Universal bounds on buffer size for packetizing fluid policies in input queued, crossbar switches

Abstract

We consider a type of on-line, traffic scheduling problem in input queued, crossbar switches. The input to a problem, at each time step, is a set of desired traffic rates. These traffic rates in general cannot be exactly achieved since they assume arbitrarily small fractions of packets can be transmitted at each time step. The goal of the traffic scheduling problem is to closely approximate the given sequence of traffic rates by a sequence of switch uses in which only whole packets are sent. The focus of this paper is bounding the costs incurred in using such an approximation, in terms of the additional buffer size required. We establish universal bounds on the additional buffer size due to sending only whole packets; these bounds do not depend on the particular distribution of the input traffic, require no speedup, and guarantee 100% throughput. Specifically, for an N × N input queued, crossbar switch, an on-line, packetizing algorithm is presented that guarantees 100% throughput with a buffer requirement of (N + 1)²/4 packets per input port with no speedup. The algorithm can he improved to run in O(N log N) time, using a fast algorithm for edge-coloring bipartite multigraphs. In the reverse direction, it is shown for an N × N input queued, crossbar switch, that any on-line, packetizing algorithm with no speedup requires a buffer size of N/e - 2 packets per input port. We also extend the main packetizing algorithm in this paper to a general class of switch architectures.

Full text

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Universal Bounds on Buffer Size for Packetizing
Fluid Policies in Input Queued, Crossbar Switches
Michael Rosenblum1
Department of Mathematics
Massachusetts Institute of Technology
Email: mrosen@math.mit.edu
Michel X. Goemans†
Department of Mathematics
Massachusetts Institute of Technology
Email: goemans@math.mit.edu
Vahid Tarokh‡
Division of Engineering
and Applied Sciences
Harvard University
Email: vahid@deas.harvard.edu
Abstract—In this paper, we consider a type of on-line, traffic
scheduling problem in input queued, crossbar switches. The
input to a problem, at each time step, is a set of desired traffic
rates. These traffic rates in general cannot be exactly achieved
since they assume arbitrarily small fractions of packets can be
transmitted at each time step. The goal of the traffic scheduling
problem is to closely approximate the given sequence of traffic
rates by a sequence of switch uses in which only whole packets
are sent. The focus of this paper is bounding the costs incurred
in using such an approximation, in terms of the additional buffer
size required.
We establish universal bounds on the additional buffer size
due to sending only whole packets; these bounds do not depend
on the particular distribution of the input traffic, require no
speedup, and guarantee 100% throughput. Specifically, for an
N × N input queued, crossbar switch, an on-line, packetizing
algorithm is presented that guarantees 100% throughput with a
buffer requirement of (N +1)2/4 packets per input port with no
speedup. The algorithm can be improved to run in O(N logN)
time, using a fast algorithm for edge-coloring bipartite multi-
graphs. In the reverse direction, it is shown for an N ×N input
queued, crossbar switch, that any on-line, packetizing algorithm
with no speedup requires a buffer size of N/e − 2 packets per
input port. We also extend the main packetizing algorithm in
this paper to a general class of switch architectures.
Keywords: Deterministic network calculus, Graph theory,
Combinatorics.
I. INTRODUCTION AND RELATED WORK
Scheduling algorithms for input queued, crossbar switches
have received growing attention due to the massive growth
of communication networks. In one approach, the designer
first ignores the packet nature of traffic and constructs a
schedule under the assumption that packets can be broken into
arbitrarily small pieces and sent at different time slots (e.g.
[1]–[4]). This schedule is referred to as a fluid policy. Next, the
designer constructs a packetized policy, which approximates
the behavior of the fluid policy in order to send packet data.
1,‡This material is based upon research supported by the National Science
Foundation under the Alan T. Waterman Award, Grant No. CCR-0139398.
Any opinions, findings, and conclusions or recommendations expressed in
this publication are those of the authors and do not necessarily reflect the
views of the National Science Foundation.
1,†This material is based upon research partly supported by the National
Science Foundation under contracts ITR-0121495 and CCR-0098018. Any
opinions, findings, and conclusions or recommendations expressed in this
publication are those of the authors and do not necessarily reflect the views
of the National Science Foundation.
Other approaches to scheduling are also possible. Much
analysis has been done in the case where the input traffic is
assumed to have certain statistical properties (e.g. [5]–[7]).
