1. Hashed and Hierarchical Timing Wheels
A paper by
George Varghese and Tony Lauck

2. Motivation

Timers are important for
 Failure recovery, rate based flow control, scheduling algorithms,
controlling packet lifetime in networks
 Timer maintenance high if
 Processor interrupted every clock tick
 Fine granularity timers are used
 # outstanding timers is high
 Efficient timer algorithms are required to reduce the
overall interrupt overhead

3. Model & Performance Measure
 Routines in the model
 Client Invoked :
 START_TIMER(Interval, Request_ID, Expiry_Action)
 STOP_TIMER(Request_ID)
 Timer tick invoked :
 PER_TICK_BOOKKEEPING
 EXPIRY_PROCESSING
 Performance Measure
 Space : Memory used by the data structures
 Latency : Time required to begin and end any of the
routines mentioned above

4. Currently Used Timer Schemes
a
a b
b c
c d
d
e
e
f
Can maintain absolute expiry maintain absolute expiry time
time or the timer interval
START_TIMER = O(n)
START_TIMER = O(1) STOP_TIMER = O(1)
STOP_TIMER = O(1) PER_TICK_BOOKKEEPING = O(1)
PER_TICK_BOOKKEEPING = O(n)

5. Tree based timers
a
a b
b c c
d
d
Can degenerate to a
maintain absolute expiry
linear list in case of equal
time
Interval timers
START_TIMER = O(log(n))
START_TIMER = O(n)
STOP_TIMER = O(1)
STOP_TIMER = O(1)
PER_TICK_BOOKKEEPING = O(1)
PER_TICK_BOOKKEEPING = O(1)

6. Simple Timing Wheel
 Keep a large timing wheel
 A curser in the timing wheel
moves one location every
0 1 time unit (just like a seconds
7 2 hand in the clock)
6
 If the timer interval is within
3 a rotation from the current
5 4 curser position then put the
timer in the corresponding
location
 Requires exponential amount
START_TIMER = O(1) of memory
STOP_TIMER = O(1)
PER_TICK_BOOKKEEPING = O(1)

7. Hashed Timing Wheel
# of rounds remaining  Say wheel has 8 ticks
 Timer value = 17
2 4  Make 2 rounds of
0 1 wheel + 1 more tick
7  Schedule the timer in
2
the bucket “1”
6 3  Keep the # rounds with
1 2
5 4 the timer
 At the expiry
2 1 1 2
processing if the #
rounds > 0 then
reinsert the timer

8. Hashed Timing Wheel
 Sorted Lists in each bucket
 The list in each bucket can be insertion sorted
 Hence START_TIMER takes O(n) time in the worst case
 If n < WheelSize then average O(1)
 Unsorted list in each bucket
 List can be kept unsorted to avoid worst case O(n) latency for
START_TIMER
 However worst case PER_TICK_BOOKKEEPING = O(n)
 Again, if n < WheelSize then average O(1)

9. Hierarchical Timing Wheel
1 3 3 5
0 1
7
2
6 3
3 7
5 4
3 5
5 1
0 1
7
Hours wheel 2
6 3
4 7 5
5 0 1
7
1 1 2 2
6 3
Minutes wheel 4
5
Seconds wheel

10. Hierarchical Timing Wheels
 START_TIMER = O(m) where m is the number of wheels
 The bucket value on each wheel needs to be calculated
 STOP_TIMER = O(1)
 PER_TICK_BOOKKEEPING = O(1) on avg.

11. Comparison
START_TIMER STOP_TIMER PER_TICK
Straight Fwd O(1) O(1) O(n)
Sequential
O(n) O(1) O(1)
List
Tree Based O(log(n)) O(1) O(1)
Simple
Wheel
O(1) O(1) O(1) High memory requirement
Hashed O(n) worst case
O(1) O(1)
Wheel (sorted) O(1) avg
Hashed Wheel O(n) worst case
O(1) O(1)
(unsorted) O(1) avg
Hierarchical
O(m) O(1) O(1)
Wheels