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Multirate Non-blocking Generalized Three-Stage Clos Switching Networks Wojciech Kabacin´ski, Senior Member, IEEE, and Fotios K. Liotopoulos, Member, IEEE
Abstract—This paper studies the non-blocking switching operation of generalized three-stage Clos switching networks in the multirate environment. The analysis presented in this paper determines the minimum number of the second-stage switches required for strictly non-blocking operation of such networks at call setup. Both the discrete and the continuous bandwidth cases are considered. For the discrete bandwidth case, sufficient and necessary conditions are derived. For the continuous bandwidth case, only sufficient conditions are given, which, in some cases, also constitute necessary conditions. The results given are, in some cases, generalizations of existing results, but they also include new results. Index Terms—Multirate switching, non-blocking switching networks, three-stage Clos switching networks.
I. INTRODUCTION
Fig. 1. Architecture of a generalized, multirate, three-stage Clos switch.
I
N 1953, Clos proposed a class of space-division three-stage switching networks and proved strictly non-blocking conditions of such networks [1]. The theory of non-blocking switching networks was later extended to time-division as well as multirate switching networks. Three-stage, time-division switching networks, composed of digital matrices, were proposed by Charransol et al. [2]. Non-blocking conditions of such switching networks, in the case of single-channel switching, were proved by Jajszczyk [3], and in the case of multichannel switching, were considered in [4] and [5]. In the case of multirate switches, an elegant model was proposed by Melen and Turner [6]. They also proved an upper bound of non-blocking conditions in the case of continuous bandwidth. This upper bound was later improved by Chung and Ross [7]. Asymmetrical switch configurations were considered in [8]. More generalized three-stage Clos switching networks were considered by Liotopoulos and Chalasani [9]. The results or . derived in those papers were limited to Both sufficient and necessary non-blocking conditions for any and were proved in [10] and [11] in the case of symmetrical and asymmetrical three-stage Clos switching networks, respectively. In some papers, call-blocking probability at call setup was also considered [12]–[14]. Paper approved by N. McKeown, the Editor for Switching and Routing of the IEEE Communications Society. Manuscript received February 15, 2000; revised May 15, 2001. This paper was presented in part at the IEEE Symposium on Computers and Communications (ISCC) 2000, Antibes-les-Pines, France, July, 2000, and at the International Conference on Telecommunications (ICT) 2000, Acapulco, Mexico, May 2000. W. Kabacin´ski is with the Poznan´ University of Technology, Institute of Electronics and Telecommunications, 60-965 Poznan´, Poland (e-mail:
[email protected]). F. K. Liotopoulos is with the Computer Technology Institute, 11851 Athens, Greece (e-mail:
[email protected]). Publisher Item Identifier 10.1109/TCOMM.2002.802564.
In this paper, non-blocking conditions for a generally asymmetrical multirate three-stage Clos switching network are considered. The results given are, in some cases, generalizations of known results, but they also include new results. The paper is organized as follows. In Section II, the model used in this paper is described. Definitions of discrete and continuous bandwidth cases are also given in this section. In Section III, nonblocking conditions for the three-stage Clos switching networks in the discrete bandwidth case are given and proved. Both sufficient and necessary conditions are considered. Section IV consists of a theorem and a proof for non-blocking operation in the continuous bandwidth case. Wide-sense non-blocking (WSN) switching networks are considered in Section V, followed by the conclusions. II. MODEL DESCRIPTION An architecture of the generalized three-stage Clos switching denotes network is shown in Fig. 1. In this figure, the number of inputs (outputs) in the first- (third-) stage switch , ,( , ), each of capacity ,( ). and represent a capacity of interstage links. Similarly, In further considerations, we will use the normalized bandwidth in the links. It means that the capacity of the link with the highest bit rate is equal to one. Links with lower bit rate have a capacity of less than one. A connection will be denoted by a triple ( , , ), where is the first-stage switch, is the third-stage switch, and is a weight representing the effective bandwidth required by the , since it has connection. In general, to be accommodated in one of the input (output) links of switch . To set up a new connection ( , , ), a second-stage switch is to be found, such that it has available bandwidth of at least
