Switching

1
Switching

• Lecture 1

2
Switch Architecture
Inter connection between N input links and N
output links
Switch Arbiter
O/p lines
I/p lines
1
2
1
3
2
3
4
4
5
5
O/p adapters
I/p adapters
Switch fabric
3
Switch Fabric
I/p lines
O/p lines
Cross bar
4
Switch Types
Single staged
Multi Staged
5
Input Queued Packets stored at the input Switch
speed line speed Any combination of packets can
be transferred across a switch as long as the
input and output terminals do not overlap
(Matching) 1-3, 2-1 allowed, 1-2, 3-2 not
allowed, 1-4, 1-5 not allowed
Output Queued Packets stored at the
output Switch speed N line speed (speedup
N) Any N packets intended for an output can be
transferred to the output simultaneously, 1-2,
3-2 allowed
6
Combined Input Output Queued Packets stored at
both input and output Speedup between 1 and N
Shared Memory Switch Packets stored in the switch
fabric every output line reads the paper as and
when they are transmitted
7
Switch Performance Metric
Throughput Rate vector comprising of rate of
transmission of packets across each output
(r1,.rN)
Delay Vector Delay between packet transmission
from input to output
8
Input Queued Switch
rij is the average number of packets arriving
at input line i for output line j
Let an output line serve one packet each slot
?i rij ? 1 for each j (output contention)
?j rij ? 1 for each i (input contention)
9
Output Queued switch
?j rij ? 1 for each i (input contention)
?i rij ? N for each j (output contention)?
Redundant constraint
10
Combined Input Output Queued Switch
?j rij ? 1 for each i (input contention)
?i rij ? k for each j (output contention)
speedup k
11
Shared memory switch
Constraints just like output queued switch
Less packet loss due to statistical multiplexing
of buffer memory Output queued switch loses
packet if any single output buffer overflows,
even if there is additional space in other output
buffers Shared memory switches share all buffers
and will overflow only if the combined buffers
overflow
12
Scheduling for Input Queued Switch
Choice of matchings determine whether any
throughput can be attained as long as the
constraints are satisfied
FIFO scheduling Every input maintains a queue of
packets Transfer the first packet of each queue
subject to matching constraints
13
HOL Contention
1
2
4
2
4
3
4
1
2
3
4
Only one packet can be transferred across the
switch
14
1
4
2
2
4
3
4
1
2
3
4
Two packets can be transferred across the switch
15
Input constraints allow (?j?j rij ? 1 )/N close
to 1 (rate of transfer of packets
across the switch per line)
Under FIFO scheduling this number is upper
bounded by 0.586 Penalty of HoL blocking Proof
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Lookahead scheduling
First, schedule among the head of line packets in
each input queue
If an input does not transfer packet but has a
packet, see whether the second packet can be
scheduled
This can be generalized to considering the first
w packets in each queue
17
1
2
4
2
4
3
4
1
2
3
4
At the first cut 2-4 is scheduled, then look at
the second packet in queue 1 and schedule 1-2
18
Lookahead performance
(?j?j rij ? 1 )/N increases with increase in w,
but the limiting value for w?? is strictly less
than 1
Queueing in High-Performance Packet
Switching Hluchyj and Karol, IEEE JSAC December
88, Vol 6, No. 9
19
Achieving 100 Throughput in Input Queued Switch
100 throughput is said to be attained if the
switch is able to sustain any arrival process as
long as the rate constraints are satisfied, This
will allow (?j?j rij ? 1 )/N close to 1
Achieving 100 Throughput in Input Queued
Switch, Mckeown et al, INFOCOM 95 Stability
properties of constrained queueing systems and
scheduling for maximum throughput in multihop
radio networks, Tassiulas and Ephremides,
IEEE Trans of Automatic Control, Vol. 37, No. 12,
pp. 1936-1949, 1992
20
Maximum weighted matching based scheduling
Every input maintains separate logical queues for
each output Weight of each queue is the queue
length Serve the packets from the queues which
form a maximum weighted matching
Weight of a matching is the sum of weights of the
queues in the matching
21
2
1
2
Schedules 1-4
4
3
4
1
2
3
4
Possible matchings are 1-2, 2-4, weight
2 1-4,
weight 3
1-2, weight 1
2-4,
weight 1
22
Maximum weighted matching based scheduling gives
priority to long queues, and also tries to
schedule a large number of queues, but may not
always schedule the largest possible number of
queues
Proof for optimality
23
Lecture 2
24
Proof for FIFO throughput
Queueing in High-Performance Packet
Switching Hluchyj and Karol, IEEE JSAC December
88, Vol 6, No. 9
Assumptions Input queues saturated, that is
packets are always waiting at the input
queues Inputs can be selected with equal
probability for each output Suppose a new packet
reaches the HOL position of an input queue, then
its destination is a specific output queue with
equal probability as any other output queue (1/N)
25
Notations
Bim Number of packets at the heads of the
input queue destined for the ith output in
the mth slot, but not selected for it.
Aim Number of packets moving to the heads of
the input queue destined for the ith output in
the mth slot
Fm Number of packets transmitted through the
switch in the mth slot (expected value F)
? Expected number of packets passing through the
switch per slot per output line (F/N)
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Bim max(0, Bim-1 Aim 1) Eqn 1
Aim is binomial (Fm-1, 1/N)
For large N, binomial (Fm-1, 1/N) is Poisson(F/N)
Thus the dynamics of Eqn 1 resembles a queueing
system (M/D/1) where Aim is the arrival process,
Poisson(?), Bim is the number of packets in the
queue.
From standard M/D/1 result, expected value of Bim
is ?2/2(1- ?)
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Fm-1 N - ?i Bim-1
Dividing both sides by N, and taking expectations
we have ? 1 expected value of Bi
? 1 - ?2/2(1- ?)
Solving for ?, ? 2 - ?2
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Proof of optimality of the maximum weighted
matching algorithm for input queues
29
Preview of Markov Process