In this input traffic model, it is often shown that the queue
lengths, considered as a stochastic process, converge to a
limiting distribution with finite expectation. The analysis of
packetizing fluid policies in this paper, however, as in most
of the related work discussed below, focuses on worst-case
bounds rather than probabilistic bounds. We give universal
bounds on buffer requirements that do not depend on any
statistical properties of the input traffic. These bounds on
required, additional buffer size are not asymptotic; they apply
to all switch sizes and to time increments of any finite duration.
In particular, one of the main results in this paper is an on-line,
packetizing algorithm that guarantees 100% throughput with
a buffer requirement of (N +1)2/4 packets per input port for
an N × N input queued, crossbar switch with no speedup.
Packetizing fluid policies can significantly simplify schedule
design. A number of authors have taken this approach in
constructing scheduling algorithms that provide certain perfor-
mance guarantees. Parekh and Gallager proposed Generalized
Processor Sharing (GPS), which creates a fluid policy that
is work-conserving and allocates bandwidth in proportion to
fixed weights on a 1×N crossbar switch [1], [2]. The fluid pol-
icy is then approximated by a packetized policy called Packet-
by-Packet Generalized Processor Sharing (PGPS). PGPS deter-
mines packet service order based on GPS’s packet-completion
times.
Bennett and Zhang [8] present another GPS-approximating,
packetized policy called Worst-case Weighted Fair Queueing.
This scheme services the packet with earliest GPS finishing
time (assuming no new packets arrive) among those that have
already begun GPS service. This scheme, however, suffers
from abrupt increases or decreases in delay (known as jitter),
as pointed out by Stamoulis and Liebeherr [9]. The jitter
problem is addressed in [9] with a modified version of GPS,
Slow-Start Generalized Processor Sharing, which creates a
smoothed version of the GPS fluid policy. This smoothed
fluid policy is then approximated by a packetized policy that
determines packet service order based on the fluid policy’s
packet-completion times.
In the related work described so far, the foundation for
each algorithm is a fluid policy created according to GPS.
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Duffield, Lakshman, and Stiliadis question “the notion that
queueing systems should closely emulate a GPS system” [10].
They point out that GPS-emulating systems allocate unused
resources to backlogged sessions in proportion to weights
that are fixed to reflect long-term bandwidth requirements
of traffic flows. These systems thus ignore the instantaneous
quality of service needs of applications in redistributing excess
bandwidth. Duffield, Lakshman, and Stiliadis present modified
fair-queuing schemes that attempt to remedy this problem, by
redistributing excess bandwidth not in proportion to weights,
but based on maximum flow delay and maximum queue
backlog [10].
Stamoulis and Giannakis [11] also point out limitations
of emulating GPS systems. In GPS, delay and bandwidth
guarantees are coupled in the fixed weights used to schedule
packets. To decouple delay and bandwidth allocations, Sta-
moulis and Giannakis propose using deterministically time-
varying weights [11].
The use of time-varying weights in GPS and the redis-
tribution of excess bandwidth based on factors other than
fixed weights result in more general fluid policies than in
pure GPS. This motivates the investigation of how well one
can packetize an arbitrary fluid policy, which may schedule
different amounts of fluid at each time step, in terms of
minimizing additional delay, buffer size, and jitter. Up to this
point, we have described related work focusing on the 1×N
switch; packetizing fluid policies has also been studied in the
context of the N × N crossbar switch.
Chang, Chen, and Huang [12] present a packetizing algo-
rithm for fluid policies on an N × N crossbar switch, in
which the same amount of fluid is scheduled at each time
step, and so can be represented by a single, rate matrix. The
entries of the rate matrix represent the fractions of bandwidth
allocated to users of the switch. Ideally, service at each time
slot should exactly mimic the behavior of the rate matrix,
but in practice only whole packets can be scheduled; thus,
the fluid behavior can only be approximated by a packetized
policy. Their packetizing algorithm, based on the Birkhoff-von
Neumann decomposition, requires initial run-time O(N4.5)
and on-line run-time O(logN) for an N ×N crossbar switch.
In this paper, we consider more general fluid policies, in which
the amount of fluid scheduled can vary at each time step.
Tabatabaee, Georgiadis, and Tassiulas [13] consider the
open problem of packetizing arbitrary fluid policies in an
N × N crossbar switch using first-come first-served, virtual
output queues. They attempt to characterize fluid policies that
can be packetized so that for each pair of input and output
ports, at each time step, the cumulative number of packets
sent between these ports differs from the cumulative fluid
scheduled between these ports by less than 1; such fluid
policies are said to be trackable. They prove that any fluid
policy for the 2 × 2 crossbar switch can be tracked, and
propose several heuristics for packetizing fluid policies on
larger switches. Bonuccelli and Clo [14] construct a simple
fluid policy for the 4×4 crossbar switch that cannot be tracked.
