheduling algorithm terminates in the former case, that is, (0, 0) is an absorbing/stopping

state.

The dynamics of the system state can be similarly explained if P schedulesQ2 in state(n1, n2)

∀n1, n2 ∈ N.

Remarks:Assigning possibly different dropping costs to packets allows the scheduler to

provide differentiated QoS to users and also to packets within the same queue. Delay-sensitive

scheduling policies like EDD are incapable of such differentiation. We illustrate this via simula-

tion results in Section VII. Incorporating buffering costsin the formulation provides additional

degrees of control. In some applications there may be an incentive not merely to meet the

packet deadlines, but transmit as many packets as possible.An example is video transmission

with packets coded using incremental redundancy (IR), where the video quality can be enhanced

if incremental information is transmitted in addition to the base frame before the deadline expires.

Incorporating a buffering cost into the model helps providesuch differentiation between queues,

and also between packets within the same queue. Also, as discussed in Section IV, the buffering

costs determine the behavior of the scheduler when packets are far off from their respective

deadlines.

III. DYNAMIC PROGRAMMING FORMULATION

Definition: Let V (n1, n2) be theexpected cost-to-goin state(n1, n2) if the optimal control

is used, that is, packets are scheduled using the policyP⋆. Thus,V (n1, n2) is the expected cost

(buffering and dropping) of successfully transmitting or dropping the HOL packets ofQ1 and

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Q2 when their times-to-expiration aren1 andn2 respectively.

If n1, n2 ∈ N\{1}, the system transitions from state(n1, n2) to state(0, n2−1) with probability

s1 and to state(n1−1, n2−1) with probability(1−s1) if Q1 is scheduled. The system transitions

to state(n1 − 1, 0) with probability s2 and state(n1 − 1, n2 − 1) with probability (1− s2) if Q2

is scheduled. Thus,V (n1, n2) is given by the recursive equation

V (n1, n2) = min{s1V (0, n2 − 1) + (1 − s1)V (n1 − 1, n2 − 1),

s2V (n1 − 1, 0) + (1 − s2)V (n1 − 1, n2 − 1)} + c1 + c2, n1, n2 ∈ N\{1}. (1)

To simplify notation, we define

α(n1, n2) = V (0, n2 − 1) − V (n1 − 1, n2 − 1), n1, n2 ∈ N\{1}

β(n1, n2) = V (n1 − 1, 0) − V (n1 − 1, n2 − 1), n1, n2 ∈ N\{1}, (2)

and expressV (n1, n2) as

V (n1, n2) = min{s1α(n1, n2), s2β(n1, n2)}+V (n1−1, n2−1)+c1+c2, n1, n2 ∈ N\{1}. (3)

If n1 = 1, n2 ∈ N\{1}, the system always transitions to state(0, n2 − 1) if Q1 is scheduled.

If Q2 is scheduled, the system transitions to state(0, 0) with probability s2 and state(0, n2 − 1)

with probability(1−s2). In the former case the HOL packet ofQ1 gets dropped with probability

(1 − s1), while in the latter case it always gets dropped. The state transitions can be similarly

described ifn1 ∈ N\{1}, n2 = 1. We have,

V (1, n2) = min{V (0, n2 − 1) + (1 − s1)λ1, (1 − s2)V (0, n2 − 1) + λ1} + c1 + c2, n2 ∈ N\{1}

V (n1, 1) = min{(1 − s1)V (n1 − 1, 0) + λ2, V (n1 − 1, 0) + (1 − s2)λ2} + c1 + c2, n1 ∈ N\{1}.

(4)

To fully characterize the cost-to-go function we specify the boundary conditions. The state(0, 0)

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is an absorbing state for the system and has zero cost associated with it.

V (0, 0) = 0

V (0, 1) = c2 + (1 − s2)λ2

V (1, 0) = c1 + (1 − s1)λ1

V (1, 1) = min{(1 − s1)λ1 + λ2, λ1 + (1 − s2)λ2} + c1 + c2. (5)

The behavior of the optimal scheduling policyP⋆ in states(n1, n2) ∀n1, n2 ∈ N\{0} is

captured by the functionγ(n1, n2) : N\{1} × N\{1} → R, which we now define:

Definition: Let γ(n1, n2) = s1α(n1, n2) − s2β(n1, n2), n1, n2 ∈ N\{1}.

Using (3) and the definition ofγ(n1, n2), P⋆ schedulesQ1 in state(n1, n2) ∀n1, n2 ∈ N\{1}

if γ(n1, n2) ≤ 0, andQ2 if γ(n1, n2) > 0.

We now state some results regarding the structural properties of the optimal scheduling policy

P⋆.

Lemma 1:

(a) V (n1, 0) =c1s1

+ (1 − s1)n1

(λ1 −

c1s1

), n1 ∈ N, s1 ∈ (0, 1]

(b) V (0, n2) =c2s2

+ (1 − s2)n2

(λ2 −

c2s2

), n2 ∈ N, s2 ∈ (0, 1].

Proof: See Appendix.

Remarks:V (n1, 0) represents the expected buffering and dropping cost incurred in the

transmission of the HOL packet ofQ1 when it hasn1 > 0 time slots remaining before the

expiry of its deadline. Since our focus is on deadline constrained packet scheduling we expect

the dropping cost to dominate the buffering cost, especially close to deadline expiration. Thus, we

would expect the total expected cost to increase as we get closer to the deadline (n1 decreases)

since the packet dropping probability increases (since thechannel is time invariant and all

transmissions are independent of past transmissions). Now, the sequenceV (n1, 0), n1 ∈ N

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is a decreasing sequence ifc1 < s1λ1 and an increasing sequence ifc1 > s1λ1. Based on the

foregoing discussion, we assume that the problem parameters satisfyc1 < s1λ1. Similarly, we

also assumec2 < s2λ2.

Lemma 2: In the states(n1, 1), n1 ∈ N one of the following is possible

(a) P⋆ schedulesQ1 for all n1 ∈ N.

(b) P⋆ schedulesQ2 for all n1 ∈ N.

(c) There exists an⋆1 > 1, such thatP⋆ schedulesQ1 for 1 ≤ n1 < n⋆

1 andQ2 for n1 ≥ n⋆1.

Proof: See Appendix.

Lemma 3: In the state(1, n2) , n2 ∈ N one of the following is possible

(a) P⋆ schedulesQ1 for all n1 ∈ N.

(b) P⋆ schedulesQ2 for all n1 ∈ N.

(c) There exists an⋆2 > 1, such thatP⋆ schedulesQ2 for 1 ≤ n2 < n⋆

2 andQ1 for n2 ≥ n⋆2.

Proof: The proof is symmetric to the proof of Lemma 2.

