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
PEGG, 7.10.2007
2
Economics of Queues

• Naors Model
• The value from getting a service is R
• The cost of waiting is 1 unit of monetary value
  per unit of time

Threshold strategy Join if R gt Waiting Time
Service
3
Naors Model
Define Social Welfare Sum of agents utilities
Thm Naor 69 The equilibrium arrival rate is
greater than the socially desired one

• 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
• Solution
• To reduce the arrival rate, impose an appropriate
  admission fee
• Analysis under assumption of Poisson arrival rate
  and exponential service time

4
Our Work

• Non stochastic model
• No assumptions on arrival distribution
• Fixed processing time
• Competitive analysis
• Worst case analysis
• Compare to the optimal solution OPT
• Competitive ratio For all input sequences ?,
  OPT(?) lt c ON (?)

5
Online Model

• Event sequence
• Packet transmission, at integral times
• Arrive events (assume distinct non-integral times)

Arrive events
Send events
Time
0
1
2
3
Arrive EventTime and value are determined by
the adversary

• Transmission events are not under adversarial
  control!

6
Results

• Homogeneous packets (as in Naors 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 competitive ratio c
  4.24
• Lower bound of 4.23 (deterministic, memory-less)
• Lower bound of 3 (deterministic)
• Agents with growing impatience
• Convex cost functions
• Implies truthful online pricing mechanism

7
Related Work Online Buffer Management

• Competitive queue policies for differentiated
  services Aiello et al 00
• Buffer overflow management in QoS switches
  Kesselman et al 01
• Competitive queuing policies for QoS switches
  Andelman et al 03
• An optimal online algorithm for packet scheduling
  with agreeable deadlines Li et al 07
• Better online buffer management Li et al 07

8
Homogeneous Packets Easy Constant Competitive
Ratio

• Online policy Accept while the queue size is at
  most ½R
• Handles at least half the traffic any reasonable
  algorithm handles
• By induction on the number of events
• Each packet gets a profit of at least ½R
• Competitive ratio 4

9
Illustration of the Benefit

• Lemma the benefit from a sequence equals the
  area between the graph of buffer heights and the
  line R1

Queue size
sent packets
R1
R
Total Benefit
Time
10
Illustration of the Benefit

• Lemma the benefit from a sequence is

Queue size
sent packets
R1
R
ds benefit
fs benefit
gs benefit
es benefit
cs benefit
bs benefit
as 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
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Lower Bound Homogeneous Packets

• Thm The competitive ratio of any online
  algorithm (deterministic or randomized) is at
  least ? 1.618

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Lower Bound Homogeneous Packets

• Proof Choose ? such that 1- ? ?(2- ?) gt ?
  1-1/? 0.38
• Adversarys Sequence
• R packets arrive at each slot
• Until ON queue size is less than or equal ?R

L
R1
L R(1- ?)
ON
a R1
a R
OPT/ ON 1/(1- ?) ?
R
R1
L R
OPT
a R1
1
13
Threshold online policy

• Threshold policy If queue size lt (1-1/?)R
  0.38R , accept, otherwise reject
• Thm The competitive ratio of the threshold
  algorithm is the golden ratio ? 1.618

14
Whats next

• Generalized model where packets have varying loss
  functions
• General algorithm for setting admission fees?
• Profit maximization
• Naor the admission fee for profit maximization
  (under Poisson arrival) is greater than the
  admission fee set to maximize social welfare
• Studying networks of queues
• Memory, Randomization and time sharing
• Do they help?

15
Comments? Questions?

• Thank you!

16
Heterogeneous Packets

• Thm There exists a memory-less online policy
  for heterogeneous packets with competitive ratio
  4.24
• Thm The competitive ratio of any deterministic
  online algorithm for heterogeneous packets is at
  least 3
• The competitive ratio of any memory-less online
  algorithm for heterogeneous packets is at least
  4.23

17
Heterogeneous Packets

• ONLINE Policy Accept a packet if Value gt 2
  queue size
• Thm the competitive ratio of the above policy is
  5¼
• Proof Sketch Of a weaker upper bound of 8
• Amortized analysis
• Map each of OPTs packets to 1/8 their value in
  ONs packets

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Some Further Sequence Relaxation
Benefit is only B

• To prove an upper bound, it suffices to consider
  sequences where
• Packets accepted by ON have the smallest possible
  value

Val 2B
B
ONs queue

• Each packet accepted by ON has benefit val- B(t)
  B(t), where B(t) is the queue size at the time
  of arrival

19
Amortized analysis Re-distribute Credit

• Re-distribute half the benefit (½ B) equally
  between packets in ON queue
• Keep the other half

Benefit B
B
Re-distribute(now credit)
ONs queue
Lemma After redistribution the credit of each
packet is at least ½ B Proof A packet gets ½ a
credit unit from every packet above it and
originally had credit which was ½ its then
position in the queue, and therefore at least ½
its current position in the queue
20
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
21
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

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Summing up

• The benefit of a packet to OPT is at most its
  value

val lt 2B
Credit ½ B
val lt 2B

• Competitive ratio 8

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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 8)
Can accept at most one
Queue
Switch
2
2
1
3
3
3

• Next arrive packets 4444 55555
• 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)

24
Whats next

• Generalized model where packets have varying loss
  functions
• General algorithm for setting admission fees?
• Profit maximization
• Naor the admission fee for profit maximization
  (under Poisson arrival) is greater than the
  admission fee set to maximize social welfare
• Studying networks of queues
• Memory, Randomization and time sharing
• Do they help?