0090-6778/02$17.00 © 2002 IEEE
´ SKI AND LIOTOPOULOS: MULTIRATE NON-BLOCKING GENERALISED THREE-STAGE CLOS SWITCHING NETWORKS KABACIN
both in one link to the first-stage switch and in one link to the third-stage switch . In order to preserve the cell order, we will assume that a connection is routed through a single link to one second-stage switch. In this case, we should have and . Let us assume that in an input link of switch (an output link . of switch ), there are connections of weights . If through an We have interstage link from switch (to switch ), connections of , are already set up, then we have weights . In an input (output) link already carrying connections, a new connection ( , , ) can be set up, iff . If these conditions are not true, the input (output) link is inaccessible by a new connection of weight . Similarly, an interstage link from switch (to switch ) already carrying connections is accessible by a new connection ( , , ) iff . Otherwise, this link is inaccessible by the new connection. A new connection is compatible with the state of a link, if this link is accessible for this connection. A new connection ( , , ) is compatible with the state of the switching network, if it is compatible with one of the input links of switch and with one of the output links of switch . Let us assume that the new connection ( , , ) is to be set up, and this connection is compatible with an input link of switch and with an output link of switch . To set up this connection, a middle-stage switch accessible by this connection is to be found. A middle-stage switch is accessible by a connection ( , , ), if one of its links to switch and one of its links to switch are both accessible by this connection. The definitions of strictly non-blocking (SNB) and WSN operations given in [15] can be extended to the multirate environment by analogy. Usually, the weights of all connections belong to a closed . Two cases are deinterval [ , ], where fined: discrete bandwidth and continuous bandwidth [7], [9]. In this paper, the strictly non-blocking operation of the generalized three-stage Clos switching network in both cases is considered. In the discrete bandwidth case, a connection of weight shall be called “quantum”. In further considerations, we will assume , where , that and . III. NONBLOCKING CONDITIONS IN BANDWIDTH CASE
THE
DISCRETE
The non-blocking conditions in the discrete bandwidth case is an integer and were given in [7]. The when was considered in [9]. We will now determine case with similar conditions for the more general case of the architecture shown in Fig. 1. Theorem 1: The three-stage switching network shown in Fig. 1 is non-blocking in the strict sense for the discrete bandwidth case, iff
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(1) Proof: Necessary conditions can be proved by showing a blocking state in the switching network with the minimum value of given by (1). The following path searching algorithm will be used. If a new connection of weight appears in the same first-stage switch as the last connection set up, we start to search a path from the interstage link through which the last connection was set up. When a new connection appears in another firststage switch, we start to search a path from the middle-stage switch next to the last engaged. , First, let us intuitively define the quantities , , , and : represents the maximum number of quanta that 1) can constitute a state of first-stage switch , compatible with a connection of weight (see also Fig. 2). represents the maximum number of quanta that 2) can constitute a state of third-stage switch , compatible with a connection of weight . represents the maximum number of quanta that can 3) fit in all third-stage switches, but switch . represents the maximum number of quanta that can 4) fit in all first-stage switches, but switch . represents the maximum number of quanta that 5) can make a trunk of internal links, connecting a first-stage switch with some second-stage switch, inaccessible by a connection of weight . represents the maximum number of quanta that 6) can make a trunk of internal links, connecting a thirdstage switch with some second-stage switch, inaccessible by a connection of weight . Let us now assume that the maximum value of (1) is obtained and . The following set of events lead to the for occupancy of all middle-stage switches: connections ( , , ), Step 1) Set up . where Step 2) Set up a connection ( , , ), where . Step 3) Set up a connection ( , , ), disconnect the connection ( , , ), and then set up connections ( , , ). Step 4) Repeat Steps 2 and 3 until connections of weight
are set up
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(a)
(b) Fig. 2.