Now, we state a key lemma that will enable us to deduce the formof the optimal scheduling

policy P⋆.

Lemma 4:γ(n1, n2 + 1) ≤ γ(n1, n2) ≤ γ(n1 + 1, n2), n1, n2 ∈ N\{1}.

Proof: See Appendix.

Equipped with Lemmas 1, 2, 3 and 4, we are ready to state a result regarding the form of

the optimal scheduling policy for the two-queue single-server deadline constrained scheduling

problem with time-invariant channels. Before that, we havea definition.

Definition: A scheduling policyP ∈ P is called aswitch-over policyif there is a non-

decreasing switch-over functionφ : Z0+ → Z0+ such that it schedulesQ1 in state(n1, n2),

n1, n2 ∈ Z0+ if n2 ≥ φ(n1) andQ2 else [14].

Theorem 1:[Optimality of Switch-over Policy] The optimal packet scheduling policyP⋆ ∈ P

is a switch-over policy for all states(n1, n2), n1, n2 ∈ N.

Proof: See Appendix.

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IV. A SYMPTOTIC BEHAVIOR OF P⋆

From Lemma 4 we have thatγ(n1, n2) is monotonically increasing inn1 for fixed n2, and

monotonically decreasing inn2 for fixed n1. Thus, limn1→∞

γ(n1, n2) and limn2→∞

γ(n1, n2) exist,

even though they may be+∞ and−∞ respectively. From Lemma 1 we have

limn1→∞

V (n1, 0) =c1s1

limn2→∞

V (0, n2) =c2s2. (6)

Lemma 5:

limn1→∞

limn2→∞

γ(n1, n2) = limn2→∞

limn1→∞

γ(n1, n2) = γ⋆.

Proof: See Appendix.

Now, from the definition ofγ(n1, n2) and (6),

limn1→∞

limn2→∞

γ(n1, n2) =s1c2s2

−s2c1s1

+ (s2 − s1) limn1→∞

limn2→∞

V (n1 − 1, n2 − 1)

limn2→∞

limn1→∞

γ(n1, n2) =s1c2s2

−s2c1s1

+ (s2 − s1) limn2→∞

limn1→∞

V (n1 − 1, n2 − 1). (7)

From (7) and Lemma 5 we conclude,

limn1→∞

limn2→∞

V (n1, n2) = limn2→∞

limn1→∞

V (n1, n2) , V ⋆. (8)

As a direct consequence,α(n1, n2) andβ(n1, n2) also have unique successive limitsα⋆ andβ⋆

respectively. In particular,

α⋆ =c2s2

− V ⋆

β⋆ =c1s1

− V ⋆. (9)

Having established the existence of a unique successive limit for the cost-to-go function, we

express the limit as a function of problem parameters, namely, ci, si andλi (i = 1, 2). From the

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definition of V (n1, n2), we note thatV ⋆ must satisfy

V ⋆ = min(α⋆, β⋆) + V ⋆ + c1 + c2

= min

(s1c2s2

− s1V⋆, s2

c1s1

− s2V⋆

)+ V ⋆ + c1 + c2. (10)

We have two cases:

(a) s1c2s2

−s1V⋆ ≤ s2

c1s1

−s2V⋆: In this case, it is optimal to scheduleQ1 for largen1, n2. We

use (10) to getV ⋆ =c1s1

+c2s2

+c1s2

. On substituting this value ofV ⋆ into the inequality,

we obtain the conditionc2s1

≤c1s2

, or equivalently,s1c1 ≥ s2c2.

(b) s1c2s2

− s1V⋆ > s2

c1s1

− s2V⋆: In this case, it is optimal to scheduleQ2 for large n1, n2.

We use (10) to getV ⋆ =c1s1

+c2s2

+c2s1

. On substituting this value ofV ⋆ in the inequality

we obtain the conditionc2s1>c1s2

, or equivalently,s2c2 > s1c1.

Thus, it is optimal to scheduleQi (i = 1, 2) for largen1, n2, if sici = max{s1c1, s2c2}. From

(a) and (b) above, the successive limit of the cost-to-go function is given by

V ⋆ =c1s1

+c2s2

+ min

{c1s2,c2s1

}. (11)

From (7), we obtain

γ⋆ =

c2s2

s1− c1 s1c1 ≥ s2c2

c2 − c1s1

s2s1c1 < s2c2.

(12)

Now, we state another key result regarding the structural properties of the optimal policyP⋆.

Theorem 2:[Saturation of Switch-over Curve]

(a) If s1c1 ≥ s2c2, there existsn⋆1, n

⋆2 ∈ N such thatφ(n1) = n⋆

2 ∀n1 ≥ n⋆1.

(b) If s1c1 < s2c2, there existsn⋆1 ∈ N such thatφ(n1) = ∞ ∀n1 > n⋆

1,

whereφ is the switch-over curve, as defined in Theorem 1.

Proof: See Appendix.

V. APPROXIMATING THE SWITCH-OVER CURVE

We have demonstrated that the optimal packet scheduling policy P⋆ is a switch-over policy.

It is hard to analytically compute the switch-over functionφ. The solution can be obtained via

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uneqlam.eps

Fig. 1. (Approximate) shape of switch-over curves for different scenarios -Clockwise:(a) s1c1 ≥ s2c2, s1λ1 < s2λ2, (b)

s1c1 ≥ s2c2, s1λ1 ≥ s2λ2, (c) s1c1 < s2c2, s1λ1 ≥ s2λ2, (d) s1c1 < s2c2, s1λ1 < s2λ2; • and◦ denote the states in which

it is optimal to scheduleQ1 andQ2, respectively. O and K respectively denote the location of the offset and the knee

numerical methods, however, such an approach is computationally expensive and not scaleable

from an implementation perspective. Thus, it is natural to seek an approximation to the switch-

over curve that can be characterized parsimoniously and computed easily.

Possible forms of the switch-over curve are depicted in Figure 1. The figure suggests that the

switch-over curve can be closely approximated by a piece-wise linear curve with two components,

one with slope0 or π/2 and the other with slopeπ/4. As shown in the figure, the piece-wise

linear approximation can be characterized by two points: the kneeand theoffset. These points

can be computed analytically in terms of the problem parameters, namelysi, ci, λi (i = 1, 2).

In this Section, we derive analytical expressions for thekneeand theoffsetunder the assump-

tion (s1λ1 − c1) > 0 and (s2λ2 − c2) > 0. For notational ease, let us denote

η1 =s2λ2 − c1s1λ1 − c1

, η2 =s1λ1 − c2s2λ2 − c2

.

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A. Offset Computation

For certain choices of problem parameters, we can have degenerate cases whereP⋆ either

schedulesQ1 in all states, or schedulesQ2 in all states. For the non-degenerate cases, theoffset

is either a point on the line(n2 = 1, n1 ∈ N) or the line(n1 = 1, n2 ∈ N).

