On blocking probability of multicast networks

Multicast is a vital operation in both broad-band integrated services digital networks (BISDN) and scalable parallel computers. We look into the issue of supporting multicast in the widely used three-stage Clos network or υ(m, n, r) network. Previous work has shown that a nonblocking υ(m, n, r) multicast network requires a much higher network cost than a υ(m, n, r) permutation network. However, little has been known on the blocking behavior of the υ(m, n, r) multicast network with only a comparable network cost to a permutation network. We first develop an analytical model for the blocking probability of the υ(m, n, r) multicast network and then study the blocking behavior of the network under various routing control strategies through simulations. Our analytical and simulation results show that a υ(m, n, r) network with a small number of middle switches m, such as m=n+c or dn, where c and d are small constants, is almost nonblocking for multicast connections, although theoretically it requires m⩾Θ(n(log r/log log r)) to achieve nonblocking for multicast connections. We also demonstrate that routing control strategies are effective for reducing the blocking probability of the multicast network. The best routing control strategy can provide a factor of two to three performance improvement over random routing. The results indicate that a υ(m, n, r) network with a comparable cost to a permutation network can provide cost-effective support for multicast communication     

A new design for wide-sense nonblocking multicast switching networks

In this paper, we propose a new design for a wide-sense nonblocking multicast switching network, which has many comparable properties to a strictly nonblocking Clos permutation network. For a newly designed four-stage N/spl times/N multicast network, its hardware cost, in terms of the number of crosspoints, is about 2(3+2/spl radic/2)N/sup 3/2/=11.66N/sup 3/2/, which is only a small constant factor higher than that of a three-stage nonblocking permutation network, and is lower than the O(N/sup 3/2/(logN/loglogN)) hardware cost of the well-known three-stage wide-sense nonblocking multicast network. In addition, the proposed four-stage nonblocking multicast network has a very simple routing algorithm with sublinear time complexity, and does not require multicast capability for the switch modules in the input stage.     

支持多时隙业务的MTS-Clos网络结构及其建模分析

多级Clos网络是一种典型的可扩展交叉互连结构,在数据通信和计算机并行网络中有着广泛应用。基于传统三级Clos网络C(m, n, r)的理论分析表明,当满足m2n-1时该网络是严格无阻塞的。该文针对多时隙业务,利用可快速实现时隙交叉的单级交换模块,构建了一种新型的MTS-Clos交换网络结构C(m, n, r)(Multiple Time Slot Clos network)。利用时隙交叉能力,该结构在保留原有网络特性的同时可提供更为理想的交换性能。采用随机分析模型,对该结构的阻塞率进行了理论分析,结果表明当中间级规模m=n+k, k是一个很小的非负整数,网络即达到无阻塞。对该结构的数值仿真有同样的结论。因此,该结构及分析结果对下一代大容量交换设备的设计,具有良好的参考价值。     

Comments on "Wide-sense nonblocking multicast Log/sub 2/(N,m,p) Networks"

Recently, Hwang and Lin derived nonblocking conditions for multicast connections in Log/sub 2/(N,m,p) networks. They improved some conditions obtained earlier by Kabacin/spl acute/ski and Danilewicz. In this letter, we identify some inaccuracy in present results concerning wide-sense nonblocking Log/sub 2/(N,m,p) multicast networks, and give the correct results.     