Worst case scenario for connection placement. (a) Switch i. (b) Switch k in the strictly non-blocking mode.
in switch
. These connections will occupy
middle-stage switches, and these switches will be inaccessible by a new connection of weight from switch . , ), Step 5) Repeat Steps 1–4 for connections ( , . These connections will occupy the next
will be occupied. Similar considerations for the third-stage switch show that the set of , but middle-stage switches no more than will be inaccessible by the connection ( , , ), where and . In the worst case these sets of middle-stage switches are disjoint, and one more switch is needed for a connection ( , , ), so
middle-stage switches, and these switches will be inaccessible by a new connection of weight to the third-stage switch . middle-stage In the above switching network, switches are occupied, and a new connection ( , , ) will . It occupy the second-stage switch numbered is available in one of the should be noted that the weight input links of switch , as well as in one of the output links of switch . Sufficiency can be proved by showing the worst state in the switching network. In order to maximize the utilization of all links and minimize capacity fragmentation of the switch, we consider that only quantum connections are already set up in ). Suppose we want to add the switching network, (i.e., . In switch at a new connection ( , , ), connections of weight can be set up in each most input links and such connections of may be set up in the remaining link. The total number of . connections is The interstage link is inaccessible by a new connection if connections of weight are already set up through this link. So the connections from switch can occupy middle-stage switches. However, in all the third-stage switches, except switch , it is possible to set up no more than connections of weight , middle-stage switches so no more than
(2) The function (2) must be maximized through all , , and and it reaches maximum for in (2), we obtain (1). Applying
, .
IV. NON-BLOCKING CONDITIONS IN THE CONTINUOUS BANDWIDTH CASE Let us now consider the three-stage switching network of Fig. 1 in the continuous bandwidth case, when
´ SKI AND LIOTOPOULOS: MULTIRATE NON-BLOCKING GENERALISED THREE-STAGE CLOS SWITCHING NETWORKS KABACIN
. Let us define the following functions as shown in (3)–(7), and (5) as shown at the bottom of the page.
if
is not an integer, or
if if
is an integer and
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Theorem 2: The three-stage switching network, shown in Fig. 1, is non-blocking in the strict sense for the continuous bandwidth case, if
(8) (3)
for for for (4) for
(6)
for for for
(7)
where conditions are as shown in (9)–(16) beginning at the bottom of the page. Proof: Sufficient conditions will be proved by showing the worst case in the switching network. Suppose we want to . Any interstage add a new connection ( , , ), link from switch will be inaccessible by the new connection of weight , if the sum of connection weights already set up . In the worst case, this through this link is greater than , sum of weights should be as small as possible, say where is close to but greater than 0. We have three cases. : The interstage link is inaccessible Case 1, by the new connection, if it carries one connection of weight . In the worst case, each connection with this weight from switch may be set up through a separate interstage link. In
for
(5)
for
for
for for all other
(9)
for all other for all other (10) (11) for
for for all other for all other for all other
(12)
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(13) (14) for
(15) for
for all other
for
(16) for for all other
one input link, there may be, at most, connections of such input links. In the last input weight . There are . link of this switch, there is a free bandwidth of weight and , then . Since In this link, we cannot set up a connection of weight . In the first-stage switch there may be connections of weight set up, and these connections may ocmiddle-stage switches. However, in all cupy third-stage switches, except switch , it is possible to set up connecno more than middle-stage tions of weight , so no more than switches will be occupied. and : The Case 2, interstage link is inaccessible by the new connection of weight , if it carries connections of a total weight greater . Since , there is no possible than way to set up a connection of such weight. But because , the interstage link will be inaccessible by the new connection of weight , if it carries two connections of weight . Similar to Case 1, we may have connections of weight in input links of switch , and connections of such a weight in the last input link of this switch. In the input links of switch there may be connections of weight , and these connections will occupy middle-stage switches, provided that all these connections can be accepted in some third-stage switch, other than switch . Since in these third-stage switches it is possible to set up, at connections of most, weight , no more than middle-stage switches will be occupied.