From the proof of Lemma 2,P⋆ schedulesQ1 in state (n1, 1), n1 ∈ N if the following

inequality holds:

(s1λ1 − c1)(1 − s1)n1−1 ≥ (s2λ2 − c2).

Note that the left hand side of the inequality if always positive. We have the following cases:

(a) (s2λ2 − c2) < 0: The inequality is trivially satisfied. Thus,P⋆ schedulesQ1 in states

(n1, 1), ∀n1 ∈ N. From Theorem 1 it follows thatP⋆ schedulesQ1 in all states(n1, n2),

n1, n2 ∈ N. Thus, we have a degenrate case.

(b) (s2λ2 − c2) ≥ 0: We can further split this case into two cases:

(i) η1 ≥ 1: The inequality is not satisfied for any choice ofn1. Thus, theoffsetdoes not

lie on the linen1 ∈ N, n2 = 1. Note that this condition is equivalent tos1λ1 < s2λ2.

(ii) η1 < 1: In this case, the largest value ofn1 that satisfies the inequality is given by

n⋆1 =

⌊1 +

log(η1)

log(1 − s1)

⌋. Thus,P⋆ schedulesQ1 for n1 ≤ n⋆

1 andQ2 for n1 > n⋆1

in states(n1, 1). Note that the condition is equivalent tos1λ1 ≥ s2λ2.

Similarly, P⋆ schedulesQ1 in state(1, n2), n2 ∈ N if the following inequality holds:

(s2λ2 − c2)(1 − s2)n2−1 ≤ (s1λ1 − c2).

The possible cases are:

(a) (s1λ1 − c2) < 0: P⋆ schedulesQ2 in all states(n1, n2), n1, n2 ∈ N.

(b) (s1λ1 − c2) ≥ 0: We can split this case further into two cases:

(i) η2 ≥ 1: P⋆ schedulesQ1 in states(1, n2) ∀n2 ∈ N. The offset lies on the line

n1 ∈ N, n2 = 1. Note that this condition is equivalent tos1λ1 ≥ s2λ2.

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(ii) η2 < 1: In states(1, n2), n2 ∈ N, P⋆ schedulesQ2 for n2 < n⋆2 andQ1 for n2 ≥

n⋆2, wheren⋆

2 is given byn⋆2 =

⌈1 +

log(η2)

log(1 − s2)

⌉. This condition is equivalent to

s1λ1 < s2λ2.

We can summarize the above computations as follows:

(a) c1 > s2λ2: P⋆ always schedulesQ1.

(b) c2 > s1λ1: P⋆ always schedulesQ2.

(c) s1λ1 ≥ s2λ2: The offsetis at the point(n⋆1, 1) wheren⋆

1 =

⌊1 +

log(η1)

log(1 − s1)

⌋.

(d) s1λ1 < s2λ2: The offsetis at the point(1, n⋆2), wheren⋆

2 =

⌈1 +

log(η2)

log(1 − s2)

⌉.

We re-emphasize that the aforementioned results hold underthe assumptions(s1λ1 − c1) > 0

and (s2λ2 − c2) > 0.

B. Knee Computation

As discussed earlier, we can have two degenrate cases, namely c1 > s2λ2 whereP⋆ schedules

Q1 in all states andc2 > s1λ2 whereP⋆ schedulesQ2 in all states. There is no need to compute

the knee for these cases and hence we preclude them from consideration in this sub-section.

Now, based on the analysis of the asymptotic behavior ofP⋆ in Section IV, we have two cases,

namely,s1c1 ≥ s2c2 and s1c1 < s2c2. We treat each case separately in the remainder of this

sub-section.

1) s1c1 ≥ s2c2: In this case, the location of thekneeis determined by the zero-crossing of

the sequenceγ(∞, n2). In fact, thekneeis the smallestn2 = n⋆2 for which P⋆ schedulesQ1.

From Lemma 4,γ(∞, n2) is a monotone non-increasing sequence. Thus, it has at most one

zero-crossing, which implies thekneeis unique. From Theorem 1, we know thatP⋆ schedules

Q2 in states(∞, n2) ∀n2 < n⋆2. In particular, from we have from (1)

V (∞, n2) = s2V (∞, 0) + (1 − s2)V (∞, n2 − 1) + c1 + c2 n2 = 2, . . . , n⋆2 − 1.

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Let ψn2 = V (∞, n2), n2 = 2, . . . , n⋆2 − 1. We get the difference equation

ψn2 = s2V (∞, 0) + (1 − s2)ψn2−1 + c1 + c2 n2 = 2, . . . , n⋆2 − 1.

From Lemma 1,V (∞, 0) =c1s1

. Sincec1 < s2λ2, P⋆ schedulesQ2 in the state(∞, 1) and we

getψ1 = V (∞, 1) =c1s1

+ c1 + c2 + (1− s2)λ2. With this initial condition, we recursively solve

the difference equation to obtain

ψn⋆2−1 =

(1 − s2)n⋆

2−1

s2(s2λ2 − c1 − c2) +

c1s1

+c2s2

+c1s2.

Now, by definition ofn⋆2, P

⋆ schedulesQ1 in state(∞, n⋆2. Thus,γ(∞, n⋆

2) ≤ 0. By definition,

γ(∞, n⋆2) = s1α(∞, n⋆

2) − s2β(∞, n⋆2), whereα(∞, n⋆

2) = V (0, n⋆2 − 1) − V (∞, n⋆

2 − 1) and

β(∞, n⋆2) = V (∞, 0)−V (∞, n⋆

2−1). We computeV (0, n⋆2−1) from Lemma 1. UsingV (∞, n⋆

2−

1) = ψn⋆2−1 and settingγ(∞, n⋆

2) ≤ 0 we get

(1 − s2)n⋆

2−1 [(s1c1 − s2c2) + s2(s2λ2 − c1)] ≤ (s1c1 − s2c2).

The kneeis set to the smallest integer value ofn⋆2 that satisfies the above. Hence,

n⋆2 =

1 +log

((s1c1−s2c2)

(s1c1−s2c2)+s2(s2λ2−c1)

)

log(1 − s2)

. (13)

2) s1c1 < s2c2: The computation of thekneeis analogous to the cases1c1 ≥ s2c2. Hence,

we provide only the final results. In this case, theknee is the largestn1 = n⋆1 for which P⋆

schedulesQ1 in state(n1,∞). The location of thekneeis determined by the zero-crossing of

the sequenceγ(n1,∞). In fact, thekneeis set to the largest value ofn⋆1 that satisfies

(1 − s1)n⋆

1−1 [(s2c2 − s1c1) + s1(s1λ1 − c2)] ≤ (s2c2 − s1c1).