Yet another result on multi-log2N networks

One-to-many connection (i.e., multicast) is an important communication primitive used in parallel processing and high-speed switching in order to simultaneously send data from an input to more than one output. We prove that for even (respectively, odd) n, a multi-log₂N network is strictly nonblocking for a one-to-many connection traffic if it is designed by vertically stacking at least (δn)/4+1((δ/2)(n-1)+1) planes of a log₂N network together, where N=2ⁿ, δ=2^[n/2], and [x] denotes the greatest integer less than or equal to x. We thus give answer to the open problem and introduce yet another strictly nonblocking multicast network. The characterized network has self-routing capability, regular topology, O(2log₂N+2log₂(log₂N)) stages, and fewer crosspoints than the Clos network for N⩾512. We then extend multi log₂N multicast networks to the fanout restricted nonblocking networks. It turns out that the multi-log₂N network nonblocking in a strict-sense for a one-to-one connection traffic is also wide-sense nonblocking for a multicast traffic in which the fanout of any connection does not exceed δ, provided that for even (respectively, odd) n, the fanout capability of each log₂N network is restricted to stage (n/2)(((n-1)/2)+1) through n-1     

Algebraic gossip: a network coding approach to optimal multiple rumor mongering

The problem of simultaneously disseminating k messages in a large network of n nodes, in a decentralized and distributed manner, where nodes only have knowledge about their own contents, is studied. In every discrete time-step, each node selects a communication partner randomly, uniformly among all nodes and only one message can be transmitted. The goal is to disseminate rapidly, with high probability, all messages to all nodes. It is shown that a random linear coding (RLC) based protocol disseminates all messages to all nodes in time ck+/spl Oscr/(/spl radic/kln(k)ln(n)), where c<3.46 using pull-based dissemination and c<5.96 using push-based dissemination. Simulations suggest that c<2 might be a tighter bound. Thus, if k/spl Gt/(ln(n))/sup 3/, the time for simultaneous dissemination RLC is asymptotically at most ck, versus the /spl Omega/(klog/sub 2/(n)) time of sequential dissemination. Furthermore, when k/spl Gt/(ln(n))/sup 3/, the dissemination time is order optimal. When k/spl Lt/(ln(n))/sup 2/, RLC reduces dissemination time by a factor of /spl Omega/(/spl radic/k/lnk) over sequential dissemination. The overhead of the RLC protocol is negligible for messages of reasonable size. A store-and-forward mechanism without coding is also considered. It is shown that this approach performs no better than a sequential approach when k=/spl prop/n. Owing to the distributed nature of the system, the proof requires analysis of an appropriate time-varying Bernoulli process.     

Towards an information theory of large networks: an achievable rate region

We study communication networks of arbitrary size and topology and communicating over a general vector discrete memoryless channel (DMC). We propose an information-theoretic constructive scheme for obtaining an achievable rate region in such networks. Many well-known capacity-defining achievable rate regions can be derived as special cases of the proposed scheme. A few such examples are the physically degraded and reversely degraded relay channels, the Gaussian multiple-access channel, and the Gaussian broadcast channel. The proposed scheme also leads to inner bounds for the multicast and allcast capacities. Applying the proposed scheme to a specific wireless network of n nodes located in a region of unit area, we show that a transport capacity of /spl Theta/(n) bit-meters per second (bit-meters/s) is feasible in a certain family of networks, as compared to the best possible transport capacity of /spl Theta/(/spl radic/n) bit-meters/s in Gupta et al. (2000), where the receiver capabilities were limited. Even though the improvement is shown for a specific class of networks, a clear implication is that designing and employing more sophisticated multiuser coding schemes can provide sizable gains in at least some large wireless networks.     

Symbol-intersecting codes

We consider codes consisting of arrays over an alphabet F, in which certain intersecting subsets of n/spl times/m coordinates are required to form codewords of length n in prescribed codes over the alphabet F/sup m/. Two specific cases are studied. In the first case, referred to as a singly-intersecting coding scheme, the user data is mapped into n/spl times/(2m-1) arrays over an alphabet F, such that the n/spl times/m subarray that consists of the left (respectively, right) m columns forms a codeword of a prescribed code of length n over F/sup m/; in particular, the center column is shared by the left and right subarrays. Bounds are obtained on the achievable redundancy region of singly-intersecting coding schemes, and constructions are presented that approach-and sometimes meet-these bounds. It is shown that singly-intersecting coding schemes can be applied in a certain model of broadcast channels to guarantee reliable communication. The second setting, referred to as a fully-intersecting coding scheme, maps the user data into n/spl times/m/spl times/m three-dimensional arrays in which parallel n/spl times/m subarrays are all codewords of the same prescribed code over F/sup m/. Bounds and constructions are presented for these codes, with the analysis based on representing the n/spl times/m/spl times/m arrays as vectors over certain algebras on m/spl times/m matrices.     

