972 IEEE COMMUNICATIONS LETTERS, VOL. 14, NO. 10, OCTOBER 2010

Queueing Analysis of Multi-Layer Contention-Tolerant Crossbar SwitchGuannan Qu, Hyung Jae Chang, Jianping Wang, Zhiyi Fang, and S. Q. Zheng

Abstract—We recently proposed Contention-Tolerant Crossbar(𝐶𝑇𝐶(𝑁)) and multi-layer 𝐶𝑇𝐶(𝑁) (𝑀𝐶𝑇𝐶(𝑁)) switch ar-chitectures. By developing queueing network model of 𝐶𝑇𝐶(𝑁)for tagged output, we proved that 𝑀𝐶𝑇𝐶(𝑁) achieves steadystate with the layer number 𝑘 ≥ 2 under Bernoulli i.i.d. uniformtraffic. In this letter, we extend the queueing network modelfor evaluating the mean cell number in each input queue andmean waiting time of 𝑀𝐶𝑇𝐶(𝑁) working in steady state underthe Bernoulli i.i.d. uniform traffic. This model is validated bysimulation results.

Index Terms—Contention-tolerant, switch, queueing analysis.

I. INTRODUCTION

IN a switch, output contentions occur when more thanone input ports have cells to be transmitted to the same

output port during the same time slot. Conventional crossbarswitches, including crossbar with crosspoint buffers switches,require complex hardware to resolve output contentions.

In our previous work [1], we proposed a new switcharchitecture called Contention-Tolerant Crossbar, denoted by𝐶𝑇𝐶(𝑁), where 𝑁 is the number of input/output ports.Similar to conventional crossbar, the fabric of the 𝐶𝑇𝐶(𝑁)is comprised by 𝑁2 crosspoints (Switching Element, SE)arranged as an 𝑁 ×𝑁 array. Each SE has three inputs, threeoutputs and two states, as shown in Fig. 1 (a). Each inputport is equipped with a scheduler 𝑆𝑖. In one time slot, if inputport 𝑖 (0 ≤ 𝑖 ≤ 𝑁 − 1) wants to transmit a cell to an outputport 𝑗 (0 ≤ 𝑗 ≤ 𝑁 − 1), 𝑆𝑖 sets the state of corresponding𝑆𝐸𝑖,𝑗 to receive-and-transmit (RT) state. The remaining SEsin the same row will be kept in cross (CR) state. If more thanone input ports set their SEs as RT in the same output line(column), the output line is configured as a pipeline, as shownin Fig. 1 (b). Cells transmitted from upstream input portswill be intercepted and buffered in downstream input ports.In this way, output contentions are tolerated in 𝐶𝑇𝐶(𝑁).Without resolving output contentions, schedulers distributedin inputs operate independently and in parallel. Compared toconventional crossbar switches, 𝐶𝑇𝐶(𝑁) is simpler and morescalable.

The switching throughput of 𝐶𝑇𝐶(𝑁) under Bernoulli i.i.d.uniform traffic was analyzed by modeling 𝐶𝑇𝐶(𝑁) withall inputs and tagged output as an open queueing network.Theoretically and experimentally, we proved that, with a single

Manuscript received January 8, 2010. The associate editor coordinating thereview of this letter and approving it for publication was V. Vokkarane.

G. Qu and Z. Fang are with the College of Computer Science andTechnology, Jilin University, Changchun, 130012, P. R. China (e-mail:qu.guannan@hotmail.com).

H. J. Chang and S. Q. Zheng are with the Department of Computer Science,University of Texas at Dallas, Richardson, TX 75083, USA.

J. Wang is with the Department of Computer Science, City University ofHong Kong, Hong Kong, P. R. China.

Digital Object Identifier 10.1109/LCOMM.2010.081910.100036

( a )

CR (cross)state

RT (receive-and-transmit ) state

Top input

SEi,j

Bottom output

Right outputRight input

Left inputLeft output

CTC(N)Layer

Input port 0

Input port 1

Input port N-1

Output port 0

Output port N-1

.

.

.

...

k

( c )

Output port j

.

.

.

Row of input port u

Row of input port i

( b )

Fig. 1. (a) A crosspoint SE and its two states; (b) each output line of𝐶𝑇𝐶(𝑁) is configured as a pipeline; (c) 𝑀𝐶𝑇𝐶(𝑁).