Hence,

n⋆1 =

1 +log

((s2c2−s1c1)

(s2c2−s1c1)+s1(s1λ1−c2)

)

log(1 − s1)

. (14)

We provide a numerical example to illustrate the computations in this section.

Numerical Example 1:Let c1 = 2, c2 = 1, λ1 = 103, λ2 = 4 × 103, s1 = 0.8 ands2 = 0.4.

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numex.eps

Fig. 2. Exact switch-over curve and approximate switch-over curve for a typical choice of parameters:λ1 = 103, λ2 = 4×103,

c1 = 2, c2 = 1, s1 = 0.8 ands2 = 0.4

• s1λ1 = 800 < 1600 = s2λ2. Thus, theoffsetis at the point(1, n⋆2) where

n⋆2 = ⌈1 + log(η2)/log(1 − s2)⌉ andη2 = (s1λ1 − c2)/(s2λ2 − c2). We getη2 = 0.4997 and

n⋆2 = 3.

• s1c1 = 1.6 > 0.4 = s2c2. Thus, thekneeis at the point(1, n⋆2) wheren⋆

2 is given by (13).

We getn⋆2 = 14.

The approximate switch-over curve and the exact switch-over (computed by solving the DP) are

depicted in Figure 2.

VI. HEURISTIC SCHEDULING POLICY

So far, we have thoroughly characterized the policyP⋆ which is the optimal packet scheduling

policy for a system with two queues and a single server under static channel conditions. In a

real wireless system, the channel conditions vary randomlywith time, owing to phenomena

like interference from other transmitter or fading due to receiver mobility. Naturally, a packet

scheduling policy must adapt to the varying channel conditions. We have assumed that the

instantaneous channel conditions are perfectly known at the transmitter. This is a reasonable

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assumption when the receivers are moving slowly. With the knowledge of channel conditions,

we assume there is a mappingsi(γi(j);Xi(j), j) : [0,∞); {Xi(j)} → [0, 1] that determines the

probability of successful packet transmission for the queue Qi if it is scheduled in time slotj,

given the instantaneous SNRγi(j) and auxiliary informationXi(j) (modulation scheme, code

rate, spreading factor etc.). Also, we assume that the channel variations are slow enough relative

to the size of the time slot so that the success probability stays constant over a time slot. LetI

denote the index set of all queues at the server and letQj be the index set of non-empty queues

that are possible candidates for scheduling. Thus,Qj = {j : Qj is empty}, Qj ⊆ I. Now, we

propose the following heuristic packet scheduling policyPh.

Repeat:

(a) If Qn = Φ, quit.

(b) If Qn = {k}, k ∈ I then scheduleQk and quit.

(c) SetA = Φ andQ = Qn. Repeat:

(i) If Q = Φ, quit.

(ii) If Q = {k}, k ∈ I then setA = A ∪ {k} andQ = Φ.

(iii) If |Q| ≥ 2, generatek and l uniformly at random such thatk, l ∈ Q andk 6= l.

(iv) Let nk(j) andnl(j) respectively denote the time-to-expiry of the HOL packet ofQk

andQj in time slotj. We have assumed thatsk(γk(j);Xk(j), j) andsl(γl(j);Xl(j), j)

are known. In addition,ck, cl, λk and λl are pre-determined parameters. With these

inputs, use the scheduling policyP⋆ to schedule one of eitherQk or Ql.

(v) If Qk is chosen, setA = A ∪ {k}. If Ql is chosen, setA = A ∪ {k}. In both cases,

setQ = Q\{k, l}.

(d) SetQn = A.

Remarks:A typical instance of the algorithm for a five queue system is illustrated in Figure

3. We note that in step (iv) of (c) above, we can replaceP⋆ by the piece-linear approximation

developed in Section V. We denote the resulting packet scheduling policy by Ph. This policy

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2 3 4 5

2 1

5 43

P⋆

P⋆

4

3

5

P⋆

R

R

5

3 5

3 5 4

P⋆

1

u uu

u u

u u

u u

u

u

u

u u

u

u

u

u

u

Fig. 3. A typical instance of the heuristic scheduling policy Ph. In the figure,R represents random pairing of queues while

P⋆ represents pairwise scheduling using policyP⋆

has computational complexity significantly lower than thatof Ph. As illustrated for a typical

choice of parameters in numerical example 1, the switch-over curves generated byP⋆ and the

piece-wise linear approximation are very close. Thus, we use Ph for performance evaluation in

all our simulation results without any noticeable performance degradation.

A. Computational Complexity

If |I| = K, the pairwise scheduling approach adopted inPh andPh terminates in atO(log2K)

steps. Theith step requires solvingO

(K

2i

)pairwise scheduling problems. Thus, the total number

of pairwise scheduling problems to be solved is upper bounded by K

(1

2+

1

22+ . . .

)= K.

In fact, the number of pairwise scheduling problems to be solved is exactly(K − 1). 3 Using

the approximation developed in Section V, we can solve each pairwise scheduling problem in

O(1) time. Thus, the packet scheduling policyPh hasO(K) computational complexity per time

3Let mk be the number of pairwise scheduling problems that need to besolved for ak queue system. Then,mk = k/2+mk/2

if k is even andmk = (k − 1)/2 + m(k+1)/2 if k is odd. It is easy to see that these equations are satisfied bymk = (k − 1)

∀k ∈ N

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slot. It is interesting to contrast this with the computational complexity of other benchmark

scheduling algorithms. The round-robin scheduler hasO(1) complexity per time slot. All index

based scheduling algorithms, maximum SNR, EDD, M-WLDF etc.haveO(K) complexity per

time slot because they involve computing the maximum of a list of K numbers. PolicyP⋆ has

complexityO(D2) for solving the pairwise scheduling problem, whereD is the maximum of the

times-to-expiry of the two queues under consideration. Thus, policyPh has complexityO(D2K)

per time slot, whereD denotes the maximum time-to-expiry over allQi, i ∈ I.

VII. SIMULATION RESULTS

In this section, we contrast the performance of the proposedscheduling policies with other

benchmark schedulers via link-level simulations. We first introduce the performance metrics used

to evaluate the performance of various scheduling policies.

We use theaverage packet drop rateas a measure of system performance. The packet drop

rate for each queue is computed as the ratio of the number of packet dropped to the total number

of packets processed by the queue (successfully transmitted or dropped) over a sufficiently long

simulation run. We denote this drop rate byPd(k) for the kth queue,Qk. The average drop rate

is simply the average of the drop rates of all queues. We denote it by Pd. Thus, for a system with

K queues,Pd =1

K

K∑

k=1

Pd(k). As discussed in Section I, apart from maximizing some metric of

system performance, it is also desirable for a scheduling policy to possess an element of fairness

so that all queues receive their fair share of service. We use∆ =

(max

1≤k≤KPd(k) − min

1≤k≤KPd(k)

)

as a measure of fairness. Thus, a small value of∆ indicates a higher degree of fairness and

vice-versa.