Optimal throughput-delay scaling in wireless networks - part I: the fluid model

Gupta and Kumar (2000) introduced a random model to study throughput scaling in a wireless network with static nodes, and showed that the throughput per source-destination pair is /spl Theta/(1//spl radic/(nlogn)). Grossglauser and Tse (2001) showed that when nodes are mobile it is possible to have a constant throughput scaling per source-destination pair. In most applications, delay is also a key metric of network performance. It is expected that high throughput is achieved at the cost of high delay and that one can be improved at the cost of the other. The focus of this paper is on studying this tradeoff for wireless networks in a general framework. Optimal throughput-delay scaling laws for static and mobile wireless networks are established. For static networks, it is shown that the optimal throughput-delay tradeoff is given by D(n)=/spl Theta/(nT(n)), where T(n) and D(n) are the throughput and delay scaling, respectively. For mobile networks, a simple proof of the throughput scaling of /spl Theta/(1) for the Grossglauser-Tse scheme is given and the associated delay scaling is shown to be /spl Theta/(nlogn). The optimal throughput-delay tradeoff for mobile networks is also established. To capture physical movement in the real world, a random-walk (RW) model for node mobility is assumed. It is shown that for throughput of /spl Oscr/(1//spl radic/(nlogn)), which can also be achieved in static networks, the throughput-delay tradeoff is the same as in static networks, i.e., D(n)=/spl Theta/(nT(n)). Surprisingly, for almost any throughput of a higher order, the delay is shown to be /spl Theta/(nlogn), which is the delay for throughput of /spl Theta/(1). Our result, thus, suggests that the use of mobility to increase throughput, even slightly, in real-world networks would necessitate an abrupt and very large increase in delay.     

Zero-error network coding for acyclic networks

Consider transmitting a set of information sources through a communication network that consists of a number of nodes. Between certain pair of nodes, there exist communication channels on which information can be transmitted. At a node, one or more information sources may be generated, and each of them is multicast to a set of destination nodes on the network. In this paper, we study the problem of under what conditions a set of mutually independent information sources can be faithfully transmitted through a communication network, for which the connectivity among the nodes and the multicast requirements of the source information are arbitrary except that the connectivity does not form directed cycles. We obtain inner and outer bounds on the zero-error admissible coding rate region in term of the regions /spl Gamma//sub N//sup */ and /spl Gamma/~/sub N//sup */, which are fundamental regions in the entropy space defined by Yeung. The results in this paper can be regarded as zero-error network coding theorems for acyclic communication networks.     

The encoding complexity of network coding

In the multicast network coding problem, a source s needs to deliver h packets to a set of k terminals over an underlying communication network G. The nodes of the multicast network can be broadly categorized into two groups. The first group includes encoding nodes, i.e., nodes that generate new packets by combining data received from two or more incoming links. The second group includes forwarding nodes that can only duplicate and forward the incoming packets. Encoding nodes are, in general, more expensive due to the need to equip them with encoding capabilities. In addition, encoding nodes incur delay and increase the overall complexity of the network. Accordingly, in this paper, we study the design of multicast coding networks with a limited number of encoding nodes. We prove that in a directed acyclic coding network, the number of encoding nodes required to achieve the capacity of the network is bounded by h/sup 3/k/sup 2/. Namely, we present (efficiently constructible) network codes that achieve capacity in which the total number of encoding nodes is independent of the size of the network and is bounded by h/sup 3/k/sup 2/. We show that the number of encoding nodes may depend both on h and k by presenting acyclic coding networks that require /spl Omega/(h/sup 2/k) encoding nodes. In the general case of coding networks with cycles, we show that the number of encoding nodes is limited by the size of the minimum feedback link set, i.e., the minimum number of links that must be removed from the network in order to eliminate cycles. We prove that the number of encoding nodes is bounded by (2B+1)h/sup 3/k/sup 2/, where B is the minimum size of a feedback link set. Finally, we observe that determining or even crudely approximating the minimum number of required encoding nodes is an /spl Nscr/P-hard problem.     