FIFO queue in each input and without speedup, the throughputof 𝐶𝑇𝐶(𝑁) is bounded by 63%. In order to improve through-put, we proposed multi-layer 𝐶𝑇𝐶(𝑁) switch [2], as shown inFig. 1 (c). 𝑀𝐶𝑇𝐶(𝑁) comprises 𝑘 parallel 𝐶𝑇𝐶(𝑁) layers.Each 𝐶𝑇𝐶(𝑁) layer has its own input and output buffers thatoperate independently. Traffic from outside of 𝑀𝐶𝑇𝐶(𝑁)can be evenly distributed over all layers. Once a cell is injectedinto a given layer, it will remain in this layer till it arrivesat its output port. Assuming the traffic arrival is Bernoullii.i.d. uniform traffic, we proved that the minimum value of 𝑘enabling each 𝐶𝑇𝐶(𝑁) layer in 𝑀𝐶𝑇𝐶(𝑁) to be steady is2.

In this letter, we extend the queueing network model ofthe 𝑘-layer 𝑀𝐶𝑇𝐶(𝑁) (𝑘 ≥ 2) to evaluate the mean cellnumber of each input queue and mean waiting time. Bysolving Discrete-Time Markov Chain state transition for eachinput queue, we obtain analytical results. Simulation resultsare used to validate our theoretical results.

II. QUEUEING MODEL AND DISCRETE TIME MARKOV

CHAIN

Without considering output buffers, we model 𝐶𝑇𝐶(𝑁) ineach layer of 𝑘-layer 𝑀𝐶𝑇𝐶(𝑁) as an open queueing net-work system, as shown in Fig. 2. Each input buffer (i.e. queue)is organized as an FIFO queue denoted by 𝑄𝑖. The Head-of-line cell (if exist) of an input queue will be transmitted tocorresponding output line within one time slot.

To simplify our work, we adopt the following assumptions:

1) The traffic model is Bernoulli i.i.d. uniform.2) When a cell arrives at an empty input queue in a time

slot, it cannot be transmitted out during the same slot.3) Each time slot is cut into two contiguous phases. Cells

arrive at input queues from outside during the first phase;During the second phase, cells which are transmitted totheir output lines either arrive at their downstream inputs(be intercepted) or arrive at their destination outputs.

1089-7798/10$25.00 c⃝ 2010 IEEE

QU et al.: QUEUEING ANALYSIS OF MULTI-LAYER CONTENTION-TOLERANT CROSSBAR SWITCH 973

a1o

aN-1o

O0

a0o

a1u

aN-1u

Q0

QN-1

.

.

.

Q1

...

...

...

...

...

ON-1O1

Fig. 2. Queueing network model of 𝐶𝑇𝐶(𝑁) in each layer of 𝑀𝐶𝑇𝐶(𝑁).

4) The input queue length is infinite.Let 𝜃𝑖 be the average throughput of 𝑄𝑖 when the queueing

network works in steady state. The traffic equation which isheld for 𝑄𝑖 [3] is

𝜃𝑖 =

{𝑎𝑜𝑖 if 𝑖 = 0;

𝑎𝑜𝑖 +∑𝑁−1

𝑗=0 (∑𝑖−1

𝑘=0 𝑝𝑘,𝑗,𝑖𝜃𝑘) if 0 < 𝑖 ≤ 𝑁 − 1,(1)

where 𝑝𝑘,𝑗,𝑖 is the probability of a cell leaving 𝑄𝑘 for 𝑄𝑖 byoutput line 𝑗; 𝑎𝑜𝑖 is the arrival rate of 𝑄𝑖 from outside. Let 𝜆be offered load from outside. For 𝑘 layers, 𝑎𝑜𝑖 = 𝜆

𝑘 .From the property of 𝐶𝑇𝐶(𝑁) which is described in [1],

one cell leaves 𝑄𝑘 for its downstream 𝑄𝑖 if and only if theyboth transmit their cells to the same output line within thesame time slot. Thus

𝑝𝑘,𝑗,𝑖 =

{𝑝𝑘,𝑗(𝑝𝑖,𝑗𝜃𝑖) if 𝑘 = 𝑖− 1;

𝑝𝑘,𝑗 [∏𝑖−1

𝑚=𝑘+1(1− 𝑝𝑚,𝑗𝜃𝑚)]𝑝𝑖,𝑗𝜃𝑖 if 0 ≤ 𝑘 < 𝑖− 1,

(2)

where 𝑝𝑖,𝑗 is the probability of a cell being transmitted tooutput line 𝑗. For uniform traffic, 𝑝𝑖,𝑗 = 1

𝑁 .Combining Equations (1) and (2), we obtain

𝜃𝑖 =𝑁𝑎𝑜𝑖

𝑁 − 𝑖𝑎𝑜𝑖. (3)

For 𝑄𝑖, the arrival rate from upstream queues 𝑎𝑢𝑖 is

𝑎𝑢𝑖 =

{0 if 𝑖 = 0;∑𝑁−1

𝑗=0 (∑𝑖−1

𝑘=0 𝑝𝑘,𝑗,𝑖𝜃𝑘) if 0 < 𝑖 ≤ 𝑁 − 1.(4)

Combining Equations (2), (3) and (4), we obtain

𝑎𝑢𝑖 =𝑖𝑎𝑜𝑖

2

𝑁 − 𝑖𝑎𝑜𝑖.