To decouple the performance of the scheduling policies fromrandomness in the packet arrival

process, we assume that arrivals occur to each queue according to a deterministic process. In

particular, exactly one packet arrives at theith queueQi everyDi > 0 time slots. The inter-

packet deadline associated with this packet isDi, that is, it must be transmitted withinDi slots

of becoming the HOL packet, else it gets dropped. This is a reasonable model for an application

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like live media streaming where packets/frames are generated at a constant rate by the source.

As discussed earlier, the instantaneous channel conditions and the choice of auxiliary parame-

ters like modulation scheme, code rate etc. are captured by amappings(γ; {X}) : [0,∞); {X} →

[0, 1] that represents the probability of successful packet transmissions for instantaneous SNR

γ and choice of auxiliary parameters{X}. We assume thats(·) remains constant over a trans-

mission time interval, a justified assumption for slowly fading channels. In a detailed system

level simulation,s(·) would be specified via actual value interface (AVI) tables [16]. In a real

system,s(·) would typically be gotten from look-up tables available at the BS, or estimated from

measurements at the receiver. For the sake of concreteness,we assume a simple functional form

for s(·), namelys(γ; {X}) = 1 − exp(−δγ), whereδ = δ({X}) is a function of the auxiliary

parameters. We do not expect the relative performance of various scheduling policies to depend

on the choice for the functional form ofs(·). We chose a value ofδ = 0.25 for our simulation

results. To compute the instantaneous SNR, we fix the mean SNRand use a 3-path channel model

(ITU Pedestrian A, 3kmph) with independently Rayleigh faded paths with Doppler spectrum.

We assume that an ideal RAKE receiver is employed at the receiver, so that the energy from all

multipaths is captured completely. All simulation resultsare generated by simulating the system

over 105 time slots. For a slot length of 2ms (HSDPA), the simulation run length is equivalent

to 200 seconds of real time.

A. Scenario A

We consider a system with four queues and deadlinesD1 = 3, D2 = 4, D3 = 5 andD4 = 6

slots respectively. The mean SNR is same for all queues and isvaried from 6dB to 18dB. The

buffering and dropping costs are set equal to1 and 103 respectively for all queues. Figure 4

depicts the fairness v/s average drop rate trade-off for different scheduling policies. The proposed

scheduling policyPh is the most fair for any given average drop rate.

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3456.eps

Fig. 4. Scenario A: Fairness (∆) v/s Average Drop Rate (Pd) trade-off for various scheduling policies for a four queuesystem

with D1 = 3, D2 = 4, D3 = 5, D4 = 6

B. Scenario B

We consider a system with two queues. We fixD2 = 5 and varyD1 from 2 to 5. We set

c1 = c2 = 1, λ2 = 103 and investigate the fairness v/s average drop rate trade-off obtained asλ1

is varied. The mean SNR is get to 10dB for both queues. The trade-off curves forD1 = 2 and

D1 = 3 are depicted in Figure 5. We then simulate the proposed scheduler Ph with the optimal

(most fair) choice ofλ1 for each value ofD1. The performance is contrasted with the EDF and

EXP-rule schedulers in Figure 6. The fairness performance of other schedulers severely degrades

asD1 decreases.

C. Scenario C

We consider a system with two queues. We fixD1 = D2 = 4. The mean SNR for the second

queue is set toγ2 =12dB while the mean SNR for the first queue (γ1) is varied from 6dB to

12dB. We also fixc1 = c2 = 1 andλ2 = 103. Then we study the fairness v/s average drop rate

trade-off obtained asλ1 is varied. The trade-off curves forγ1 =6dB andγ1 =8dB are depicted

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optlam.eps

Fig. 5. Fairness (∆) v/s Average Drop Rate (Pd) trade-off for the scheduling policyPh for two different values ofD1 for a

two queue system, asλ1 is varied

2qdiffdl.eps

Fig. 6. Scenario B: Fairness (∆) v/s Average Drop Rate (Pd) trade-off for various scheduling policiesPh for different values

of D1 for a two queue system

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optlam2.eps

Fig. 7. Fairness (∆) v/s Average Drop Rate (Pd) trade-off for the scheduling policyPh for two different values ofγ1 for a

two queue system, asλ1 is varied

2qdiffsnr.eps

Fig. 8. Scenario C: Fairness (∆) v/s Average Drop Rate (Pd) trade-off for various scheduling policiesPh for different values

of γ1 for a two queue system

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in Figure 7. We then simulate the proposed schedulerPh with the optimal (most fair) choice of

λ1 for each value ofγ1. The performance is contrasted with the EDF and EXP-rule schedulers

in Figure 8. The proposed scheduler provides almost uniformfairness over all choices ofγ1 and

γ2, while the performance of other schedulers degrades asγ1 decreases.

VIII. C ONCLUSIONS

We investigated the problem of downlink packet scheduling in a time-slotted wireless commu-

nication network. We focused on a scenario where packets have associated deadlines, as is the

case in numerous real-time applications like multimedia communication and mobile computing.

We constructed a parsimonious reduction of the scheduling problem and formulated the reduced

problem in a dynamic programming framework. We achieved a tremendous reduction in the

size of the state-space of the DP by using the notion of inter-packet deadlines. We derived key

structural properties of optimal solution to the DP and showed that the optimal policy is of

switch-overtype. We constructed a piece-wise linear approximation to the optimal switch-over

curve. Finally, we proposed a heuristic scheduling policy based on randomization and pairwise

scheduling (using the switch-over policy). We contrasted the performance of the proposed

scheduler with other benchmark scheduling policies via simulations under several different

scenarios and demonstrated that the proposed scheduler yields a significantly better trade-off

between user fairness ans system performance. Future work involves incorporating power control

and adaptive modulation and coding into the scheduling framework.

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[13] D. Bertsekas,Dynamic Programming and Optimal Control, vol. 1 & 2, 2nd Ed., Athena Scientific, 2000.

[14] J. Walrand,An Introduction to Queueing Networks, Prentice Hall, Englewood Cliffs, NJ, 1988.

[15] J. Huang, R.A. Berry and M.L. Honig, “Wireless scheduling with hybrid ARQ”,

www.ece.northwestern.edu/ rberry/harq.pdf.

[16] S. Hamalainen, P. Slanina, M. Hartman, A. Lappetelainen, H. Holma and O. Salonaho, “A novel interface between link and

system level simulations”,Proc. ACTS Mobile Telecommunications Summit, Aalborg, Denmark, Oct. 1997, pp. 599-604.