This untrackable fluid policy can be extended to larger switch
sizes.
The result of Bonuccelli and Clo [14] that for many switch
sizes there exist untrackable fluid policies, has motivated
us to consider a relaxed version of tracking, focusing on
buffer size. The present work tackles the problem of bounding
additional buffer size when packetizing fluid policies for input
queued, crossbar switches. The contributions of this paper are
described next.
Section II formulates the problem to be considered in this
paper and establishes the notation. In particular, we define
backlog, which quantifies how much additional buffer space
is used by a packetized policy than is used by the fluid
policy it is emulating. The term “buffer requirement” refers
to this additional buffer space, or backlog. In Section III, we
prove tight bounds on the buffer requirements for packetizing
arbitrary fluid policies on 1 × N input queued, crossbar
switches. The bounds are universal in that they do not depend
on the distribution of the fluid input and assume no speedup.
Specifically, for a 1 × N input queued, crossbar switch, a
greedy, on-line, packetizing algorithm is presented that guar-
antees 100% throughput with a buffer requirement of N/e
packets per input port with no speedup. We also prove that
this packetizing algorithm is optimal for minimizing the buffer
requirement in the worst-case. These results for the 1×N input
queued, crossbar switch were proved independently in a recent
paper by Adler et al. [15]. In Sections IV and V, we treat
N × N input queued, crossbar switches. We present an on-
line, packetizing algorithm that guarantees 100% throughput
with a buffer requirement of (N + 1)2/4 packets per input
port with no speedup. Moreover, it is shown that any on-
line, packetizing algorithm with no speedup has a buffer
requirement of at least N/e − 2 packets per input port. In
Section V we show for any ? > 0 how to modify this algorithm
to have O(1
requirement of (1 + ?)(N + 1)2/4 packets per input port.
Section VI extends the results of Section IV to more general
switch architectures than the crossbar switch, using a resource
allocation model of Tassiulas [6]. Finally, Section VII presents
our conclusions and directions for future research.
?N logN) run-time per fluid arrival, with a buffer
II. NOTATION AND DEFINITIONS
We consider the N × M input queued, crossbar switch
operating in discrete time with no speedup. All packets are
assumed to have the same size. The line rate (capacity) at
all input and output ports is one packet per time step. This
switch allows one to send, in one time step, packets from any
of the N input ports to any of the M output ports. The only
constraints are that in one time step, at most one packet can
leave a single input port, and at most one packet can arrive at
a single output port. The switch uses virtual output queueing
to avoid head-of-line blocking. We first define the notion of
a fluid policy, which represents the ideal, packet-scheduling
behavior desired.
A Fluid Policy for an N ×M crossbar switch is a sequence
of (fictitious) switch uses specifying the number of fractional
packets that ideally would be sent and received at every time
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step. This is represented by a sequence of N × M doubly
sub-stochastic matrices1{F(k)}k>0. Then, F(k)
fraction of a packet that should be sent from input i to output
j at time step k. We next define a packetized policy, which
should approximate the fluid policy.
A Packetized Policy for an N × M crossbar switch is a
sequence of switch uses where only whole packets are sent
and received at each time step. It is represented by a sequence
of N × M sub-permutation2matrices, {P(k)}k>0. The value
of P(k)
ij
is 1 if and only if a packet is sent from input i to
output j at time step k; otherwise it has value 0.
It is convenient to record, for each input port i and output
port j, the difference between the cumulative number of
fractional packets scheduled by the fluid policy up to and
including time k, and the cumulative number of whole packets
sent by the packetized policy up to and including time k. This
information is stored in the N × M matrix C(k), for k ≥ 0.
In particular, C(0):= 0, and C(k):= C(k−1)+ F(k)− P(k)
for k ≥ 1.
A packetizing algorithm takes a fluid policy as input, and
outputs a packetized policy with the property that it never
sends a packet before the fluid policy has sent some part of
that packet. In other words, for all k, the packetized step
P(k)is only allowed to have value 1 in entries (i,j) for
which C(k−1)
ij
+ F(k)
ij
> 0; this ensures that all entries of the
matrix C(k)are greater than −1. We consider only on-line,
packetizing algorithms, that is algorithms which have access
only to F(1),...,F(k)in computing P(k). See Figures 1 and
2 for examples of fluid policies, packetized policies, and their
cumulative differences.