Case 3, or and : Before we move to the proof of this case, let us intuitively define some variables. represents the maximum number of conwhere , that can be nections of weight set up in one input link of switch . represents the available bandwidth in 2) the input link of switch after setting up connections of weight . When this available is equal bandwidth is less than , then to 0, since this bandwidth cannot be used by any new connection. represents the number of connections of weight 3) which have to be set up through the interstage link to make this link inaccessible by a connection of weight . represents the number of interstage 4) links inaccessible by a connection of weight , when connections of weight are set up through each of these links. represents available bandwidth in the link 5) , then with new connection of weight . If . represents the remaining bandwidth after setting 6) and up all previous connections of weights . , only one connection of such a weight If may be set up. In the other case, at least two connections must be set up in an of total weight greater than interstage link. In one input link, we can have no more than 1)
´ SKI AND LIOTOPOULOS: MULTIRATE NON-BLOCKING GENERALISED THREE-STAGE CLOS SWITCHING NETWORKS KABACIN
connections of total weight , so this number of connections is greater than given by (3). This means that there can be connections of total weight greater than set up in one input link. So, at most, connections of such weight may be set up in the first-stage in switch . There is still free bandwidth of weight the last input link, but it cannot be used by a connection of . In each weight greater than input links, there is free bandwidth of weight of the , but when , this bandwidth cannot be used by the next connection. This means that in switch , we connections of have , and these connections occupy weight greater than middle-stage switches. , the remaining bandwidth When in each link can be used by the next connection. Several such in one interstage link connections of weight may lead to a state in which this link will be inaccessible by a new connection (the minimum number of these connections is ). The next interstage link will be inaccessible by denoted by cona new connection, if it carries nections of weight . This means that the next interstage links from switch will be inaccessible by a new connection. In switch , we now have input links with available bandwidth , and one link with available bandwidth of (function indicates whether the bandwidth quantity is less than or not, so it determines whether this bandwidth can be used by a connection or not). Connections of such weights may occupy a bandwidth of in an is greater than , then this interinterstage link. If stage link will also be inaccessible by the new connection from switch . Whether this interstage link is accessible or not can . Therefore, we be calculated by function may have interstage links from switch , which are inaccessible by a new connection of weight . These middle-stage switches. links will fully occupy , this remaining bandFor width may be divided among more than one connection. However, these connections may occupy no more than interstage links. This means that
interstage links may be inaccessible by a new connection of weight , and they will occupy middle-stage switches. Similarly, as in Cases 1 and 2, all connections in switch have to be accepted by all third-stage switches except switch . The
number of connections of weight greater than can be accepted is given by
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which
These connections will occupy no more than middle-stage switches. Combining these cases together, we can write that middle-stage switches will be inaccessible by a new connection of weight , where conditions exist as shown in (17)–(20) at the bottom of the next page, and this value must be maximized through all , . On the other hand, any interstage link to switch will be inaccessible by a new connection of weight , if the sum of connection weights already set up through this link is greater . Similar considerations, as for switch , show that than middle-stage switches will be inaccessible by the new connection of weight , and this value must be . However, in switch , maximized through all , we cannot set up more connections than can be accepted in all first-stage switches except switch . Hence, no more than middle-stage switches can be occupied. This implies that another set of middle-stage switches is inaccessible by the new connection, and are defined similarly, as in (17) where and (20), respectively. To set up the connection ( , , ), one more switch is needed in the middle stage. Therefore, we have
(21) The function (21) must be maximized through all , , , , and , and it reaches maximum for . in (21), we obtain formula (8). Applying It should be noted that, if the maximum is obtained for an , , or and such that and for a , such that , or and , then Theorem 2 provides necessary conditions as well. Such conditions can be proved by showing a set of events leading to the occupancy of the number of middlestage switches given by (8). This set of events can be constructed in a similar way, as in the proof of Theorem 1. Space-Division Non-blocking Switching Networks: Multirate switching networks are a more generalized case of space-division and multi-channel switching. Theorems 1 and 2 should then include already-known results. Let us consider the space-division three-stage , , , Clos switching network with , , all , and . , For the continuous bandwidth case, we have
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which is Case 1. We obtain , and now be rewritten as
, , . Therefore, (1) and (8) can Considering all cases, similar to the space-division case [see (3)], we obtain the results given in [16]. Similar considerations can be applied for the continuous bandwidth case, but in this case, the results obtained only constitute an upper bound.