• A sequence of random variables X1 , X2,.,Xn ,..
  such that
• Xi1 is independent of X1,.Xi-1 given Xi
• Pr(Xi1 ai1/ Xi ai, ., X1 a1) Pr(Xi1
  ai1/ Xi ai )
• Discrete time discrete state markov chain
• So a markov chain evolution can be specified by
• Initial states
• Transition probabilities Pr(Xi1 ai1/ Xi ai
  ) pi,i1

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0.5
1
0
0.5
0.5
0.5
State A communicates with state B if there is a
positive probability path from A to B
A set of states is closed if all states in the
set communicate with each other, and no state in
the set communicates to any state outside the
set, e.g., 0, 1
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A state has a period d if the process can only
return to this state in intervals which are
multiples of d
More precisely, di is the g.c.d. of k pi ik gt
0, Where pi ik is Pr(Xk i/ X0 i)
d0 1
d0 2
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Period of any two communicating states are equal
A state is aperiodic if its period is 1
Consider a function of the states f(x) Ef(x) ?x
p(x)f(x)
33
Consider the input queueing system
Let the system have random arrivals, i.e., the
number of arrivals for the different sessions is
random
Arrival process for each input-output pair is
i.i.d (independent and identically
distributed) Number of arrivals of i-j in slot t
is independent of the number of arrivals of any
other pair in past or future slots, and also
independent of the number of arrivals of any
other pair in the same slot
Probability that a packet arrives for i-j in any
slot t is qij
34
Under this assumption the queue length process at
the nodes (values of the Bij(t) s for all pairs
i-j) constitute a markov process
Under maximum weighted matching, this markov
chain consists of one closed set with periodicity
1 (check it!)
Let f(x) be the total number of packets in the
input queues in state x Ef(x) ?x p(x)f(x)
A scheduling strategy is said to be optimum if it
attains finite expected queue length as long as
any other strategy attains it.
35
Then the probability distribution of the markov
chain converges to a probability distribution
with finite expectation
Assume that the expected arrival rates satisfy
the rate constraints for the input queued switch,
we will show that the negative drift condition
is satisfied
36
The function V(x) we choose is ?ixi2
Notations B(t) Vector of queue lengths
in slot t (column vector) A(t) Vector
of arrivals in slot t B(t) Vector of
departures in slot t
B(t1) B(t) A(t1) D(t1)
V(B(t 1))- V(B(t)) (B(t 1) - B(t))T (B(t
1) B(t))
(A(t1) - D(t1))T (A(t1)
D(t1) B(t))
(A(t1) - D(t1))T (A(t1) D(t1)) (A(t1)
- D(t1))T B(t)
37
EV(B(t 1))- V(B(t))/ B(t) x E(A(t1) -
D(t1))T (A(t1) D(t1))/ B(t) x E(A(t1)
- D(t1))T B(t)/ B(t) x
The first term in the summation can be upper
bounded by a positive constant since the expected
arrivals in slot t 1 are finite and independent
of the queue lengths in slot t, the expected
departure from each pair is at most 1 and at
least 0
38
E(A(t1))T B(t)/ B(t) x aTx a ?i ?i Mi
(using rate constraints) where M is a
matching vector and ?i ?i ? 1, ?i ?
0 aTx ?i ?i weight of matching i under x
? (?i ?i ) weight of maximum weight
matching under x
E(D(t1))T B(t)/ B(t) x weight of maximum
weight matching under x
E(A(t1) - D(t1))T B(t)/ B(t) x ? ((?i ?i )
- 1) weight of maximum weight matching under x
39
Depending on state x, this can become as highly
negative as desired
Thus the negative drift condition holds except
for a finite number of states.
40
Performance Difference Between Input and Output
Queues
Note that input queued switch has additional
rate constraints at the output (as compared to
output queued switches)
For output line stability these constraints will
eventually arise in output queued switches as
well.
Thus throughput wise under maximum weighted
matching input and output queued switches perform
similarly.
41
Delay performances can be different because of
the nature of constraints.
Can an input queued switch emulate an output
queued switch?
Lecture 12, TCOM 799, Fall 01
42
Lecture 13

