Order Optimal Delay for Opportunistic Scheduling In Multi-User Wireless Uplinks and Downlinks Michael J. Neely University of Southern California *Sponsored in part by NSF OCE Grant (DIGITAL OCEAN) 1 2 N S 2 (t) S N (t) Num. Users N Avg. Delay S 1 (t) {ON, OFF} Allerton 2006

N 1 2 The System Model: N Users, 1 Server Discrete Time System: Timeslots t = {0, 1, 2, …} q1q1 A i (t) = Arrivals to Queue i during slot t [ i.i.d over slots, E[A i (t)] = i ] Q i (t) = Current Num. Packets in queue i Uplink user 1user N Downlink 12 N S i (t) = Current Channel State ({ ON, OFF }) [ i.i.d. over slots, Pr[S i (t) = ON] = q i ] q2q2 qNqN  i (t) = Packets Transmitted over link i on slot t

N 1 2 The System Model: N Users, 1 Server Discrete Time System: Timeslots t = {0, 1, 2, …} q1q1 Uplink user 1user N Downlink 12 N q2q2 qNqN Q i (t)A i (t)  i (t) Q i (t+1) = max[Q i (t) -  i (t), 0] + A i (t) Scheduling Constraints: Can serve at most one “ON” link per slot:  i (t) {0,1} i=1 N  i (t) 1,,  i (t)=0 if S i (t)=OFF

N 1 2 q1q1 q2q2 qNqN Model is central to channel-aware (“opportunistic”) scheduling. This model is investigated in [Tassiulas, Ephremides 93]: Results of [Tas, Eph 93]: 1) Capacity Region  2) LCQ Algorithm (“Largest Connected Queue”) 3) Delay Optimality for Symmetric Systems The Capacity Region  : Set of all rate vectors ( 1,.., N ) that can be stabilized. Example: (N=2)  is the set of all ( 1, 2 ) such that: 1 q 1, 1 q q 1 + (1-q 1 )q 2 1 2

N 1 2 q1q1 q2q2 qNqN Model is central to channel-aware (“opportunistic”) scheduling. This model is investigated in [Tassiulas, Ephremides 93]: Results of [Tas, Eph 93]: 1) Capacity Region  2) LCQ Algorithm (“Largest Connected Queue”) 3) Delay Optimality for Symmetric Systems The Capacity Region  : Set of all rate vectors ( 1,.., N ) that can be stabilized. General Case for N: ( 1,.., N )  if and only if i i I 1 - (1-q i ) for each of the 2 N -1 non-empty subsets I of {1,.., N}

An isolated set of delay-optimality results: q q q -Largest Connected Queue (LCQ) [Tassiulas and Ephremides 93]: Proof uses stochastic coupling and exploits symmetry… For Symmetric Systems: -Rate Allocation in Gaussian Multiple Access Channels [Yeh 2001, Yeh and Cohen 2003] -Multi-Server Systems: [Yeh 2001, Ganti, Modiano, Tsitsiklis 2002]

An isolated set of delay-optimality results: q q q -Largest Connected Queue (LCQ) [Tassiulas and Ephremides 93]: Proof uses stochastic coupling and exploits symmetry… For Symmetric Systems: -Rate Allocation in Gaussian Multiple Access Channels [Yeh 2001, Yeh and Cohen 2003] -Multi-Server Systems: [Yeh 2001, Ganti, Modiano, Tsitsiklis 2002] The actual delay that is achieved is unknown (even for these symmetric cases) O(N)? O( N )? O(1)?

An isolated set of delay-optimality results: For Heavy Traffic: The actual delay that is achieved is unknown (even for these symmetric cases) O(N)? O( N )? O(1)?  = fraction is away from capacity region boundary q q q Shakkottai, Srikant, Stolyar 2004  1 (Heavy Traffic) An exponential Scheduling Rule approaches delay optimality ( 

Related: Delay for N x N Switch Scheduling: 12 3N 1 N -[Leonardi, Mellia, Neri, Marsan 2001]: O(N/(1-  )) Delay bound (MWM Sched.) -[Neely, Modiano 2004]: O(log(N)/(1-  ) 2 ) Delay bound (Frame Based Sched.)

Related: Delay for N x N Switch Scheduling: 12 3N 1 N Some Interesting Queue Grouping Approaches (mainly to reduce complexity): -Mekkittikul, McKeown (1998) -Shah (2003) -Wu, Srikant (wireless, 2006)

Related: Delay for N x N Switch Scheduling: 12 3N 1 N Some Interesting Queue Grouping Approaches (mainly to reduce complexity): -Mekkittikul, McKeown (1998) -Shah (2003) -Wu, Srikant (wireless, 2006) -Leonardi et al. (2001) + =

Related: Delay for N x N Switch Scheduling: 12 3N 1 N Some Interesting Queue Grouping Approaches (mainly to reduce complexity): -Mekkittikul, McKeown (1998) -Shah (2003) -Wu, Srikant (wireless, 2006) -Leonardi et al. (2001) + = O(1) Delay when  < 1/2 (half loaded)

q q q What is the optimal delay (as a function of N) for the N user wireless problem with varying channels? Time Varying Channels make analysis more complex, cannot use same approaches as switch problems… Previous Upper and Lower Bounds: (N users) N (1-  ) O( ) 1 (1-  ) O( ) E[Delay] “Single-Queue Bound” [Neely, Modiano, Rohrs 03]

q q q What is the optimal delay (as a function of N) for the N user wireless problem with varying channels? Our Results: (part 1) If scheduling doesn’t consider queue backlog (such as stationary randomized scheduling) then: 1)E[Delay] is at least linear in N 2) Uniform Poisson Traffic: E[Delay] > N 2r N (1-  ) r N = 1-(1-q) N (max possible output rate)