On The General Relation Between /spl alpha/ and Q (Correspondence)

The relation between the quality factor Q and the attenuation constant /spl alpha/ of a transmission line has been known as follows: /spl alpha/ = /spl beta/ / /2Q where /spl beta/ is the phase constant. Recently from the following relation of propagation constant at resonance /spl Gamma/(/spl omega//sub 0/) + /spl part//spl Gamma/ / /spl part//spl omega/ /spl Delta//spl omega//spl cong/ i/spl beta/(/spl omega//sub 0/), where /spl Gamma/(/spl omega//sub 0/) = /spl alpha/(/spl omega//sub 0/) + i/spl beta/(/spl omega//sub 0/). Yeh derived a general relation between Q and /spl alpha/, namely, /spl alpha/ = /spl upsi//sub p/ / /spl upsi//sub g/ /spl beta/ / /2Q where /spl upsi//sub p/, and /spl upsi//sub g/ are the phase velocity and group velocity of the wave respectively. This general relation can be derived very simply from the generally accepted definition of /spl alpha/ and Q.     

Asymptotic distribution of the number of isolated nodes in wireless ad hoc networks with Bernoulli nodes

Nodes in wireless ad hoc networks may become inactive or unavailable due to, for example, internal breakdown or being in the sleeping state. The inactive nodes cannot take part in routing/relaying, and thus may affect the connectivity. A wireless ad hoc network containing inactive nodes is then said to be connected, if each inactive node is adjacent to at least one active node and all active nodes form a connected network. This paper is the first installment of our probabilistic study of the connectivity of wireless ad hoc networks containing inactive nodes. We assume that the wireless ad hoc network consists of n nodes which are distributed independently and uniformly in a unit-area disk, and are active (or available) independently with probability p for some constant 0

相似文献   

An analytical model on network blocking probability

We present a new analytical model on the blocking probability of the three-stage Clos (1953) network. Due to the effect of approximations, a common problem with previously proposed analytical models is that they may not be very accurate in some cases. In particular, the blocking probability in these models contradicts the well-known deterministic nonblocking condition for the Clos network. The most notable feature of the newly proposed model is that it can more accurately describe the blocking behavior of the network and is consistent with the deterministic nonblocking condition     

Performance analysis of k-fold multicast networks

Multicast involves transmitting information from a single source to multiple destinations, and is an important operation in high-performance networks. A k-fold multicast network was recently proposed as a cost-effective solution to providing better quality-of-service functions in supporting real-world multicast applications. To give a quantitative basis for network designers to determine the suitable value of system parameter k under different traffic loads, in this paper, we propose an analytical model for the performance of k-fold multicast networks under Poisson traffic. We first give the stationary distribution of network states, and then derive the throughput and blocking probability of the network. We also conduct extensive simulations to validate the analytical model, and the results show that the analytical model is very accurate under the assumptions made. The analytical and simulation results reveal that by increasing the fold of the network, network throughput increases very fast when the fanouts of multicast connections are relatively small, compared with the network size.     