• Computationally Simple Algorithms for Maximum
  possible throughput in input queued switches

43
Implementational complexity
Maximum weighted matching will require the
arbiter to know the instantaneous queue lengths
and then computing the maximum weighted
matching.
Will there be a loss in throughput if the
scheduling is as per the maximum weighted
matching in a previous slot?
Not, as long as the delay is finite and the
maximum number of arrivals are upper bounded in a
slot
44
The proof for throughput optimality uses
properties of maximum weight matching in
computing the following value
E(D(t1))T B(t)/ B(t) x weight of maximum
weight matching under x
If D(t1) is not a maximum weight matching, then
(D(t1))T B(t) differs from the weight matching
by at most a constant provided the maximum number
of arrivals in any slot is finite.
The result follows.
45
Computational complexity
A maximum weighted matching can now be computed
once in a certain interval, and used throughout
the interval without any loss in throughput.
O notation
Maximum weighted matchings can be computed in
bipartite graphs in O(N3 log N)
46
Other low complexity matchings
Maximum size matchings Low throughput for
nonuniform arrival rates
Computation can become simpler for maximum weight
matchings with weight no longer the queue lengths

• "A Practical Scheduling Algorithm to Achieve 100
  Throughput in Input-Queued Switches." Adisak
  Mekkittikul, and Nick McKeown IEEE Infocom 98,
  Vol 2, pp. 792-799, April 1998, San Francisco.