We say that a packetizing algorithm tracks if for any given
fluid policy, it outputs a packetized policy with the following
property: The packetized policy sends each packet at or before
the time step in which the fluid policy has scheduled 1 unit of
fluid to it [13]. In other words, for all k, the maximum entry
of the matrix C(k)is less than 1.
Bonuccelli and Clo [14] construct a fluid policy for the
4×4 crossbar switch that cannot be tracked. Their construction
implies that for any N × M input queued, crossbar switch
with N,M ≥ 4, there exists a fluid policy that cannot be
tracked. This negative result motivates the study of packetizing
algorithms that satisfy a relaxation of the tracking property
defined above.
Backlog, defined next, quantifies how much additional
buffer space is used by a packetized policy than is used by
the fluid policy it is emulating. For time step k, and for a set
of pairs of input and output ports, we define their backlog to
be the positive part of the difference between the cumulative
amount of fluid scheduled between these pairs by the fluid
policy, and the cumulative number of packets sent between
ij
represents the
1A non-negative valued matrix is doubly sub-stochastic if all its row sums
and column sums are ≤ 1. If the row sums and column sums all equal one,
the matrix is doubly stochastic.
2A {0,1}-valued matrix is a sub-permutation matrix if all its row sums
and column sums are ≤ 1. A permutation matrix is a sub-permutation matrix
with a 1 in every row and every column.
these pairs by the packetized policy. This can be found by
summing the entries of the positive part3of the cumulative
difference matrix, (C(k))+, corresponding to these pairs of
input/output ports. In particular, the backlog at input port i is
the sum of entries in row i of (C(k))+. The total backlog of
all input/output ports is the sum of all the entries of (C(k))+.
See Figure 1 for an example.
For a set of pairs of input and output ports, we say that a
packetizing algorithm gets at most b-backlogged if for all time
steps k, their backlog is no more than b. In other words, for
all k, the sum of corresponding entries in (C(k))+is at most
b. For example, we say that a packetizing algorithm gets at
most b-backlogged per input port if in each row of (C(k))+,
the sum of entries is at most b at all time steps.
III. TIGHT BOUNDS ON BACKLOG FOR THE 1 × N
CROSSBAR SWITCH.
Consider the 1×N crossbar switch, that is, a switch with a
single input port and N output ports. The definition of backlog
for the 1 × N crossbar switch reduces to the following: For
a set of output ports, at time k, their corresponding backlog
is the positive part of the difference between the cumulative
fluid scheduled to leave these ports by the fluid policy up to
and including time k, and the cumulative number of packets
sent through these output ports by the packetized policy up to
and including time k.
We show that for the 1 × N crossbar switch, the greedy
algorithm, which is defined in part B of this section, for
building a packetized policy from any given fluid policy is
optimal in terms of minimizing the worst-case backlog over
all possible sets of output ports. First, we prove that for every
on-line, packetizing algorithm, there exists an adversarial fluid
policy that gives a lower bound linear in N on the total backlog
required to packetize any fluid policy and a lower bound
logarithmic in N on the backlog needed at any output port.
Next, we show that the greedy algorithm achieves both lower
bounds. The results in this section were proved independently
in a recent paper by Adler et al. [15].
A. A Lower Bound on Backlog for the 1×N Crossbar Switch
We design an adversarial fluid policy for a given on-line,
packetizing algorithm by, at each time step, scheduling fluid
equally to a subset of the output ports to which the algorithm’s
packetized policy has never sent.
In addition to keeping track at each time step k of the fluid
step F(k), packetized step P(k), and cumulative difference
C(k), we keep track of a subset of output ports Sk ⊆
{1,2,...,N}. Sk will include all the ports sent to by the
packetized policy up to and including time step k; however, Sk
may contain additional output ports. Also, we will maintain
for all k ≥ 0 that
Sk⊆ Sk+1
(1)
3The positive part of a matrix M is denoted M+, where M+
max{Mij,0}.
ij
:=
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Fluid Policy F(k)
1
2
00
0
22
0
22
1
2
00
0
22
0
22
1
2
00
1
2
00
1
22
0
00
2
1
22
0
00
2
Fig. 1. Three steps of a fluid policy (first column), packetized policy (second column), and their cumulative difference (third column). Note that since a
packetized step can only schedule packets for which some fluid has arrived, P(3)cannot be a permutation matrix; it can only be a sub-permutation matrix.