Let the steady-state probability vector of 𝑄𝑖 be Π𝑖 =[𝜋0, 𝜋1, 𝜋2, ..., 𝜋𝑛, ...]. To solve Π𝑖, we must consider twopossible cases:

1) 𝑄𝑖 (𝑖 ∕= 0): 𝑞𝑖(𝑡) is defined as the state of 𝑄𝑖 which isthe number of cells in 𝑄𝑖 at the end of time slot 𝑡. Obviously,𝑄𝑖 intercepts the cell from its upstream nodes if and only ifit is busy. Therefore, the probability of a cell arriving at 𝑄𝑖

from upstream under the condition that 𝑄𝑖 is not idle is

𝜉𝑖 =𝑎𝑢𝑖𝜃𝑖

=𝑖

𝑁𝑎𝑜𝑖 .

Analyzing this queueing model requires the construction ofa state-dependent Discrete-Time Markov Chain (DTMC) for𝑄𝑖. The transmission diagram is illustrated as Fig. 3.

0 21 ...

Fig. 3. The DTMC state transition diagram for 𝑄𝑖 (𝑖 ∕= 0).

The state transition probabilities 𝑃𝑥,𝑦 = 𝑃𝑟[ 𝑞𝑖(𝑡) =𝑦 ∣ 𝑞𝑖(𝑡− 1) = 𝑥 ] are

𝑃𝑥,𝑦 =

⎧⎨⎩

1− 𝑎𝑜𝑖 𝑥 = 𝑦 = 0;𝑎𝑜𝑖 𝑥 = 0, 𝑦 = 1;(1− 𝑎𝑜𝑖 )(1 − 𝜉𝑖) 𝑦 = 𝑥− 1, 𝑥 ≥ 1;𝑎𝑜𝑖 (1− 𝜉𝑖) + 𝜉𝑖(1− 𝑎𝑜𝑖 ) 𝑥 = 𝑦, 𝑥 ≥ 1, 𝑦 ≥ 1;𝑎𝑜𝑖 𝜉𝑖 𝑦 = 𝑥+ 1, 𝑥 ≥ 1;0 𝑜𝑡ℎ𝑒𝑟𝑠.

(5)To simplify the expression, we define:{

𝜌0 =𝑃0,1

𝑃1,0=

𝑎𝑜𝑖

(1−𝑎𝑜𝑖 )(1−𝜉𝑖)

;

𝜌 =𝑃𝑘−1,𝑘

𝑃𝑘,𝑘−1=

𝑎𝑜𝑖 𝜉𝑖

(1−𝑎𝑜𝑖)(1−𝜉𝑖)

.(6)

From global balance equations, the steady state solutions canbe obtained recursively as:

𝜋𝑛 =

{1−𝜌

1−𝜌+𝜌0𝑛 = 0;

(1−𝜌)𝜌0𝜌𝑛−1

1−𝜌+𝜌0𝑛 > 0.

(7)

2) 𝑄0: Cells only arrive at 𝑄0 from outside, thus, 𝑎𝑢0 = 0.The DTMC state transition of 𝑄0 which is illustrated in Fig.4.

0 1

Fig. 4. The DTMC state transition diagram for 𝑄0.

The transition probabilities are

𝑃𝑥,𝑦 =

{1− 𝑎𝑜𝑖 𝑦 = 0;𝑎𝑜𝑖 𝑦 = 1.

(8)

According to the definition in Equation (6), we have{𝜌0 =

𝑃0,1

𝑃1,0=

𝑎𝑜𝑖

1−𝑎𝑜𝑖;

𝜌 = 0.

Thus, the solutions are

𝜋𝑛′ =

{ 11+𝜌0

𝑛′ = 0;𝜌0

1+𝜌0𝑛′ = 1.

(9)

Comparing Equations (5) and (8), we can see 𝑄0 is a case of𝑄𝑖 that only has state 0 and 1. Let 𝜌𝑛−1 = 1 when 𝜌 = 0 and𝑛 = 1. Solution (9) is a special case of (7).

974 IEEE COMMUNICATIONS LETTERS, VOL. 14, NO. 10, OCTOBER 2010

0

5

10

15

00.2

0.40.6

0.810

1

2

3

4

5

6

7

8

Input number (N=16)Offered load (λ)

Mea

nce

llnum

ber

Theoretical results

Simulation results

Fig. 5. Mean cell number of each input queue (𝑁 = 16).

aio

O0

Qi ...