47
Choose a maximum weight matching where the
Weight of a pair is now the sum of the queue
lengths at the input and the output ports for the
pair.
This is also a maximum size matching.
Simpler algorithms for computing maximum size
matchings can be used to compute this maximum
weight matching in O(N2.5 )
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Proof for optimality
Consider a symmetric matrix T (N2 X N2) such that
maxmatching M MTB(t) is not upper
bounded by a constant .
Then the scheduling policy which schedules the
matching M which maximizes the above every slot
is also throughput optimal
T can be chosen suitably to reduce the complexity
of computing such a optimal matching
49
Examples of T
Identity matrix Maximum weight matching with
weight queue length
Tij 2 if i j 1 if ?i/N? ?j/N?
1 if i mod N j mod N 0
otherwise Maximum weight matching with weight of
a pair is now the sum of the queue lengths at the
input and the output ports for the pair
Can there be a T to represent maximum size
matching?
50
Proof for optimality
The function V(x) we choose is XTX
V(B(t 1))- V(B(t)) (B(t 1) - B(t))T T (B(t
1) B(t))
(A(t1) - D(t1))T T(A(t1)
D(t1) 2B(t))
(A(t1) - D(t1))T T(A(t1) D(t1)) 2
(A(t1) - D(t1))T TB(t)
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EV(B(t 1))- V(B(t))/ B(t) x E(A(t1) -
D(t1))T T(A(t1) D(t1))/ B(t) x
2E(A(t1) - D(t1))T TB(t)/ B(t) x
The first term in the summation can be upper
bounded by a positive constant since the expected
arrivals in slot t 1 are finite and independent
of the queue lengths in slot t, the expected
departure from each pair is at most 1 and at
least 0
52
E(A(t1))T TB(t)/ B(t) x aT Tx a ?i ?i
Mi (using rate constraints) where M is a
matching vector and ?i ?i ? 1, ?i ?
0 aT Tx ?i ?i Mi Tx ? (?i ?i )
maxmatching M MTx
E(D(t1))T TB(t)/ B(t) x maxmatching M
MTx
E(A(t1) - D(t1))T T B(t)/ B(t) x ? ((?i ?i
) - 1) maxmatching M MTx
This can be as negative as desired for suitable
x, result holds
53
Linear complexity algorithms

• Linear complexity algorithms for maximum
  throughput in radio networks and input queued
  switches" Leandros TassiulasIEEE Infocom 98,
  Vol 2,April 1998, San Francisco.