At time step 3, the backlog at input ports 1,2,3,4 (corresponding to the rows of C(3)) is 1,1/2,1/2,1 respectively. The total backlog at time step 3 is 3.
Packetized Policy P(k)






Cumulative Differences C(k)










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

1
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0
0
1
2
0
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1
2
1
2
0
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2
1
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1


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1
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0
−1
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1
2
−1
2
1
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0
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2
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2
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


Fluid Policy F(k)
?
0
4
?
00
Packetized Policy P(k)
?
010
?
000
Cumulative Differences C(k)
?
−4
?
−4
1
5
1
5
1
1
5
1
4
1
3
1
2
1
5
1
4
1
3
1
2
1
5
1
4
1
3
0
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,
10000
?
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−4
5
1
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1
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Fig. 2.An example of an adversarial fluid policy for the 5 × 1 crossbar switch.
and |Sk| = k. The 1 × N incidence vector χSkhas value 1
for all components in Skand has value 0 elsewhere.
Given an on-line, packetizing algorithm, we iteratively con-
struct the adversarial fluid policy {F(k)}, for k ∈ {1,2,...,N−
1}.
First, we set S0:= ∅.
At each iteration k, we define
1
N − (k − 1)([1,1,...,1] − χSk−1).
Then we run the packetizing algorithm to determine which
output port j it sends a whole packet to at time step k. If
j / ∈ Sk−1, we set Sk := Sk−1∪ {j}; otherwise, we choose
the least i ≥ 1 not in Sk−1, and set Sk:= Sk−1∪ {i}.
An example of such an adversarial fluid policy for the 5×1
crossbar switch is given in Figure 2.
Let Hmdenote the mth harmonic number; that is, H0:=
0, and for m ≥ 1, we have Hm := Hm−1+ 1/m. Having
constructed {F(k)}1≤k≤N−1, it is not difficult to prove that
for any m : 1 ≤ m ≤ N, just after time step N − 1, the
backlog of the set of m most-backlogged output ports is at
least m(HN− Hm). This is true immediately after iteration
N − m of the above construction, since the m output ports
not in SN−meach have cumulative difference equal to HN−
Hm. The sum of cumulative difference values of this set of m
output ports remains lower bounded by m(HN−Hm) at each
subsequent time step, since a total of exactly 1 unit of fluid
is scheduled among these m output ports (by (1)) and at most
one packet is sent by the packetized policy per time step.
F(k):=
Since we will refer to this summation often in the next
section, we define for fixed N, and m : N ≥ m > 0,
g(m) := m(HN− Hm).
It is easy to verify that for m : N ≥ m > 0,
1) g(m) ≥ 0,
2) m(g(m + 1) + 1) = (m + 1)g(m),
3) lnN − 1 ≤ g(1) ≤ lnN,
4) N/e − 2 ≤ maxm:N≥m>0g(m) ≤ N/e.
We have proved the following theorem:
Theorem 1: For the 1 × N crossbar switch, for every on-
line, packetizing algorithm, there exists an adversarial fluid
policy with the following property: For any m : N ≥ m > 0,
there exists a set S of entries of C(N−1)with |S| = m such
that
?
i∈S
C(N−1)
i
≥ g(m).
(2)
In particular, there is a set of output ports with backlog at least
N/e − 2. Also, there is a single output port with backlog at
least lnN − 1. We now turn to a converse, which in light of
the above theorem shows optimality of the greedy algorithm
for minimizing the worst case backlog over all possible sets
of pairs of input/output ports.
B. An Upper Bound on Backlog for the 1×N Crossbar Switch
We prove the following theorem for the greedy algorithm,
which constructs a packetized policy that at time k + 1 sends
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a packet to output port l := argmaxi
Theorem 2: For the 1 × N crossbar switch and for any
fluid policy, the greedy packetizing algorithm produces a
packetized policy satisfying the following set of inequalities:
For all k and for any non-empty set S of entries of C(k), we
have
?
C(k)
i
+ F(k+1)
i
?
.4
?
i∈S
C(k)
i
≤ g(|S|).
(3)
This implies the greedy packetizing algorithm is at most lnN-
backlogged per output port and at most N/e-backlogged over
all output ports.
