✬ ✩ Dept. of Electrical Engineering, Yale University Throughput and Delay Optimal Resource Allocation in Multiple Access Fading Channels DIMACS Network Information Theory Workshop March 18, 2003 Edmund M. Yeh Department of Electrical Engineering Yale University Joint work with Aaron Cohen, Brown University ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Acknowledgments Many thanks to Professor Robert Gallager and Professor Emre Telatar for their advice and encouragement. ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Multiple Access Communications • Multiple access (many to one): multiple senders transmit to one receiver (possibly) over fading channels. • Ex: cellular telephony, satellite networks, local area networks. ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Central Problems • Contention/interference - resource sharing. • Bursty sources ⇒ random number of active senders. • Network/MAC layer QOS issues - throughput, delay. • Physical layer issues - channel modelling, coding, detection. ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Need for Cross-Layer Approach • Multiple access network theory (ALOHA, CSMA) - concentrates on source burstiness and delay; poor modelling of noise and interference. • Multiple access information theory - concentrates on channel modelling and coding; ignores random arrival of messages and delay. • Need more unified cross-layer framework: – Random packet arrivals affect resource sharing. – Choice of modulation and coding affects QOS issues. – Random fading affects resource allocation. – Gallager (85), Ephremides and Hajek (98). ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University New Approach • Goal: – Combine information-theoretic limits with QOS issues. – Establish fundamental bounds on throughput/delay performance. • Implementation: – Random arrivals, information-theoretic optimal coding. – Power control and rate allocation as function of fading and queue states to optimize throughput and delay ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Previous Work • Telatar and Gallager (95) – Achievable multiple access scheme with feedback. – Poisson arrivals; no queueing; single-user decoding; processor sharing system. • Telatar (95) – Analogy between MAC and multi-processor queue. – Each user has fixed pool of bits to send. – Optimal processor assignment to minimize average packet delay. • Yeh (01) – Poisson arrivals; queueing. – Optimal rate allocation from C to min. average packet delay. – Longer Queue Higher Rate (LQHR) policy strongly delay optimal. ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Multiple Access Fading Channel • Continuous-time M -user Gaussian multiple access fading channel with bandwidth W : M � � Y ( t ) = H i ( t ) X i ( t ) + Z ( t ) . i =1 • { Z ( t ) } : white Gaussian noise, density N 0 / 2. • Slowly-varying and flat-fading (under-spread) channel. ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Multiple Access Fading Channel • Block fading model, block length = T . • T large enough for reliable communication at a fixed fade. • { H ( t ) = ( H 1 ( t ) , . . . , H M ( t )) } modulated by finite-state ergodic Markov chain. • Transmitter i has (long-term) average power constraint P i , and (short-term) peak power constraint ˆ P i . ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Information-theoretic Capacity Region C ( h , p ) (Ahlswede, Liao, Cover, Wyner 1971-75) • Fixed h = ( h 1 , . . . , h M ) and p = ( p 1 , . . . , p M ). • C ( h , p ) = set of r ∈ R M + such that � i ∈ S h i p i � � � r i ≤ W log 1 + , ∀ S ⊆ { 1 , . . . , M } . N 0 W i ∈ S • Reliable communication possible inside C ( h , p ), impossible outside C ( h , p ), for any coding and modulation scheme. • Polymatroid structure (Tse and Hanly 98). ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Two-User Capacity Region C ( h , p ) r 2 ✻ r A Dominant face ❅ � ❅ � ✠ ❅ ❅ r B ✲ r 1 0 ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Multiple Access Channel with Random Arrivals H 1 ( t ) ✲ ✲ A 1 ( t ) ✛ Transmitter 1 P 1 ( t ) , R 1 ( t ) H 2 ( t ) ✲ ✲ A 2 ( t ) ✛ Transmitter 2 P 2 ( t ) , R 2 ( t ) . . . . . . H M ( t ) ✲ ✲ A M ( t ) ✛ Transmitter M P M ( t ) , R M ( t ) ✲ H ( t ) , U ( t ) Controller ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Arrivals and Unfinished Work • { A i ( t ) } = ergodic packet arrival process to transmitter i . • User i packets i.i.d. ∼ F Z i ( · ) , E [ Z i ] < ∞ . • U i ( t ) = number of untransmitted bits in queue i at time t . ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Power Control and Rate Allocation • Controller: ( H ( t ) , U ( t )) �→ ( P ( t ) , R ( t )). • Two stages: 1. Power control policy P : p = P ( h , u ) s.t. for all i , E [ P i ( H , U )] ≤ ¯ P i , P i ( h , u ) ≤ ˆ P i for all ( h , u ). 2. Rate allocation policy R : r = R ( h , p , u ) ∈ C ( h , p ) . ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Main Results • Stability region S of all bit arrival rates for which all queues can be kept finite. • For given power control policy, find throughput optimal rate allocation policy. • In symmetric scenario, find delay optimal rate allocation policy for any symmetric power control policy. ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Stability Region S • λ i = lim t →∞ A i ( t ) /t = packet arrival rate to queue i . • ρ i = λ i E [ Z i ] = bit arrival rate to queue i . � t 1 • Define f i ( ξ ) = lim sup t →∞ 0 1 { U i ( τ ) >ξ } dτ . t • System stable if f i ( ξ ) → 0 as ξ → ∞ for all i . • S = set of all ρ = ( ρ 1 , . . . , ρ M ) for which can stabilize system. ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Stability Region S • Assume { A i ( t ) } modulated by finite-state ergodic Markov chain. Theorem 1 S = C ( P , ˆ P ) = information-theoretic capacity region under power control (Tse and Hanly 98). • C ( P , ˆ P ) = � P∈F C ( P ). • F = {P : E [ P i ( H )] ≤ P i , ∀ i ; P i ( h ) ≤ ˆ P i , ∀ h , ∀ i } . • C ( P ) = E [ C ( H , P ( H ))]. ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Stability Theorem • Achievability : ρ ∈ int( S ): knowing ρ and statistics of { H ( t ) } , can stabilize system using stationary P , R depending only on current channel state. • Converse: ρ / ∈ S : cannot stabilize system, even with non-stationary policy with knowledge of queue state and/or knowledge of future events, so long as � t 1 p i ( τ ) ≤ ˆ lim sup p i ( τ ) dτ ≤ P i ∀ i ; P i , ∀ τ, ∀ i. t t →∞ 0 ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Throughput Optimal Resource Allocation • Find “universal” power/rate policy to stabilize system even if ρ not known, as long as ρ ∈ int( S ). • Must use both H ( t ) and U ( t ). • Suppose know ρ ∈ C ( P ) = E [ C ( H , P ( H ))]. • Assume { H i ( kT ) } i.i.d. for each i , { A i (( k + 1) T ) − A i ( kT ) } i.i.d. for each i . • Assume E [( A i (( k + 1) T ) − A i ( kT )) 2 ] < ∞ . ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University No Work Conservation r 2 ✻ r A Dominant face ❅ � ❅ � ✠ ❅ ❅ r B ✲ r 1 0 ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Throughput Optimal Rate Allocation Theorem 2 Given P ∈ F , throughput optimal rate allocation policy is M r ∗ = R ∗ ( h , P ( h , u ) , u ) = arg � max u i r i (1) r ∈C ( h , P ( h , u )) i =1 • Idea appeared in Tassiulas and Ephremides ’92; McKeown, et al. ’96; Tassiulas ’97; Neely et al. ’02. • Here, motivated by delay optimality results. ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Longest Queue receives Highest Possible Rate (LQHPR) • Due to polymatroidal nature of C ( h , P ( h , u )), solution to (1) has special form. • Order queues u [1] ≥ u [2] ≥ · · · ≥ u [ M ] . � � h [ i ] P [ i ] ( h , u ) r ∗ [ i ] = W log 1 + � j<i h [ j ] ( t ) P [ j ] ( h , u ) + N 0 W • Longest Queue receives Highest Possible Rate (LQHPR). • LQHPR ⇔ adaptive successive decoding : u [ M ] decoded first, u [1] decoded last. ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Two-User Rate Allocation r 2 ✻ r A Dominant face ❅ � ❅ ✠ � ❅ ❅ r B ✲ r 1 0 • u 1 ≥ u 2 : r B u 1 < u 2 : r A ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Proof of Stability Theorem • Stability of Markov chains based on negative Lyapunov drift. i U 2 • V ( U ) = � i . • Show there exists compact set Γ ⊂ R M s.t. for some ǫ > 0, E [ V ( U ( t + T )) − V ( U ( t )) | U ( t )] ≤ − ǫ whenever U / ∈ Γ . ✫ ✪
✬ ✩ Dept. of Electrical Engineering, Yale University Delay Optimal Resource Allocation • Beyond stabilization, keep queues as short as possible. • Find feasible P and R to minimize lim t →∞ E [ � M i =1 U i ( t )] (average bit delay) for ρ ∈ int( S ). ✫ ✪
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