Randomized algorithms
Choose the schedule randomly with a probability
distribution which depends on the queue lengths
54
Maximum weighted matching chooses the matching M
which attains maxmatching M Mx when the queue
length vector is x
The randomized algorithm chooses schedules with a
certain probability, the distribution is such
that the probability of choosing the maximum
weighted matching is at least ?, where ? can be
pre-selected as any number between 0 and 1.
Example choice include each pair
independently with probability 0.5 if the
choice is not a matching dont include any edge
probability of choosing any matching is 2-(NN)
55
Let this schedule be I
Let I(t) I if B(t)I ?B(t)I(t-1)
I(t-1) otherwise
The policy will be to schedule I(t) in each slot
Markov process representation is Y(t) (B(t),
I(t))
B(t1) B(t) A(t1) I(t)
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Proof of optimality
Let IB(t) be the maximum weighted matching vector
for queue lengths B(t)
V(Y) V1(Y) V2(Y) V1(Y) ?ibi2 V2(Y)
((IB-I)T B)2
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V1(Y(t 1))- V1(Y(t)) (B(t 1) - B(t))T (B(t
1) B(t))
(A(t1) - I(t))T (A(t1) I(t)
2B(t))
(A(t1) - I(t))T (A(t1) I(t)) 2 (A(t1) -
I(t))T B(t)
(A(t1) - I(t))T B(t) (A(t1) IB(t) )T B(t)
(IB(t) I(t))T B(t)
E(A(t1) - IB(t) )T B(t)/ Y(t) Y ? ((?i ?i
) - 1) wt of maximum Matching under B(t) ?
(1/N)(?i ?i ) - 1)? V1(Y)
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E(IB(t) - I(t1))T B(t)/ Y(t) Y ? V2(Y)
E(A(t1) - I(t))T (A(t1) I(t))/ Y(t)
constant
EV1(Y(t 1))- V1(Y(t))/ Y(t) ? 2(1/N)((?i ?i
) - 1)? V1(Y) 2? V2(Y) constant
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EV2(Y(t 1))/ Y(t) Y 0. PI(t1) IB(t1)
EV2(Y(t 1))/ Y(t)Y, IB(t1) ? I(t1)
PI(t1) ? IB(t1)
?(1-?) E((IB(t1) I(t1))T B(t1))2 / Y(t),
IB(t1) ? I(t)
E((IB(t1) I(t1))T B(t1))2 / Y(t), IB(t1)
? I(t1) E((IB(t1) I(t1))T
(B(t) A(t1) I(t))2 / Y(t), IB(t1) ?
I(t1)
E((IB(t1))T B(t) - (I(t1))T B(t)
(IB(t1) I(t1))T (A(t1) I(t)))2 / Y(t),
IB(t1) ? I(t1)
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(IB(t1))T B(t) ? (IB(t))T B(t)
(I(t1))T B(t)? I(t)T B(t)
E((IB(t1))T B(t) - (I(t1))T B(t) (IB(t1)
I(t1))T (A(t1) I(t)))2 / Y(t), IB(t1) ?
I(t1) ? E((IB(t))T B(t) - (I(t))T B(t)
(IB(t1) I(t1))T (A(t1) I(t)))2 / Y(t),
IB(t1) ? I(t1)
E ((IB(t) I(t))T B(t))2 / Y(t), IB(t1) ?
I(t1) E (IB(t1) I(t1))T (A(t1)
I(t)))2 / Y(t), IB(t1) ? I(t1) 2E((IB(t)
I(t))T B(t)) (IB(t1) I(t1))T (A(t1)
I(t))) / Y(t), IB(t1) ? I(t1)
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E ((IB(t) I(t))T B(t))2 / Y(t), IB(t1) ?
I(t1) ((IB(t) I(t))T B(t))2
V2(Y)
E (IB(t1) I(t1))T (A(t1) I(t)))2 /
Y(t), IB(t1) ? I(t1) ? constant
E((IB(t) I(t))T B(t)) (IB(t1) I(t1))T
(A(t1) I(t))) / Y(t), IB(t1) ? I(t1) ?
constant ((IB(t) I(t))T B(t)) constant
? V2(Y)
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E((IB(t1) I(t1))T B(t1))2 / Y(t), IB(t1)
? I(t1) ? V2(Y) constant constant ? V2(Y)
EV2(Y(t 1))/ Y(t), IB(t1) ? I(t1)
PI(t1) ? IB(t1) ?(1-?) (V2(Y) constant
constant ? V2(Y))
EV2(Y(t 1))/ Y(t) Y ?(1-?) (V2(Y)
constant constant ? V2(Y))
EV2(Y(t 1)) - V2(Y(t )) / Y(t) Y ?-?V2(Y)
(1-?) (constant constant ? V2(Y))
63
EV(Y(t 1))- V(Y(t))/ Y(t) ? 2(1/N)((?i ?i )
- 1)? V1(Y) -?V2(Y) (2-const.?)? V2(Y)
constant
64
Shared Memory Switches
65
Resource management constraints
Packets are stored in the switch fabric
As soon as packets arrive at the input, they are
read in the switch memory and stored there.
Each output line serves packets intended for it
as and when it is available.
There are N logical queues
66
Output lines serve packets independent of each
other.
Scheduling among the outputs is not an issue
Different queues share the same physical memory.
So memory management rather than scheduling is
the issue
67
How does memory management affect performance?
Performance metric Throughput or packet drop
Let N 2 Suppose the entire switch memory
consists of packets for output 1 Packets cleared
from the memory at rate 1 per slot. If the memory
had packets for both outputs, packets would be
cleared at rate 2
Load balancing reduces packet drop! Memory can be
managed to balance the load
68
Memory management options

• When a packet arrives
• It can be accepted
• It can be rejected
• It can be accepted while dropping some other
  packet (pushout)

The objective is to choose the optimal course of
action so as to minimize packet drops
Some architectures do not allow pushout!
69
Optimal memory management in presence of pushout
Optimal Buffer Sharing I. Cidon, L.
Georgiadis, R. Guerin, A. Khamisy IEEE JSAC,
Vol. 13, No. 7, September 1995
70
Optimal Strategy for N 2
Optimal memory management strategy accepts
packets whenever the buffer has space without any
push outs
If the buffer is full, then the arrival for
output port j is accepted pushing out one for the
other port, if the number of packets for port j
is below a certain threshold, j 1,2 The
thresholds for different ports can be different,
but their sum equals the total memory B
Can the queue lengths for both ports be above
their respective thresholds?
If the service rates are equal, then the
threshold is lower than B/2 for the one with the
higher arrival rate
71
Optimal strategy for arbitrary N
Known for identical arrival and transmission
rates for all ports.
Accept a packet if the buffer is not full.
If the buffer is full, reject any arrival for the
largest queue, accept arrivals for any other
packet while dropping packets from the largest
queue
Proofs involve markov decision process
72
Lecture 4
73
Optimal policy when push out is not allowed
Sharing memory optimally G. Foschini, B.
Gopinath, IEEE transactions on Communications,
Vol 31, No. 3, March 1983
Load balancing still reduces the packet drop.
Since push out is not allowed, load can be
balanced by rejecting new arrivals of overloaded
queues even when the buffer has space
74
Optimal Strategy for N 2