We distinguish three cases. and Case 1)
are the minima. In this case, V. WSN SWITCHING NETWORKS
. are the minima. In this case, . and are the minima. In this case, Case 3) . and cannot be both It is obvious that minima at the same time. Therefore, for the space-division net. work, we can deduce that , , , , For , and , we obtain an asymmetrical three-stage Clos network. Considering the discrete bandwidth case, we have Case 2)
and
The number of middle-stage switches in non-blocking switching networks may be reduced by using the concept of WSN switching networks. For multirate networks, an algorithm with a “functional division” of middle-stage switches may be considered [12]. In this algorithm, middle-stage switches are divided into two groups. The first group of switches is used for , and the second group setting up connections with . of switches is used only by connections with Theorem 3: The three-stage switching network shown in Fig. 1 is non-blocking in the wide sense for the discrete bandwidth case, when an algorithm with a functional division is used, if and only if (22) where
(23)
for
for for all other for all other
and
for all other
and
(17)
and
(18) (19) for
(20) for for other
´ SKI AND LIOTOPOULOS: MULTIRATE NON-BLOCKING GENERALISED THREE-STAGE CLOS SWITCHING NETWORKS KABACIN
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and Gao’s results, switches are sufficient, while from our Theorem 4, it can be calculated that only 44 switches are sufficient.
is the group of switches used for connections with
VI. CONCLUSIONS
(24)
, and is the group of switches used for connections with , , , , , and are defined in Theorem 1. in (1), Proof: Formula (23) is obtained by letting . and (24) is derived from (1) by assuming In a similar way, the following theorem can be proved for the continuous bandwidth case. Theorem 4: The three-stage switching network shown in Fig. 1 is non-blocking in the wide sense for the continuous bandwidth case, when an algorithm with a functional division is used, if (25) where
In this paper, we studied the SNB operation (at call setup) of generalized three-stage Clos switching networks in the multirate environment. Both the continuous and the discrete bandwidth cases were considered. For the discrete bandwidth case, Theorem 1 provides sufficient as well as necessary non-blocking conditions. For single-channel interstage links ( ), integer, and a symmetrical switching network, our results are the same as in [7]. In the continuous bandwidth case, only sufficient conditions were proved in Theorem 2, however, in special cases, they also constitute necessary conditions. In Theorems 3 and 4, conditions for WSN switching networks were given for the discrete and continuous bandwidth, respectively. The results obtained contain some generalizations of existing results, but new results and analysis are also provided. For WSN switching networks, an algorithm assuming a functional division of middle-stage switches was considered, and new, generalized analytical results concerning the required middle-stage switches for WSN operation under this algorithm have been derived. These results, in some cases, improve already existing results. REFERENCES
(26) is the group of switches used for connections with
,
(27)
, and is the group of switches used for connections with , , , and are defined in Theorem 2. It should be noted that the algorithm with a functional decomposition of middle-stage switches results in a reduction of and . the required switching elements for some values of For instance, in the discrete bandwidth case, WSN symmetrical networks require less middle-stage switches than SNB, for . In the continuous bandwidth case, this reduction also depends on and . Hwang and Gao have proposed a similar algorithm for WSN multirate networks [17]. When using this algorithm and conditions given in Theorems 1 and 2, the number of middle-stage switches for WSN networks given in Theorems 3 and 4 may be , , further reduced. For example, for and (continuous bandwidth case), according to Hwang