Design of wide-sense nonblocking multicast ATM switches

Nonblocking multicast asynchronous transfer mode (ATM) switches can simplify the call admission control process and increase the external links' utilization. We derive the wide-sense nonblocking condition for multicast ATM switches based on a general Clos network. We also propose a routing algorithm to achieve wide-sense nonblocking. It is illustrated by an example that the number of required middle stages in our switch is significantly less than that of strictly nonblocking multicast switches     

Capacity of ad hoc wireless networks with infrastructure support

We determine the asymptotic scaling for the per user throughput in a large hybrid ad hoc network, i.e., a network with both ad hoc nodes, which communicate with each other via shared wireless links of capacity W bits/s, and infrastructure nodes which in addition are interconnected with each other via high capacity links. Specifically, we consider a network model where ad hoc nodes are randomly spatially distributed and choose to communicate with a random destination. We identify three scaling regimes, depending on the growth of the number of infrastructure nodes, m relative to the number of ad hoc nodes n, and show the asymptotic scaling for the per user throughput as n becomes large. We show that when m /spl lsim/ /spl radic/n/logn the per user throughput is of order W//spl radic/n log n and could be realized by allowing only ad hoc communications, i.e., not deploying the infrastructure nodes at all. Whenever /spl radic/n/log n /spl lsim/ m /spl lsim/ n/log n, the order for the per user throughput is Wm/n and, thus, the total additional bandwidth provided by m infrastructure nodes is effectively shared among ad hoc nodes. Finally, whenever m /spl gsim/ n/log n, the order of the per user throughput is only W/log n, suggesting that further investments in infrastructure nodes will not lead to improvement in throughput. The results are shown through an upper bound which is independent of the routing strategy, and by constructing scenarios showing that the upper bound is asymptotically tight.     

More results on the weight enumerator of product codes

We consider the product code C/sub p/ of q-ary linear codes with minimum distances d/sub c/ and d/sub r/. The words in C/sub p/ of weight less than d/sub r/d/sub c/+max(d/sub r//spl lceil/(d/sub c//g)/spl rceil/,d/sub c//spl lceil/(d/sub r//q)/spl rceil/) are characterized, and their number is expressed in the number of low-weight words of the constituent codes. For binary product codes, we give an upper bound on the number of words in C/sub p/ of weightless than min(d/sub r/(d/sub c/+/spl lceil/(d/sub c//2)/spl rceil/+1)), d/sub c/(d/sub r/+/spl lceil/(d/sub r//2)/spl rceil/+1) that is met with equality if C/sub c/ and C/sub r/ are (extended) perfect codes.     

Order statistics from a logistic distribution and applications to survival and reliability analysis

Joint moments involving arbitrary powers of order statistics are the main concern. Consider order statistics u/sub 1/ /spl les/ u/sub 2/ /spl les/ /spl middot//spl middot//spl middot/ /spl les/ u/sub k/ coming from a simple random sample of size n from a real continuous population where u/sub 1/ = x/sub r(1):n/ is order-statistic #r/sub 1/, u/sub 2/ = x/sub r(1)+r(2):n/ is order statistic #(r/sub 1/ + r/sub 2/), et al., and u/sub k/ = x/sub r(1)+/spl middot//spl middot//spl middot/+r(k):n/ is order statistic #(r/sub 1/ +/spl middot//spl middot//spl middot/+ r/sub k/). Product moments are examined of the type E[u/sub 1//sup /spl alpha/(1)/ /spl middot/ u/sub 2//sup /spl alpha/(2)//sub /spl middot/ /spl middot//spl middot//spl middot//spl middot//u/sub k//sup /spl alpha/(k)/] where /spl alpha//sub 1/, ..., /spl alpha//sub k/ are arbitrary quantities that might be complex numbers, and E[/spl middot/] denotes the s-expected value. Some explicit evaluations are considered for a logistic population. Detailed evaluations of all integer moments of u/sub 1/ and recurrence relations, recurring only on the order of the moments, are given. Connections to survival functions in survival analysis, hazard functions in reliability situations, real type-1, type-2 /spl beta/ and Dirichlet distributions are also examined. Arbitrary product moments for the survival functions are evaluated. Very general results are obtained which can be used in many problems in various areas.