What is the optimal delay (as a function of N) for the N user wireless problem with varying channels? Our Results: (part 2) For any  such that  < 1 Av. Delay log(1/(1-  )) (1-  ) O( ) Independent of N Holds for Symmetric Systems and a large class of Asymmetric ones q q q r N = 1-(1-q) N (max possible output rate)

What is the optimal delay (as a function of N) for the N user wireless problem with varying channels? Our Results: (part 2) For any  such that  < 1 Av. Delay log(1/(1-  )) (1-  ) O( ) Independent of N We use a form of queue grouping together with Lyapunov drift And statistical multiplexing q q q r N = 1-(1-q) N (max possible output rate)

Intuition about Queue Grouping: q q q N user System, Uniform Poisson inputs: r N = 1-(1-q) N (max possible output rate) Compare to a single-queue system with Pr[ON] = q Pr[serve]=q Can show that any work conserving scheduling policy in multi- queue system yields delay that is stochastically smaller than single- queue system. Leads An easy upper bound on delay… (GI/GI/1 queue)

Intuition about Queue Grouping: q q q N user System, Uniform Poisson inputs: r N = 1-(1-q) N (max possible output rate) Compare to a single-queue system with Pr[ON] = q Pr[serve]=q Single Queue Upper Bound on Avg. Delay: (GI/GI/1 queue) PoissonBernoulli E[Delay] = 1 - tot /2 q - tot Only works for  tot < q (i.e.,  <  where  = q/r N ) 1 (1-  /  ) O( ) =

Queue Grouping Approach: Form K Groups: {G 1, G 2, …, G K } 1 2 N M1 M1+1 G1G1 G2G2 GKGK Q sum, k (t) = Q i (t) i G k

G1G1 G2G2 GKGK Q sum, k (t) = Q i (t) i G k sum, k = i i G k The Largest Connected Group (LCG) Algorithm: Every slot t, observe the queue backlogs and channel states, and select the group k {1, …, K} that maximizes 1 k (t)Q sum, k (t). Then serve any non-empty connected queue in that group (breaking ties arbitrarily). Define: 1 k (t) = { 1, if group G k has at least one non-empty connected queue. 0, else

G1G1 G2G2 GKGK q1q1 qNqN q2q2 1 q min, 1 2 q min, 2 K q min, K sum, 1 sum, 2 sum, N Define:  K = Capacity region of the K-queue System sum, k = i i G k q min, k = min {q i } i G k Theorem: If there is an  > 0 such that: ( sum, 1 + , sum, 2 +  sum, K +   K Actual N-queue SystemComparison K-queue System Then LCG stabilizes the system and yields average delay:

G1G1 G2G2 GKGK q1q1 qNqN q2q2 1 q min, 1 2 q min, 2 K q min, K sum, 1 sum, 2 sum, N Define:  K = Capacity region of the K-queue System sum, k = i i G k q min, k = min {q i } i G k Theorem: If there is an  > 0 such that: ( sum, 1 + , sum, 2 +  sum, K +   K Actual N-queue SystemComparison K-queue System If arrivals are independent and Poisson, then we have:

Theorem: If there is an  > 0 such that: ( sum, 1 + , sum, 2 +  sum, K +   K If arrivals are independent and Poisson, then we have: Proof Concept: Use the following Lyapunov function: 1)LCG comes within additive constant of minimizing: (Lyapunov drift) 2) (tricky part) Prove there exists another algorithm that yields: (h() linear)

Application to Symmetric Systems: r N = 1-(1-q) N (max possible output rate) q q q Q N-1 (t) Q N (t) Q 1 (t) Q 2 (t) q For any loading  such that 0 <  < 1: log(2/(1-  )) log(1/(1-q)) Choose K = For simplicity assume N = MK (K groups of equal size M) Then  = r N (1-  )/(2K), … Plug this into the theorem… tot =  r N

Application to Symmetric Systems: r N = 1-(1-q) N (max possible output rate) For any loading  such that 0 <  < 1: log(2/(1-  )) log(1/(1-q)) Choose K = For simplicity assume N = MK (K groups of equal size M) tot =  r N E[W] 2K r N (1-  ) = log(1/(1-  )) (1-  ) O( ) Then LCG => q q q Q N-1 (t) Q N (t) Q 1 (t) Q 2 (t) q

Application to Asymmetric Systems: tot =  r max i=1 N (1-q i ) r max = 1 - (max possible output rate) q1q1 q2q2 q N-1 Q N-1 (t)  Q N (t)  Q 1 (t)  Q 2 (t) qNqN  tot = 1 + … + N Form variable length groups by iteratively packing individual streams until total rate of the group exceeds tot /N. Then: sum, k < tot /N + max for all groups k

Application to Asymmetric Systems: For any loading  such that 0 <  < 1: log(2/(1-  )) log(1/(1-q min )) Choose K = tot =  r max E[W] log(1/(1-  )) (1-  ) O( ) For any N K, LCG => > i=1 N (1-q i ) r max = 1 - (max possible output rate) Assume max < (1-  )r max /(3K) q1q1 q2q2 q N-1 Q N-1 (t)  Q N (t)  Q 1 (t)  Q 2 (t) qNqN 

Conclusions: Order-Optimal Delay for Opportunistic Scheduling in a Multi-User System (N users) -Backlog-unaware scheduling: Delay grows at least linear with N -Backlog-aware scheduling: It is possible to achieve O(1) delay (independent of N) -The first explicit bound for optimal delay in this setting -Queue Grouping is a useful tool for analysis and design