Many multistage interconnection networks (MINs) [7] have been proposed for communication systems, such as Clos, Benes, etc. The Shuffle-Exchange Network ...
Balance Routing Traffic in Generalized Shuffle-Exchange Network Zhen Chen, Zeng-Ji Liu, Zhi-Liang Qiu, Peng Chen, and Xiao-Ming Tao Abstract—Under equiprobable address of uniform packet traffic at input port, we find, the GSEN will route such traffic unevenly, an abnormal phenomenon caused by using unbalanced routing tags. We address and formulate this problem. To balance the routing traffic, the idea to use routing tag according to a probability is explained. By using Moore-Penrose pseudoinverse in matrix analysis, a methodology to solve this problem is proposed. An instance is given, and the simulation is done to verify our idea, which shows a big performance improvement by our methodology. Index Terms—Multistage Interconnection Network, Generalized Shuffle-Exchange Network, Tag-based Routing Algorithm, Moore-Penrose Pseudoinverse, Linear System, Matrix Analysis.
I. Introduction Many multistage interconnection networks (MINs) [7] have been proposed for communication systems, such as Clos, Benes, etc. The Shuffle-Exchange Network is a very common topology in MINs, which is widely used in telecommunication and parallel computing, especially in the current optical communication as a virtual topology. But the port number of available shuffle-exchange network is restricted by a power. K. padmanabham [1] and L. N. Bhuyan et al [6] are the few to break such limitation and proposes the generalized shuffle-exchange network (GSEN) with composite number of ports. A tag-based routing algorithm is also presented in [1], which is the focus of interest here. In this paper, we discuss the problem of uneven traffic caused by the tag-based routing algorithm in GSEN. To handle this problem, we formulate such problem with matrix analysis. By using Moore-Penrose pseudoinverse, we propose a methodology to solve this problem. An example to use such methodology is given, and the simulation is done to consolidate our idea.
II. Background A. Generalized Shuffle-Exchange Network Definition 1. (Generalized Shuffle-Exchange Network) [2] A Generalized shuffle-exchange network GSEN (k , r , n 1) is an indirect network with switch elements aligned in n 1 stages, labeled 0,
, n . Each stage consists of r switch elements, labeled with 0,
, r 1 . Every
switch element is a k k crossbar. Thus there are totally N ' k r links on one side of a stage, labeled 0, , N ' 1 . The parameters k , r , n of a GSEN satisfy the following equation:
log(kk r ) log k N ' n 1
Eq. 1
The switch elements in the adjacent stages are connected in the general shuffle pattern [1] defined as follows:
(i ) ( ki
ki ) mod N ', 0 i N ' 1 N '
Eq. 2
A tag-based routing algorithm is one that sets up a path from an input to an output by using a control tag L . Each digit of the k -ary representation of the tag (lnln1
1
l0 )k is used to control
one switch element in the path, where the digit ln is used to control the first switch element from input, digit ln1 the next, and so on. We consider higher order shuffles-exchange networks with non-binary switching elements, an example of GSEN (k , r , n 1) is given in Fig. 1, where k 3 , r 5 , n 2 , i.e., GSEN (3,5,3) . 0 1 2
0 1 2
3 4 5
3 4 5
6 7 8
6 7 8
9 10 11
9 10 11
12 13 14
12 13 14
Fig. 1. An 15×15 Shuffle-exchange network using ternary shuffles and switching elements. Let N ' N M , with N k
n
and k M (k 1) N . A tag-based routing algorithm has
been given in [1], cited as follows: Theorem 1. In GSEN (k , r , n 1) , a path from input i to output j can be set up by using the control tag L in the way that each switch element in the path is controlled by a single k -ary digit of the tag. The Tag L is given by
L1 ( j kMi) mod N '
Eq. 3
In addition, other control tags (and paths) may be available, specified by
LP L1 ( p 1) N ' ,
If LP kN , 1 p k .
Eq. 4
The routing tag used by source switch 0 in GSEN (3,5,3) to reach the destination switch q
(q 0,1, 2,3, 4) is given as follows: L(0) 00 or 12 L(4) 11
L(1) 01 or 20
L(2) 02 or 21
L(3) 10 or 22
B. The Uneven Traffic in GSEN In the scenario of GSEN (3, 5, 3) , the routing tags generated from the Theorem 1 will make the traffic uneven as we know, i.e., the times of symbol 2, symbol 1 and symbol 0 occurring in every location in the representation of routing tag are not the same. In general, the uneven distribution of traffic in GSEN (k , r , n 1) will deteriorate the performance issue because some internal links will encounter congestion. Thus we need a balanced routing algorithm to make them even. Our idea is to assign a probability to every routing tag, to make the times of symbol 2, symbol 1 and symbol 0 occurring in every location in the representation of routing tag equal. In essence, this is a combinatorial problem, while we solve this problem by using matrix analysis, which is 2
interesting per se. Here we put emphasis on the engineering meaning instead of mathematical meaning.