• Accept a packet for port j if
• Buffer has space
• Number of packets waiting for port j is less than
  a threshold mj

The policy reserves B m1 memory units for
output 2 and B m2 memory units for output 1
75
Optimal Strategy for N 3

• Accept a packet for port j if
• Buffer has space
• Number of packets waiting for port j is less than
  a threshold mj
• ?i?j xij ? mij even after accepting the packet

The policy reserves memory units for individual
outputs as also the combinations
76
Proof for optimality
In general, every policy will have a set of
states ? and will admit packets only if it does
not move out of the set.
The objective will be to find the set which
reduces the blocking.
The system can be modeled by a continuous time
markov chain if arrivals are Poisson and
departures exponential, with the state vector
consisting of queue lengths of individual queues
77
System Model
Queue j has Poisson arrivals at the rate ?j
exponential service at rate ?j Utilization ?j
?j / ?j
Steady state distribution ?(a,. ) ?1a. ?N../
?a.. ?1a. ?N.
Reversibility
78
Lecture 8
79
Optimal memory management in presence of pushout
Queue j has Poisson arrivals at the rate ?j
exponential service at rate ?j
Loss minimization is equivalent to throughput
maximization
Whenever a packet is served, system gets a reward
of 1 unit.
The objective is to maximize the total reward
80
System state queue length vector (x1 x2 .. xN)
Control action memory management
ej A vector with 1 in the ith position and 0 in
the rest
I(x) 0 if x 0 1 otherwise
J(x1 x2 .. xN) ??j I(xj ? 0) ?? ?i
max(J(x), J(x ei), J(x ei - ej)) if
?i xi ? B ??j I(xj ? 0)
?? ?i max(J(x), J(x ei - ej)) if
?i xi B
81
Optimal Strategy for N 2
Properties for the cost function J(x, y)
Monotonicity and Boundedness in x 0 ? J(x1,
y) J(x, y) ? 1, 0 ? x ? B - 1
Monotonicity and Boundedness in y 0 ? J(x,
y1) J(x, y) ? 1, 0 ? y ? B - 1
Concavity along x J(x1, y) J(x, y) ? J(x, y)
- J(x-1, y), 1 ? x ? B - 1
82
Concavity along y J(x, y1) J(x, y) ? J(x, y)
- J(x-1, y), 1 ? x ? B - 1
Concavity along the line x y b, 2 ? b ? B
J(x1, y-1) J(x, y) ? J(x, y) - J(x-1, y1),
1 ? x ? B 1, 1 ? y ? B 1
83
Derivation of Optimal strategy from the properties
Monotonicity and boundedness implies that the
optimal decision is to accept a packet for a port
if the buffer has space
The concavity property on the straight line x y
B implies that the function J(x, y) has either
a unique maxima on the line or two consecutive
maximas. These correspond to the replacement
thresholds.
84
Nature of Thresholds
If ?1 ? ?2 ?1 ?2 J(x, y) ?J(y, x), y ? x
85
Approximate computation of thresholds
J(x, y) ? f(x,y) d c1?1x c2?2y
d (?1 ?2)/(1- ?)
cj ?j/(1 - ?(1 - ?j (1- ?j )))
?j roots of quadratic equation
The maximum of f(x, B-x) gives the thresholds.
86
Optimal Strategy for N for symmetric arrival rates
Properties for the cost function J(x, y)
Monotonicity and Boundedness 0 ? J(x ej )
J(x) ? 1,
Symmetry J(x) J(y), where y is a permutation of
x
Balancing J(x) ? J(x ei - ej) if ith
component of x is less than the jth component
Drop from the longest queue