Proof: We show by induction that the greedy algorithm
for determining a packetized policy from a fluid policy main-
tains (3) at each time step. Let S be any non-empty subset of
{1,2,...,N}.
The set of inequalities (3) are true initially since C(0)= 0
and for all non-empty S, g(|S|) ≥ 0.
For the inductive step, assume that (3) holds at time step k.
We will show it holds for time step k+1 regardless of the fluid
policy’s choice of F(k+1). We set P := P(k+1),C := C(k),
and F := F(k+1)for clarity of exposition.
By definition, the greedy algorithm chooses the 1×N matrix
P to be all 0’s except at entry l := argmaxi(Ci+ Fi), where
P has value 1. We consider two cases, depending on whether
l ∈ S or not.
Case 1: l ∈ S
Since?N
(Ci+ Fi)
− 1
i=1Fi≤ 1, we have:
??
i∈S
?
≤
?
g(|S|).
i∈S
Ci
≤
Since the left hand side equals the sum over output ports
i ∈ S of the cumulative difference at time step k + 1, this
proves (3) in this case.
Case 2: l / ∈ S
We have the following inequalities:
??
|S| + 1
|S|
i∈S
(Ci+ Fi)
?
≤
?
i∈{l}∪S
g(|S| + 1) + 1.
(Ci+ Fi)
≤
The first inequality holds since by the greediness of the
algorithm, we have for all i,Ci + Fi ≤ Cl + Fl. The
second inequality holds by the inductive hypothesis and since
?N
m(g(m + 1) + 1) = (m + 1)g(m), we get from the above
inequality that
?
i=1Fi≤ 1.
Since for any m : N ≥ m > 0, we have
i∈S
(Ci+ Fi) ≤ g(|S|).
4If for all i, C(k)
to any output port at time k + 1.
i
+ F(k+1)
i
≤ 0, then the greedy algorithm doesn’t send
Since the left hand side equals the sum over output ports i ∈ S
of the cumulative difference at time step k+1, this proves (3)
in this case.
The induction is complete.
IV. BOUNDS ON BACKLOG FOR THE N × N CROSSBAR
SWITCH
A. A Lower Bound on Backlog for the N×N Crossbar Switch
A corollary of the first theorem in the previous section gives
the following bound for the N × N crossbar switch:
Corollary: For the N × N crossbar switch, for every on-
line, packetizing algorithm, and any input port i, there exists
an adversarial fluid policy such that for any m : N ≥ m >
0, there exists a set of m entries in row i of C(N−1)with
backlog at least g(m). In particular, there exists a set of pairs
of input/output ports in row i with backlog at least N/e − 2.
The proof is the natural generalization of the proof for
the 1 × N crossbar switch from Section III. We construct an
adversarial fluid policy by, at each time step, using only output
ports to which the algorithm’s packetized policy has never sent
from input i.
It is not clear how to generalize the proof of the second
theorem in Section III, in which the greedy packetizing
algorithm is shown to achieve the lower bound resulting from
an adversarial fluid policy, to the N×N crossbar switch. First,
the 1×N and N×N crossbar switches differ in that there is no
clear choice for what the greedy packetizing algorithm should
be for the N × N crossbar switch. A second obstacle is that
in the N ×N crossbar switch, it may be impossible to send a
permutation matrix at every packetized step since only packets
for which some fluid has arrived can be sent in a packetized
step; for example, see Figure 1 and the accompanying caption.
In light of this fact, it is not initially clear whether backlog
can be kept finite at all for the N × N crossbar switch. The
next section proves that it can.
B. An Upper Bound on Backlog for the N × N Crossbar
Switch
We present an on-line, packetizing algorithm that from any
fluid policy, builds a packetized policy that is at most [(N +
1)2/4−1]-backlogged per input port for the N ×N crossbar
switch.
Algorithm 1: The algorithm builds a packetized policy
{P(k)}k>0 from a given fluid policy {F(k)}k>0. At each
iteration, the algorithm computes P(k+1)based on C(k)and
F(k+1). This computation is described below, where we set
P := P(k+1),C := C(k), and F := F(k+1)for clarity of
exposition. The algorithm maintains the following invariant
for all time steps k:
Invariant 1: For all k, the sum of positive entries in any
row or column of C(k)is at most (N + 1)2/4 − 1.
There are three main steps in the packetizing algorithm.
First, the algorithm dominates5C + F by a matrix B with
non-negative entries and with all row sums and column sums
5Matrix D?dominates matrix D if for all i,j,D?
ij≥ Dij.
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