III. The Methodology to Solve This Problem A. The Methodology To balance routing traffic, our idea is to assign a probability to every routing tag, which denotes the likelihood to use such routing tag when there are several candidate routing tags to select from. By wisely assigning the probability, in GSEN (3,5,3) for example, one can make the times of symbol 2, symbol 1 and symbol 0 occurring in every location in the representation of routing tag equal, hence a balanced routing traffic. The following subsection gives the detailed formulation of our idea. B. The Formulation of The Problem There
X P0
are 0
P0
total 1
P0
kn
unknown
Pk 1
k -1
variables 0
Pk -1
1
in
such
Pk -1
k -1
linear
equation
denote
the
system.
N
Let
unknown
variables, which represent the probability assigned to every tag denoted as the subscript. There are total r Tag-guided Traffic Equations.
Tr N X N 1 [1/ r
1/ r ]'
Eq. 5
r
1 i j p r 0 i j p r
Matrix T is defined as tij
where p 0,1,
, k 1 . A basic form of
Matrix T is shown as follows:
1 0 0 1 T 0 0
0 1 0 0 0 1 1 0 0
Also there are total n (k 1) Tag-guided Balance Equations.
Bn( k 1)N X N 1 0n( k 1)1 Matrix B is defined as
3
Eq. 6
Br N
1 1 1 1
1 1 1
1 k
1
0 0
k
1
k
1 0 0 k
0
-1 -1
k
1
k
0 1 1 k
0
1 1 0
0
1 -1
k
k
1
k
1 0
k
1 0 1 0
k
Combined the Equation (3) and (4) into one, thus we get the matrix
T A , B
1 1 C r1 , the Matrix has the form of A r ( n1)( k 1) N . r 0n( k 1)1 A r n( k 1) N X N 1 C [1/ r
1/ r 0
0]'
Eq. 7
n( k 1)
r
C. Moore-Penrose Pseudoinverse in Matrix Analysis [4] [5] Moore-Penrose pseudoinverse is used to solve the rectangular system. The undetermined linear system of Eq. (7) involves more unknowns than equations. Thus the solution is not unique. It also need to know if the solution is compatible, then such solution can be used to construct the balance routing tags. By using Moore-Penrose pseudoinverse, the Eq. (7) can be solved as follows.
X A1 * C
Eq. 8
D. An instance for our Methodology We revisit the GSEN (3,5,3) , which is shown in Fig. 1. In GSEN (3,5,3) , N ' 15 , N 9 , M 6 , k 3 , r 5 , n 2 . The routing tags used by input port 0 in GSEN (3,5,3) to reach the destination switch q (q 0,1, 2,3, 4) have been given above as an example.
1 0 0 0 0 1 0 0 0 1 1 1 -1 -1 -1 0 0 0 0 1 0 0 0 0 1 0 0 1 1 1 0 0 0 -1 -1 -1 , C [1 1 1 1 1 0 0 0 0]' , T 0 0 1 0 0 0 0 1 0 , B 1 -1 0 1 -1 0 1 -1 0 0 0 0 1 0 0 0 0 1 1 0 -1 1 0 -1 1 0 -1 0 0 0 0 1 0 0 0 0
4
1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 A 0 0 0 0 1 0 0 0 0 . 1 1 1 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 1 1 0 1 1 0 1 1 0 1 0 1 1 0 1 1 0 1 The equation AX C is solved by using Moore-Penrose Pseudoinverse, here is a solution: t
2 1 2 1 3 1 2 1 2 . X 15 15 15 15 15 15 15 15 15 According to this solution, we construct an illustration of the balanced routing tag as follows. The set of balanced routing tags may not be only this one. There are many other choices too.
L(0) 00 L(1) 01 L(2) 02 L(3) 10 L(4) 11
00 20 02 22 11
L(0) 00 (2) 12(1) L(1) 01 (1) 20(2) L(2) 02 (2) 21(1) L(3) 10 (1) 22(2) L(4) 11 (3)
12 20 21 22 11
For any pair of source and destination, if one use the tag of three sets with the same probability, then the times of symbol 2, symbol 1 and symbol 0 occur in every location in the representation of routing tag are the same, hence the traffic is balanced. To consolidate the observation, we give the following theorem. Theorem 2. The routing tags generated by our methodology make the traffic even. Remarks: In a GSEN (3,5,3) , there are three input ports belonging to the same source switch. If each input port of the source switch uses different set of routing tags respectively, the traffic is balanced. Some Math Language such as MATLAB and MATHEMATICA can help us to compute the Moore-Penrose pseudoinverse of a matrix.
