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A message is blocked at a given channel when all the adaptive virtual channels of the remaining dimensions to be visited and also the deterministic virtual channels of the lowest dimension still to be visited are busy [11]. The probability of blocking depends on the number of output channels and, thus, on the virtual channels that a message can use to advance toward its destination. When a message has entirely crossed 7% dimensions, it can select any available virtual channel of the lowest dimension and any available adaptive virtual channel of the remaining (n — w—1) dimensions to make its next hop. If Busy, denotes the probability that all adaptive virtual channels of a physical channel are busy and Busygq is the probability that all adaptive and deterministic virtual channels are busy, the probability of blocking, Pb,.;, can therefore be written as More generally, the probability that a message has entirely crossed y dimensions can be written as

Figure 1 A message is blocked at a given channel when all the adaptive virtual channels of the remaining dimensions to be visited and also the deterministic virtual channels of the lowest dimension still to be visited are busy [11]. The probability of blocking depends on the number of output channels and, thus, on the virtual channels that a message can use to advance toward its destination. When a message has entirely crossed 7% dimensions, it can select any available virtual channel of the lowest dimension and any available adaptive virtual channel of the remaining (n — w—1) dimensions to make its next hop. If Busy, denotes the probability that all adaptive virtual channels of a physical channel are busy and Busygq is the probability that all adaptive and deterministic virtual channels are busy, the probability of blocking, Pb,.;, can therefore be written as More generally, the probability that a message has entirely crossed y dimensions can be written as

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probabilities Busy, and Busy,q are given in terms of P,.;) as (see [23] for more details):
probabilities Busy, and Busy,q are given in terms of P,.;) as (see [23] for more details):
Fig. 1. Transition diagram to compute virtual channel occupancy probabilities.
Fig. 1. Transition diagram to compute virtual channel occupancy probabilities.
The steady-state probability vector, P = (P(,j)),0<7< V and 1 <i <7, of the bivariate Markov chain satisfies the following equations:
The steady-state probability vector, P = (P(,j)),0<7< V and 1 <i <7, of the bivariate Markov chain satisfies the following equations:
4 VALIDATION OF THE MODEL Since only one deterministic virtual channel per physical channel is required to ensure deadlock-freedom in the hypercube [11], the probability that all adaptive virtual channels of a physical channel are busy, Busy,, and the probability that all adaptive and deterministic virtual channels are busy, Busy,g, are expressed as [19] A discrete-event simulator, operating at the flit level, has been developed in order to validate the above analytical model. Each simulation experiment was run until the network converged to its steady state. It has been noted that, in the presence of self-similar traffic, the convergence of simulations to a steady state is often very slow; this phenomenon has also been reported in [24]. The network cycle time in the simulator is defined as the transmission time of a single flit to cross from one node to the next. The message latency is defined as the mean amount of time from the generation of a message until the last data flit reaches the processing element at the destination node.
4 VALIDATION OF THE MODEL Since only one deterministic virtual channel per physical channel is required to ensure deadlock-freedom in the hypercube [11], the probability that all adaptive virtual channels of a physical channel are busy, Busy,, and the probability that all adaptive and deterministic virtual channels are busy, Busy,g, are expressed as [19] A discrete-event simulator, operating at the flit level, has been developed in order to validate the above analytical model. Each simulation experiment was run until the network converged to its steady state. It has been noted that, in the presence of self-similar traffic, the convergence of simulations to a steady state is often very slow; this phenomenon has also been reported in [24]. The network cycle time in the simulator is defined as the transmission time of a single flit to cross from one node to the next. The message latency is defined as the mean amount of time from the generation of a message until the last data flit reaches the processing element at the destination node.
Fig. 3. Latency predicted by the model and simulation: (a) 16-ary 2-cube, m = 32,64 flits, V = 3, H =0.88, p=0.6 and (b) 16-ary 2-cube, m = 32,48 flits, V =5, H = 0.56, p= 0.8. Fig. 2. Latency predicted by the model and simulation: (a) 8-ary 2-cube, m = 32,64 flits, V =3, H =0.75, p =0.62 and (b) 8-ary 2-cube, m = 32.48 flits, V = 6, H = 0.58, 9p = 0.7.
Fig. 3. Latency predicted by the model and simulation: (a) 16-ary 2-cube, m = 32,64 flits, V = 3, H =0.88, p=0.6 and (b) 16-ary 2-cube, m = 32,48 flits, V =5, H = 0.56, p= 0.8. Fig. 2. Latency predicted by the model and simulation: (a) 8-ary 2-cube, m = 32,64 flits, V =3, H =0.75, p =0.62 and (b) 8-ary 2-cube, m = 32.48 flits, V = 6, H = 0.58, 9p = 0.7.
Fig. 4. Latency predicted by the model and simulation: (a) 8-ary 3-cube, m = 32,64 flits, V = 3, H = 0.86, p = 0.67 and (b) 8-ary 3-cube m = 32,48 flits, V = 4, H = 0.61, p = 0.75.
Fig. 4. Latency predicted by the model and simulation: (a) 8-ary 3-cube, m = 32,64 flits, V = 3, H = 0.86, p = 0.67 and (b) 8-ary 3-cube m = 32,48 flits, V = 4, H = 0.61, p = 0.75.
Fig. 5. Latency predicted by the model and simulation (a) 2-ary 10-cube, m = 32,64 flits, V = 2, H =0.8, p=0.65 and (b) 2-ary 6-cube, m = 32,48 flits, V = 3, H = 0.75, p = 0.45.
Fig. 5. Latency predicted by the model and simulation (a) 2-ary 10-cube, m = 32,64 flits, V = 2, H =0.8, p=0.65 and (b) 2-ary 6-cube, m = 32,48 flits, V = 3, H = 0.75, p = 0.45.
Fig. 6. Comparison of network performance under self-similar traffic and under Poisson traffic, m = 32 flits: (a) 8-ary 2-cube, V = 3, H = 0.83, p = 0.82, (b) 8-ary 3-cube, V = 4, H = 0.88, p = 0.78, (c) 2-ary 8-cube, V = 2, H = 0.9, p = 0.75.
Fig. 6. Comparison of network performance under self-similar traffic and under Poisson traffic, m = 32 flits: (a) 8-ary 2-cube, V = 3, H = 0.83, p = 0.82, (b) 8-ary 3-cube, V = 4, H = 0.88, p = 0.78, (c) 2-ary 8-cube, V = 2, H = 0.9, p = 0.75.
Related topics:
Distributed ComputingComputer NetworksComputer HardwarePerformance ModelAdaptive RoutingContext ModelingSimulation experimentConcurrent ComputingDifference OperatorSimulation Experiments
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