IX. A PPENDIX

A. Proof of Lemma 1

When the system is in state(n1, 0), that is,Q2 is empty, there is no scheduling decision to

be made, and the HOL packet ofQ1 is transmitted in successive time slots till it is received

correctly atR1, or dropped due to expiration of its deadline. Thus, we have

V (n1, 0) = s1V (0, 0) + (1 − s1)V (n1 − 1, 0) + c1, n1 > 1

= (1 − s1)V (n1 − 1, 0) + c1, n1 > 1.

With the initial conditionV (1, 0) = c1 + (1 − s1)λ1, we have a simple difference equation of

the formx(n) = ax(n− 1) + b, x(1) = x1, a < 1, the solution to which is given by

x(n) =b

1 − a+ an−1

(x1 −

b

1 − a

). (15)

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Using a = (1 − s1) and b = c1 in (15), we conclude (a). Similarly, if the system is in state

(0, n2), Q1 is empty andQ2 is always scheduled. Thus,

V (0, n2) = (1 − s2)V (0, n2 − 1) + c2, n2 > 1.

Using the initial conditionV (0, 1) = c2 + (1− s2)λ2, and witha = (1− s2) andb = c2 in (15),

we conclude (b).

B. Proof of Lemma 2

From (4) we have,

V (n1, 1) = min{(1 − s1)V (n1 − 1, 0) + λ2, V (n1 − 1, 0) + (1 − s2)λ2} + c1 + c2.

Thus,P⋆ schedulesQ1 in state(n1, 1) if

V (n1 − 1, 0) ≥s2λ2

s1

.

We use Lemma 1 to computeV (n1 − 1, 0) which yields the condition,

(s1λ1 − c1)(1 − s1)(n1−1) ≥ (s2λ2 − c2). (16)

For notational ease, letη1 =s2λ2 − c1s1λ1 − c1

. Sincec1 > s1λ1, we have three possibilities

(a) (s2λ2 − c1) < 0: The inequality in (16) is trivially satisfied. Thus,P⋆ always schedules

Q1 in state(n1, 1) ∀n1 ∈ N.

(b) (s2λ2 − c1) ≥ 0: We further split this into two sub-cases:

(i) η ≥ 1: The inequality in (16) is never satisfied. Thus,P⋆ always schedulesQ2 in

state(n1, 1) ∀n1 ∈ N.

(ii) η < 1: There exists an⋆1 > 1 such thatP⋆ schedulesQ1 in state(n1, 1) for n1 ≤ n⋆

1

andQ2 else. The choice ofn⋆ is given by

n⋆1 =

⌊1 +

log(η)

log(1 − s1)

⌋.

All three (mutually exhaustive) possibilities reduce to one of the three cases stated in the lemma.

Hence, we claim the lemma is true.

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C. Proof of Lemma 4

The proof is through the principle of mathematical induction. We begin with the base case,

that is,n1 = n2 = 2.

1) Base Case:γ(2, 3) ≤ γ(2, 2) ≤ γ(3, 2)

From definitions, we have

γ(2, 3) = s1V (0, 2) − s2V (1, 0) + (s2 − s1)V (1, 2).

γ(2, 2) = s1V (0, 1) − s2V (1, 0) + (s2 − s1)V (1, 1).

γ(3, 2) = s1V (0, 1) − s2V (2, 0) + (s2 − s1)V (2, 1).

We will show thatγ(2, 2) − γ(2, 3) = s1[V (0, 1) − V (0, 2)] + (s2 − s1)[V (1, 1) − V (1, 2)] ≥ 0.

We make three observations:

(a) P⋆ schedulesQ1 in state(1, 1) if s1λ1 ≥ s2λ2.

(b) P⋆ schedulesQ2 in state(1, 2) if s1λ1 ≥ s2[c2 + (1 − s2)λ2].

(c) By assumption,s1λ2 > c2. Thus,λ2 > c2 + (1 − s2)λ2.

Now, we have three cases:

(a) s1λ1 ≥ s2λ2: From observation (a) and (c),P⋆ schedulesQ1 in both states(1, 1) and(1, 2).

Thus,V (1, 1) = (1 − s1)λ1 + λ2 + c1 + c2, V (1, 2) = (1 − s1)λ1 + (1 − s2)λ2 + c1 + 2c2

andγ(2, 2) − γ(2, 3) = s2(1 − s1)(s2λ2 − c2) ≥ 0.

(b) s1λ1 < s2[c2 + (1 − s2)λ2]: From observation (b) and (c),P⋆ schedulesQ2 in both states

(1, 1) and(1, 2). Thus,V (1, 1) = λ1 + (1− s2)λ2 + c1 + c2, V (1, 2) = (1− s2)2λ2 + λ1 +

c1 + (2 − s2)c2 andγ(2, 2) − γ(2, 3) = s2(1 − s2)(s2λ2 − c2) ≥ 0.

(c) s2[c2 +(1−s2)λ2] ≤ s1λ1 < s2λ2: From observation (b) and (c),P⋆ schedulesQ1 in state

(1, 2) andQ2 in state(1, 1). Thus,V (1, 1) = λ1+(1−s2)λ2+c1+c2, V (1, 2) = (1−s1)λ1+

(1− s2)λ2 + c1 + 2c2 andγ(2, 2)− γ(2, 3) = s1(1− s2)(s2λ2 − c2) + (s2 − s1)(s1λ1 − c2).

Using s1λ1 ≥ s2[c2 + (1 − s2)λ2, we getγ(2, 2) − γ(2, 3) ≥ s2(1 − s2)(s2λ2 − c2) ≥ 0.

By similar arguments, we can showγ(2, 2) ≤ γ(3, 2). Hence, the base case is proved.

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2) Inductive Step:We now assume that for somen1, n2 > 2, the following holds:

γ(n1 − 1, n2) ≤ γ(n1 − 1, n2 − 1) ≤ γ(n1, n2 − 1). (17)

We will show that the inductive hypothesis (17) impliesγ(n1, n2+1) ≤ γ(n1, n2) ≤ γ(n1+1, n2).

We will focus on proving the right side of the inequality, that is, γ(n1 + 1, n2) ≥ γ(n1, n2). The

proof for the left side of the inequality is analogous.

By definition, P⋆ schedulesQ1 in state (n1, n2) ∀n1, n2 ∈ N\{1} if γ(n1, n2) ≤ 0 and

schedulesQ2 else. We have the following three cases:

(a) γ(n1, n2−1) ≤ 0: From (17),γ(n1 −1, n2 −1) ≤ 0. Thus,P⋆ schedulesQ1 in both states

(n1 − 1, n2 − 1) and (n1, n2 − 1).

