Competitive Queue Management for Latency Sensitive Packets

1. Competitive Queue Management for Latency Sensitive Packets Amos Fiat, Yishay Mansour and Uri Nadav Tel Aviv University Dagsthul Meeting on Fair Division

2. Dagsthul Meeting on Fair Division

3. Naor’s Model • Service time is exponentially distributed • New customers arrive according to a Poisson distribution • Getting the service worth R units of monetary value • Waiting 1 unit of time costs 1 unit of monetary value Dominating threshold strategy: Join ifR > E[ Waiting Time ] Queue Service Naor: In observable queues, individual decisions are not socially preferred Dagsthul Meeting on Fair Division

4. Naor’s Model Why? • A customer who joins the queue may cause future customers to spend more time in the system • The individual's objective does not take this into consideration • To reduce the arrival rate, set an appropriate admission fee Thm [Naor 69]: The equilibrium arrival rate is greater than the socially desired one Dagsthul Meeting on Fair Division

5. Our Work • Non stochastic model • Competitive analysis • Compare to the optimal solution OPT • Competitive ratio: For all input sequences , OPT() < c * ON () Dagsthul Meeting on Fair Division

6. Results • Homogeneous packets (as in Naor’s model: equal valued packets) • Lower bound  = 1.618 (the golden ratio) (even for randomized algorithms) • Matching upper bound (deterministic). • Heterogeneous packets (Not necessarily equal valued packets): • Deterministic algorithm with comptitive ratio c, 4 < c < 8 • Lower bound of 3 (deterministic) • Implies truthful online pricing mechanism Dagsthul Meeting on Fair Division

7. Related Work in Operations Research • On the regulation of queue size by levying tolls. [Naor, Econometrica 69] • On the Optimality of First Come Last Served Queues. [Hassin 85] Optimal social welfare without admission fee • Book: To Queue or not to Queue: Equilibrium Behavior in Queueing Systems. [Hassin and Haviv, 06] Dagsthul Meeting on Fair Division

8. Related Work – Online Buffer Management • Competitive queue policies for differentiated services [Aiello et al. Journal of Algorithms 05] • Buffer overflow management in QoS switches [Kesselman et al. STOC 01] • Competitive queuing policies for QoS switches [Andelman et al. SODA 03] • Better online buffer management [Li et al. SODA 07] Dagsthul Meeting on Fair Division

9. Time 0 1 2 3 Online Model • Event sequence • Packet transmission, at integral times • Arrive events (for simplicity assume distinct non-integral times) Arrive events: Send events: Arrive Event:Time and value are determined by the adversary • Transmission events are not under adversarial control! Dagsthul Meeting on Fair Division

10. Homogeneous Packets: Easy Online policy: Accept while the queue size is at most ½R • Handles at least half the traffic • By induction on the number of events • Each packet gets a profit of at least ½R • Competitive ratio 4 Dagsthul Meeting on Fair Division

11. Illustration of the Benefit # sent packets Lemma: the benefit from a sequence is Queue size #sent packets R+1 R f’s benefit d’s benefit b’s benefit g’s benefit e’s benefit a’s benefit c’s benefit Total Benefit f g d e e e c c c c c b b b b b b a a a a a a a Time Dagsthul Meeting on Fair Division

12. Lower Bound Homogeneous Packets Thm: The competitive ratio of any online algorithm (deterministic or randomized) is at least  Dagsthul Meeting on Fair Division

13. (α R)R - ½(α R)2 = ½R2α (2- α ) ½R2 Lower Bound Homogeneous Packets Proof: Choose such that 1-  = (2- )=>  = 1-1/ = 0.38 Sequence: • R packets arrive at each slot • until ON queue size is less than or equal R L R+1 L R(1- ) ON α R+1 α R OPT/ ON = 1/(1- ) =  R R+1 OPT L R α R+1 1 Dagsthul Meeting on Fair Division

14. Adaptation to Lower Bound on Randomized Algorithms • Oblivious adversary • OPT (σ) < c * E[ ON(σ) ] • ON’s queue size at time t is a random variable • Sequence: Feed R packets at each time slot until the first time t0such that E[queue size at t0] <  R Dagsthul Meeting on Fair Division

15. Threshold online policy Threshold policy: If queue size < (1-1/)R, accept, otherwise reject Thm: The competitive ratio of the threshold algorithm is at least  * In this talk we consider threshold ½ and prove competitive ratio 2 Dagsthul Meeting on Fair Division

16. Sequence Relaxation • To prove an upper bound, it suffices to consider sequences where • No packets arrive after ON’s queue is empty Queue size ON OPT T1 T2 • Consider a packet arriving between T1 and T2 – if this packet were to arrive after T2 ON would only lose relative to OPT • At point T2 we return to initial state Dagsthul Meeting on Fair Division

17. f(t) v(t) Potential Function • Denote by B(t) the queue size at time t • Prior to arrival and prior to send at timet Queue size at time t Queue size R+1 Event sequence 5 6 7 8 1 2 3 4 Time 0 t Dagsthul Meeting on Fair Division