ON-1

aiu

i

u

...

Fig. 6. Cells arrive at 𝑄𝑖 from outside and from Ω𝑢.

III. NUMERICAL RESULTS

A. Mean Cell Number

The steady-state probability vector of each input queue hasbeen solved as a function of layer number 𝑘, input number 𝑖and offered load 𝜆. Let 𝑞𝑖 be the mean number of cells in 𝑄𝑖

at the end of one time slot. We can obtain

𝑞𝑖 =

∞∑𝑛=0

𝑛𝜋𝑛 =𝜌0

(1− 𝜌)(1− 𝜌+ 𝜌0).

Figure 5 shows the graphical results for 𝑞𝑖 as a function ofoffered load 𝜆 and input number 𝑖 with switch size 𝑁 = 16and layer number 𝑘 = 2. We can see that the simulation resultsvalidate the theoretical results.

B. Mean Waiting Time

We introduce a recursive method to compute the meanwaiting time experienced by cells in the queueing networksystem. First we define the queueing subnetwork of 𝑄𝑖 as:

Ω𝑖 = {𝑄𝑠 : 0 ≤ 𝑠 ≤ 𝑖, 0 ≤ 𝑖 ≤ 𝑁 − 1}.The cells passing through 𝑄𝑖 (0 < 𝑖 ≤ 𝑁 − 1) come fromtwo sources: from outside and from its upstream subnetworkΩ𝑢 (0 ≤ 𝑢 ≤ 𝑖 − 1), as shown in Fig. 6. All cells will spendmean waiting time in 𝑄𝑖, yet cells from upstream 𝑄𝑢 haveexperienced additional mean waiting time on getting throughΩ𝑢 when they arrive at 𝑄𝑖.

From Little’s law, we can obtain mean waiting time of cellsin 𝑄𝑖 is

𝜔𝑖 =𝑞𝑖𝜃𝑖.

The mean waiting time for Ω𝑖 can be solved recursively asfollows:

𝜔′𝑖 =

{𝜔𝑖 if 𝑖 = 0;

𝜔𝑖 +∑𝑁−1

𝑗=0 (∑𝑖−1

𝑘=0 𝜔𝑘 × 𝑝𝑘𝑗𝑖𝜃𝑘𝜃𝑖

) if 0 < 𝑖 ≤ 𝑁 − 1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

Mea

nw

aitin

gtim

e(t

ime

slots

)

N=16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

Mea

nw

aitin

gtim

e(t

ime

slots

)

N=64

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

Offered load (λ)

Mea

nw

aitin

gtim

e(t

ime

slots

)

N=128

Theoretical results

Simulation results

Fig. 7. Theoretical results and simulation results of mean waiting time.

A cell completes its travel when it arrives at its destinationoutput in its 𝐶𝑇𝐶(𝑁) layer, no matter from which queueingsubnetwork or by which output line. Under the uniformtraffic assumption, we can consider one output, e.g. 𝑂𝑗 . Theprobability of a cell arriving at 𝑂𝑗 within one time slot is

𝛿𝑗 = 1−𝑁−1∏𝑖=0

(1− 𝑝𝑖𝑗𝜃𝑖).

If cell 𝑐 arrives at 𝑄𝑗 at time slot 𝑡, the probability that itleaves queueing network from Ω𝑖 is

𝛾𝑖 =

{1𝛿𝑗𝑝𝑖𝑗𝜃𝑖

∏𝑁−1𝑘=𝑖+1(1− 𝑝𝑘𝑗𝜃𝑘) if 0 ≤ 𝑖 < 𝑁 − 1;

1𝛿𝑗𝑝𝑖𝑗𝜃𝑖 if 𝑖 = 𝑁 − 1.

Thus, the mean waiting time of cells passing through thewhole queueing network is

𝑊 =

𝑁−1∑𝑖=0

𝜔′𝑖𝛾𝑖.

Figure 7 shows the theoretical results and simulation resultswith layer number 𝑘 = 2 and switch size 𝑁 = 16,𝑁 = 64 and𝑁 = 128, respectively, which validate our theoretical results.

REFERENCES

[1] G. Qu, H. J. Chang, J. Wang, Z. Fang, and S. Q. Zheng, “Contention-tolerant crossbar packet switches without and with speedup,” in IEEEInternational Conference on Communication (ICC), May 2010.

[2] G. Qu, H. J. Chang, J. Wang, Z. Fang, and S. Q. Zheng, “Contention-tolerant crossbar packet switches,” to appear in International J. Commun.Syst., 2010.

[3] T. G. Robertazzi, Computer Networks and Systems: Queueing Theory andPerformance Evaluation, 3rd edition, p. 103. Springer-Verlag, 2000.