(b) γ(n1 −1, n2 −1) > 0: From (17),γ(n1, n2 −1) > 0. Thus,P⋆ schedulesQ2 in both states

(n1 − 1, n2 − 1) and (n1, n2 − 1).

(c) γ(n1 − 1, n2 − 1) ≤ 0 andγ(n1, n2 − 1) > 0: P⋆ schedulesQ1 in state(n1 − 1, n2 − 1)

andQ2 in state(n1, n2 − 1).

Now,

γ(n1, n2 − 1) − γ(n1 − 1, n2 − 1) =

s2[V (n1 − 2, 0) − V (n1 − 1, 0)] + (s2 − s1)[V (n1 − 1, n2 − 2) − V (n1 − 2, n2 − 2)],

and

γ(n1 + 1, n2) − γ(n1, n2) =

s2[V (n1 − 1, 0) − V (n1, 0)] + (s2 − s1)[V (n1, n2 − 1) − V (n1 − 1, n2 − 1)].

From Lemma 1 we obtain,

V (n1 − 1, 0) − V (n1, 0) = s1V (n1 − 1, 0) − c1

V (n1 − 2, 0) − V (n1 − 1, 0) =1

1 − s1[V (n1 − 1, 0) − V (n1, 0)] .

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For case (a),

V (n1, n2 − 1) = s1α(n1, n2 − 1) + V (n1 − 1, n2 − 2) + c1 + c2

= s1V (0, n2 − 2) + (1 − s1)V (n1 − 1, n2 − 2) + c1 + c2

V (n1 − 1, n2 − 1) = s1α(n1 − 1, n2 − 1) + V (n1 − 2, n2 − 2) + c1 + c2

= s1V (0, n2 − 2) + (1 − s1)V (n1 − 2, n2 − 2) + c1 + c2.

We combine the above equations to get

γ(n1 + 1, n2) − γ(n1, n2) = (1 − s1)[γ(n1, n2 − 1) − γ(n1 − 1, n2 − 1)] ≥ 0.

The inequality follows from the inductive assumption (17).

For case (b),

V (n1, n2 − 1) = s2β(n1, n2 − 1) + V (n1 − 1, n2 − 2) + c1 + c2

= s2V (n1 − 1, 0) + (1 − s2)V (n1 − 1, n2 − 2) + c1 + c2

V (n1 − 1, n2 − 1) = s2β(n1 − 1, n2 − 1) + V (n1 − 2, n2 − 2) + c1 + c2

= s2V (n1 − 2, 0) + (1 − s2)V (n1 − 2, n2 − 2) + c1 + c2.

In this case we can show that

γ(n1 + 1, n2) − γ(n1, n2) = (1 − s2)[γ(n1, n2 − 1) − γ(n1 − 1, n2 − 1)] ≥ 0.

The inequality follows from the inductive assumption.

For case (c),

V (n1, n2 − 1) = s2β(n1, n2 − 1) + V (n1 − 1, n2 − 2) + c1 + c2

= s2V (n1 − 1, 0) + (1 − s2)V (n1 − 1, n2 − 2) + c1 + c2

V (n1 − 1, n2 − 1) = s1α(n1 − 1, n2 − 1) + V (n1 − 2, n2 − 2) + c1 + c2

= s1V (0, n2 − 2) + (1 − s1)V (n1 − 2, n2 − 2) + c1 + c2.

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In this case we can show that

γ(n1 + 1, n2) − γ(n1, n2) = (1 − s2)γ(n1, n2 − 1) − (s2 − s1)γ(n1 − 1, n2 − 1) ≥ 0.

The inequality follows because for case (c),γ(n1, n2−1) > 0 andγ(n1−1, n2−1) ≤ 0. Hence,

we have shownγ(n1 + 1, n2) − γ(n1, n2) > 0 for all three cases. The proof forγ(n1, n2) −

γ(n1, n2 + 1) ≥ 0 is analogous and can be constructed in similar fashion by considering three

mutually exhaustive cases. We omit the details.

We now invoke the principle of mathematical induction to claim that the statement of the

lemma is true for all states(n1, n2), n1, n2 ∈ N\{1}.

D. Proof of Theorem 1

When the system is in state(n1, 0) ∀n1 ∈ N or (0, n2) ∀n2 ∈ N, there is no scheduling

decision to be made andQ1 andQ2 are respectively scheduled. We thus focus on the non-trivial

part of the state space, which is the states(n1, n2) ∀n1, n2 ∈ N, where the scheduler needs to

make a scheduling decision.

Let us focus on the states(n1, n2) ∀n1, n2 ∈ N\{1}. Fix n1 = n′1, say. From Lemma 4,

γ(n′1, n2) is a monotone non-increasing function ofn2. Thus, we have three cases in the state

(n′1, n2) ∀n2 ∈ N\{1}.

(a) γ(n′1, n2) > 0 ∀n2 ∈ N\{1}. Thus,P⋆ schedulesQ2 in the states(n′

1, n2) ∀n2 ∈ N\{1}.

(b) γ(n′1, n2) ≤ 0 ∀n2 ∈ N\{1}. Thus,P⋆ schedulesQ1 in the states(n′

1, n2) ∀n2 ∈ N\{1}.

(c) There is an⋆2 = n⋆

2(n′1) ≥ 1 such thatγ(n′

1, n2) ≤ 0 for n2 ≥ n⋆2 and γ(n′

1, n2) ≤ 0 for

n2 < n⋆2. Thus,P⋆ schedulesQ2 in state(n′

1, n2) for n2 < n⋆2 andQ1 for n2 ≥ n⋆

2.

Hence, there is a switch-over functionφ such thatφ(n′1) = ∞ in case (a),φ(n′

1) ≤ 2 in case

(b) andφ(n′1) = n⋆

2(n′1) in case (c).

From Lemma 4 we know thatγ(n1, n2) is a monotone non-decreasing function ofn1. In

particular,γ(n′1 + 1, n2) ≥ γ(n′

1, n2) ∀n2 ∈ N\{1}. Also, γ(n′1 + 1, n2) is a monotone non-

increasing function ofn2. If case (a) is true,γ(n′1 + 1, n2) > 0 ∀n2 ∈ N\{1}. It is easy to

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show that ifP⋆ schedulesQ2 in state(n1, 2), it necessarily schedulesQ2 in state(n1, 1). Thus,

φ(n′1 + 1) = ∞. If case (b) is true, eitherγ(n′

1 + 1, n2) ≤ 0 ∀n2 ∈ N\{1}, or there is an⋆2 > 1

such thatγ(n′1 +1, n2) > 0 for n2 < n⋆

2 andγ(n′1 +1, n2) ≤ 0 for n2 ≥ n⋆

2. Thus,φ(n′1 +1) ≤ 2

or φ(n′1 +1) = n⋆

2(n′1 +1). The latter is also true for case (c). From the monotone non-increasing

property ofγ(n1, n2) as a function ofn2 we conclude thatn⋆2(n

′1 + 1) ≥ n⋆

2(n′1). Thus, in cases

(a) and (c),φ(n′1 + 1) ≥ φ(n′

1). In case (b), ifP⋆ schedulesQ1 in state(n′1 + 1, 1) then it

necessarily schedulesQ1 in state(n′1, 1), from Lemma 2. Thus,φ(n′

1 + 1) ≥ φ(n′1) in case (b).