[1] C. Clos, “A study of non-blocking switching networks,” Bell Syst. Tech. J., pp. 406–424, 1953. [2] P. Charransol, J. Hauri, C. Athenes, and D. Hardy, “Development of a time division switching network usable in a very large range of capacities,” IEEE Trans. Commun., vol. COM-27, pp. 982–988, July 1979. [3] A. Jajszczyk, “On non-blocking switching networks composed of digital symmetrical matrices,” IEEE Trans. Commun., vol. COM-31, pp. 2–9, Jan. 1983. [4] W. Kabacin´ski, “On non-blocking switching networks for multichannel connections,” IEEE Trans. Commun., vol. 43, pp. 222–224, Feb.-Apr. 1995. [5] G. Niestegge, “Non-blocking multirate switching networks,” in Proc. 5th ITC Seminar, Lake Como, Italy, 1987. [6] R. Melen and J. S. Turner, “Non-blocking multirate networks,” SIAM J. Comput., vol. 18, no. 2, pp. 301–313, 1989. [7] S.-P. Chung and K. W. Ross, “On non-blocking multirate interconnection networks,” SIAM J. Comput., vol. 20, no. 4, pp. 726–736, 1991. [8] M. Collier and T. Curran, “The strictly non-blocking condition for threestage networks,” in Proc. 14th Int. Teletraffic Congress (ITC’94), Antibes-les-Pines, France, 1994, pp. 635–644. [9] F. K. Liotopoulos and S. Chalasani, “Strictly non-blocking operation of 3-stage Clos switching networks ,” in Performance Modeling and Evaluation of ATM Networks. London, U.K.: Chapman & Hall, 1996, vol. II. [10] W. Kabacin´ski, “Non-blocking three-stage multirate switching networks,” in Proc. Sixth IFIP Workshop on Performance Modeling and Evaluation of ATM Networks, Ilkley, U.K., 1998, pp. 26/1–26/10. [11] , “Non-blocking asymmetrical three-stage multirate switching networks,” in Proc. Int. Conf. Communication Technology, vol. 1, Beijing, China, 1998, pp. S11-11/1–S11-11/5. [12] S. C. Liew, M.-H. Ng, and C. W. Chan, “Blocking and non-blocking multirate Clos switching networks,” IEEE/ACM Trans. Networking, vol. 6, pp. 307–318, Mar. 1998. [13] M. Stasiak, “Combinatorial considerations for switching systems carrying multi-channel traffic streams,” Ann. Telecommun., vol. 51, no. 11-12, pp. 611–625, 1996. [14] E. Valdimarsson, “Blocking in multirate interconnection networks,” IEEE Trans. Commun., vol. 42, pp. 2028–2035, Feb.-Apr. 1994. [15] V. E. Beneˇs, Mathematical Theory of Connecting Networks and Telephone Traffic. New York: Academic, 1965.
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[16] F. K. Liotopoulos and W. Kabacin´ski, “Multirate, Non-blocking, Generalized 3-stage Clos Switching Networks,” Computer Technol. Inst., Tech. Rep. TR99/12/01, 1999. [17] B. Gao and F. K. Hwang, “Wide-sense non-blocking for multirate 3-stage Clos networks,” Theor. Comput. Sci., vol. 182, pp. 171–182, 1997.
Wojciech Kabacin´ski (A’94–SM’01) received the M.Sc., Ph.D., and D.Sc. degrees in communication from Poznan´ University of Technology, Poznan´, Poland, in 1983, 1988, and 1999 respectively. Since 1983, he has been with the Institute of Electronics and Telecommunications, Poznan´ University of Technology, where he currently is an Associate Professor. His scientific interests cover broadband switching networks and photonic switching. Dr. Kabacin´ski has published three books, 78 papers and has 10 patents. He is a member of the IEEE Communications Society and the Association of Polish Electrical Engineers.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 9, SEPTEMBER 2002
Fotios K. Liotopoulos (S’92–M’96) graduated from the University of Patras, Patras, Greece in 1988 and received the M.Sc. and Ph.D. degrees from the University of Wisconsin-Madison, WI, in 1991 and 1996, respectively. Currently, he is a Researcher and R&D Unit Manager at the Research Academic Computer Technology Institute (R.A.C.T.I.) in Athens, Greece. He also teaches at the Greek Open University. His research interests include digital communications, broadband switching, electro-optical switching, ATM networks, parallel and distributed architectures, multiprocessor/multicomputer systems, performance analysis, and design optimization. Dr. Liotopoulos is a member of ACM, AMS, Sigma Xi, Eta Kappa Nu, the Technical Chamber of Greece, the IEEE Technical Committee of Communications Switching and Routing, and is on the Editorial Board of Wireless Communications Mobile Computing Journal.