IV. Simulation In this section, we make a simulation for GSEN (3,5,3) , the performance improvement in term of throughput and delay is shown in Fig.2 and Fig. 3. For simulation, 95% confidence interval is used and we use the following conditions [3]. (1) The switch element is input queueing with the queue size of one buffer. And the MIN operates in a synchronous packet-switched mode. (2) With the same probability, each input at the first stage generates packets, which are uniformly distributed over all destination switches. (3) If two or more packets contend for the same output in a switch element, each packet has an equal probability to win the contention. And a blocked packet is resubmitted to the original 5
destination after a network cycle. (4) A network cycle is composed of two phases. In phase 1, sending buffers check the availability of the buffer in succeeding stage. In phase 2, each buffer sends a packet or enters into the blocked state depending on the availability. A successful packet will transfer to the next stage in one cycle. Fig.2 and 3 plot the throughput and delay data obtained from the simulation of GSEN (3,5,3) with two different sets of routing tags used by routing algorithm, i.e., the unbalanced routing tags, and the balanced routing tags remarked previously. By using the balanced routing tags, the curves clearly illuminate that we attain the performance improvement by 15% in the throughput and the lower delay characteristic under the condition of full load case.
Fig. 2. Throughput characteristics of GSEN (3,5,3) with two different sets of routing tags used by routing algorithm.
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Fig. 3. Delay characteristics of GSEN (3,5,3) with two different sets of routing tags used by
V. Conclusion The GSEN offers the flexibility to construct the MIN with an arbitrary composite port number. But the cost pays for such flexibility is the unbalanced routing traffic under the tag-based routing algorithm. Thus the main issue of our work is to balance the routing traffic. Then we propose a methodology to handle this problem. For easy explanation, an instance is given and performance improvement is illustrated by simulation results, which verified our idea. But there are still many things require further research. Some characteristics of the GSEN under the balanced routing algorithm lie concealed in the matrix A need to be discovered, and how to find integer solutions of such undetermined system. In the future, we will continue to investigate some interesting properties of GSEN, such as multicast, fault-tolerant routing etc., and expect to give a performance analytical model.
Reference [1] K. Padmanabham, “Design and analysis of even-sized binary shuffle-exchange networks for multiprocessors,” IEEE Trans. Parallel and Distributed Systems, vol. 2, no. 4, pp. 385–397, Jan. 1991. [2] Z. Chen, Z. Liu and Z. Qiu, “Bidirectional Shuffle-Exchange Network and Tag-Based Routing Algorithm”, IEEE Communications letters, vol. 7, no. 3, pp. 121-123, March 2003. [3] Youngsong Mun and Hee Yong Youn, “Performance analysis of finite buffered multistage interconnection networks,” IEEE Transactions on Computers, Vol. 43, No. 2, pp. 153 -162, Feb. 1994 [4] Rao C. R., Mitra S. K., Generalized Inverse of Matrices and its Applications, John Wiley and Sons, New York, 1971. [5] N. J. Pullman, Matrix Theory And Its Applications, Selected Topics, Academic Press, 1976. [6] L. N. Bhuyan, and Dharma P. Agrawal, “Design and Performance of Generalized Interconnection Networks,” IEEE Trans. on Compu., Vol. C-32, No.12, Dec. 1983. [7] Mischa Schwartz, Broadband Integrated networks，Prentice-Hall Inc., 1996.
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广义混洗网络的平衡路由算法1 陈震，刘增基，邱智亮，陈鹏，陶晓明 (综合业务网国家重点实验室，西安电子科技大学，陕西 西安 710071)
摘要--在输入业务流量为均匀到达，目的地址均匀分布的情况下，广义混洗网络 的内部链路流量是不均衡的。这种现象的原因是使用了不均衡的路由标记。本文 研究并解决了这个问题。为了平衡广义混洗网络内部链路的流量，我们思想是给 每个路由标记一定的使用概率。通过采用矩阵分析的 Moore-Penrose 逆，我们提 出了一套解决该问题的方法。为了说明我们的思想，我们给出了一个具体的实例。 仿真结果表明使用该方法获得的标记可以有效的改善网络性能。 关键词：多级连接网络，广义混洗网络，标记路由算法，Moore-Penrose 逆，线 性方程组，矩阵分析。
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国家 863 项目资助（2002AA103062 和 2002AA121061） 8