The arguments presented so far are true forn1 > 1. For n1 = 1, we invoke Lemma 3. If

P⋆ schedulesQ1 in state (1, n2) ∀n2 ∈ N, then φ(1) = 1. Sinceφ(n1) ≥ 1 by definition,

φ(2) ≥ φ(1). If P⋆ schedulesQ2 in state(1, n2) ∀n2 ∈ N, thenφ(1) = ∞. From Lemma 2,P⋆

schedulesQ2 in state(n1, 1) ∀n1 ∈ N\{1}. Thus,φ(n1) = ∞ ∀n1 ∈ N. Lastly, suppose there

is a n⋆2 > 1 such thatP⋆ schedulesQ2 for n2 < n⋆

2 andQ1 for n2 ≥ n⋆2. Then,φ(1) = n⋆

2. It

can be shown easily thatP⋆ necessarily schedulesQ2 in state(2, n2) if it schedulesQ2 in state

(1, n2). Thus,φ(2) ≥ φ(1).

Hence, for states(n1, n2), ∀n1, n2 ∈ N, we have shown the existence of a non-decreasing

switch-over functionφ such that the optimal policyP⋆ schedulesQ1 in state(n1, n2) if n2 ≥

φ(n1), else it schedulesQ2. We conclude that the optimal policy isP⋆ is a switch-over policy.

E. Proof of Theorem 2

(a) We will use the method of contradiction. Suppose the claim is false. Then, for any choice

of n1 ∈ N, say n′1, there exists an′′

1 > n′1 such thatφ(n′′

1) 6= φ(n′1). It follows from

Theorem 1 thatφ(·) is a monotone non-decreasing function. Thus,φ(n′′1) > φ(n′

1). Also,

φ(n1) ≥ φ(n′′1) > φ(n′

1) ∀n1 ≥ n′′1. We can find such ann′′

1 for every choice ofn1 =

n′1 ∈ N. Hence, by suitably choosing ann′′

1 for eachn′1, we can construct a monotone

strictly increasing sub-sequenceφ(nk). Clearly, this sub-sequence converges to+∞. Since

φ(n) is a monotone non-decreasing sequence, it converges to a limit, and every sub-

sequence ofφ(n) converges to the same limit. It follows that the sequenceφ(n) converges

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to +∞. By definition, φ(n1) is the value ofn2 at whichP⋆ switches over fromQ2 to

Q1 in state(n1, n2) for fixed n1. Alternatively, it is the value ofn2 for which γ(n1, n2)

changes sign from positive to negative, for fixedn1. Now, limn1→∞

φ(n1) = +∞ implies that

limn1→∞

γ(n1, n2) > 0 ∀n2 ∈ N. Thus, limn1→∞

limn2→∞

γ(n1, n2) = γ⋆ > 0. However, from (12)

in Section IV, for the cases1c1 ≥ s2c2 we haveγ⋆ = c2s2

s1

− c1 ≤ 0. This leads us to a

contradiction. We conclude that the claim is indeed true.

(b) We will again employ the method of contradiction. Suppose the claim is false. Then, for any

choice ofn1 ∈ N, sayn′1, there existsn′′

1 > n′1 such thatφ(n′′

1) <∞. From monotonicity

of φ(·), φ(n1) < ∞ ∀n1 ≤ n′′1. Since such a choice ofn′′

1 is possible for eachn′1, we

conclude thatφ(n1) < ∞ ∀n1 ∈ N. By definition, for eachn1 ∈ N, there exists a finite

n2 = φ(n1) such thatP⋆ schedulesQ1 in state(n1, n2) ∀n2 ≥ φ(n1). Alternatively, for

eachn1 ∈ N, γ(n1, n2) changes sign from positive to negative atn2 = φ(n1). In particular,

this is true for limn1→∞

. This implies limn1→∞

limn2→∞

γ(n1, n2) = γ⋆ < 0. However, from (12)

in Section IV, for the cases1c1 > s2c2 we haveγ⋆ = c2 − c1s1

s2> 0. This leads us to a

contradiction. We conclude that the claim is indeed true.

F. Proof of Lemma 5

We define

an2 , limn1→∞

γ(n1, n2) ∀ n2

bn1 , limn2→∞

γ(n1, n2) ∀ n1. (18)

Thus,an2 exists∀ n2, andbn1 exists∀ n1.

We know thatγ(n1, n2) ≥ γ(n1, n2 + 1), therefore{an1} is a decreasing sequence. Thus, it

has a limita⋆. Also, γ(n1 + 1, n2) ≥ γ(n1, n2), therefore{bn2} is an increasing sequence, and

has a limitb⋆. Using (18) we have

limn2→∞

an2 = limn2→∞

limn1→∞

γ(n1, n2) = a⋆

limn1→∞

bn1 = limn1→∞

limn2→∞

γ(n1, n2) = b⋆.

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We have to show thata⋆ = b⋆. We have the following inequalities

limn1→∞

γ(n1, n2) = an2 ≥ a⋆

γ(n1, n2) ≥ bn1 ∀n1

limn1→∞

γ(n1, n2) ≥ limn1→∞

bn1

an2 ≥ b⋆.

The first inequality follows because the{an2} is non-increasing. The second inequality follows

becauseγ(n1, n2) is non-increasing inn2 and has a limit. The third inequality follows directly

from the second, and the fourth follows by definition. Since the above set of inequalities are

true for anyn2, we can conclude thata⋆ ≥ b⋆. Now, since{bn1} is a non-decreasing sequence

andγ(n1, n2) is non-decreasing inn1, we have the following set of inequalities:

limn2→∞

γ(n1, n2) = bn1 ≥ b⋆

γ(n1, n2) ≥ an2 ∀n2

limn2→∞

γ(n1, n2) ≥ limn2→∞

an2

bn1 ≥ a⋆.

Since the above hold for anyn1, we concludeb⋆ ≥ a⋆. Sincea⋆ ≥ b⋆ andb⋆ ≥ a⋆, we conclude

a⋆ = b⋆.

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