18. Potential function Thm: The potential is always non negative Corollary: Competitive ratio ≤ 2 OPT ON Proof: By induction on number of events in sequence • Packet transmission (integral times) • Packet arrival (non-integral time) Dagsthul Meeting on Fair Division

19. f(t-ε) v(t-ε) Packet Transmission, integral time t • On a send event mass shifts from v to f f(t-ε) + v(t-ε) = f(t+ε) + v(t+ε) Queue size R+1 Event sequence 5 6 7 8 1 2 3 4 0 t=5 Dagsthul Meeting on Fair Division

20. f(t+ε) v(t+ε) Packet Transmission, integral time t • On a send event mass shifts from v to f f(t-ε) + v(t-ε) = f(t+ε) + v(t+ε) Queue size R+1 Event sequence 5 6 7 8 1 2 3 4 0 t=5 Dagsthul Meeting on Fair Division

21. Packet arrival event at time t ON • No send event atεtime aboutt: f(t-ε) =f(t+ε) ; f*(t-ε) = f*(t+ε) OPT • Both ON and OPT decline, v(t-ε) = v(t+ε) • ON accepts, OPT either accepts or declines, • v(t+ε) ≥ v(t-ε) + R/2 • v*(t+ε) ≤ v*(t-ε) + R • ON declines, OPT accepts • Special treatment Dagsthul Meeting on Fair Division

22. ON declines, OPT accepts Lemma:At any time s where ON queue is not empty, 2 f(s) ≥ f*(s), Proof: • ON queue size is at most R/2 Queue size R+1 f(s) s Dagsthul Meeting on Fair Division

23. ON declines OPT accepts • Previous lemma and proof that 2 v(t+ε) ≥ v*(t+ε), gives thatΨ ≥ 0 • ON queue size = R/2 ; v(t) = R+ (R-1) + … + R/2 (On Declines) • OPT queue size < R ; v*(t) < R+ (R-1) + … + R/2 + …+1 R+1 ON α R α R R R+1 OPT α R 1 Dagsthul Meeting on Fair Division

24. Heterogeneous Packets ONLINE Policy: Accept a packet if Value > 2 * queue size Thm: the competitive ratio of the above policy is at least 4 and at most 8 Proof Sketch: • Amortized analysis • Map each of OPT’s packets to 1/8 their value in ON’s packets Dagsthul Meeting on Fair Division

25. If v < w then Some Further Sequence Relaxation • To prove an upper bound, it suffices to consider sequences where • Packets accepted by ON have the smallest possible value Benefit is only B Val = 2B B ON’s queue • Each packet accepted by ON has benefitval- B(t) = B(t), where B(t) is the queue size at the time of arrival Dagsthul Meeting on Fair Division

26. ½B B +½ +½ +½ Amortized analysis – Re-distribute Credit • Re-distribute half the benefit (½B) equally between packets in ON queue • Keep the other half Benefit = B B Re-distribute(now “credit”) ON’s queue Lemma: After redistribution the credit of each packet is at least ½B Proof: A packet gets ½ a credit unit from every packet above it and originally had credit which was ½ it’s then position in the queue, and therefore at least ½ its current position in the queue Dagsthul Meeting on Fair Division

27. Mapping OPT packets to ON packets • Map every packet in OPT queue to half a packet in ON queue • Choose oldest un-mapped half packet ON OPT Dagsthul Meeting on Fair Division

28. Mapping OPT packets to ON packets ON OPT Lemma: the mapping is well defined Proof[sketch]: • When a packet is accepted by OPT, its value is at most 2B(t) (If ON declines, true, if ON accepts, also true by minimal value assumption • Hence, OPT queue size is at most 2B(t) • A packet in OPT is not transmitted prior to the packet it is mapped to • By induction on the number of packets accepted by OPT Dagsthul Meeting on Fair Division

29. Summing up • The benefit of a packet to OPT is at most its value val < 2B Credit: ½ B val < 2B • Competitive ratio 8 Dagsthul Meeting on Fair Division

30. Lower Bound on Heterogeneous Packets Thm: The competitive ratio of every deterministic algorithm is at least 3 Thm: Define the next sequence during the first slot: Must be accepted(or competitive ratio is ∞) Can accept at most one Queue Switch 3 2 3 2 1 3 • Next arrive packets {4;4;4;4} ; {5;5;5;5;5} • Sequence stops when ONLINE takes no packet of a certain class • ONLINE can accept 1,2,3… • OPT accepts all the packets of the last class offered (or 5,5,5,5,5) Dagsthul Meeting on Fair Division

31. Priority Truthful QoS • All QoS buffer management rely on truthful report of priority class • Online pricing mechanism • Charge Queue Size • Approximates social welfare • Does not require prior knowledge of the highest value possible IP Packet Dagsthul Meeting on Fair Division

32. Future research • Consider a generalized model where packets can have varying latency sensitive penalty functions • Profit maximization • Naor: the admission fee for profit maximization (under Poisson arrival) is greater than the admission fee set to maximize social welfare • Multiple queues • Studying networks of queues Dagsthul Meeting on Fair Division