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Performance modelling of optical packet switched networks with the Engset traffic model

Harald Øverby

Open Access Open Access

Abstract

Stochastic processes have been widely employed in order to assess the network layer performance of Optical Packet Switched (OPS) networks. In this paper we consider how the Engset traffic model may be applied to evaluate the blocking probability in asynchronous bufferless OPS networks. We present two types of the Engset traffic model, i.e. the Engset lost calls cleared traffic model and the Engset overflow traffic model. For both traffic models, the time-, call-, and traffic congestion are derived. A numerical study shows that the observed blocking probability is dependent on the choice of traffic model and performance metric.

©2005 Optical Society of America

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Figures (8)
Fig. 1.
Fig. 1. A generic optical packet switch with F input/output fibres, and W wavelengths per fibre.

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Fig. 2.
Fig. 2. A state diagram of an input wavelength, which changes between two states. In the idle state, no packets are arriving on the input wavelength, while in the busy state, the input wavelength is transmitting a packet to the tagged output port. The holding times are negative exponential distributed.

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Fig. 3.
Fig. 3. State transition diagram of the Engset LCC. The states denote the number of busy wavelengths at a tagged output port. The tagged output port is congested in the red state.

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Fig. 4.
Fig. 4. State transition diagram of the Engset OFL. State (i,j) denotes the number of output wavelengths at the tagged output port currently busy (i), and the number of input wavelengths transmitting a packet that has been dropped (j). The tagged output port is congested in the red states.

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Fig. 5.
Fig. 5. The blocking probability as a function of the normalized system load (A) at a tagged output wavelength. F=4.

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Fig. 6.
Fig. 6. The blocking probability as a function of the normalized system load (A) at a tagged output fibre. F=4, W=16.

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Fig. 7.
Fig. 7. The blocking probability as a function of the number of input/output fibres (F) at a tagged output wavelength. A=0.8.

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Fig. 8.
Fig. 8. The blocking probability as a function of the number of input/output fibres (F) at a tagged output fibre. A=0.8, W=16.

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Tables (1)

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Table 1. Overview of the number of input/output wavelengths in the NWC and FOWC scenario.

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Equations (20)

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Z = σ 2 A T = S α ( 1 − α ) S α
Z = 1 − A T S = 1 − A N S = 1 − A F
∂ Z ∂ F = A F 2 > 0
∂ 2 Z ∂ F 2 = − 2 A F 3 < 0
Q j ( s − j ) λ = Q j + 1 ( j + 1 ) μ ( 0 ≤ j ≤ N − 1 )
Q j = S j β j ∑ k = 0 N S k β k
E L C C ( N , S , β ) = Q N
B L C C ( N , S , β ) = Q N ( S − N ) λ ∑ j = 0 N Q j ( S − j ) λ = E L C C ( N , S − 1 , β )
C L C C ( N , S , β ) = A − Y A = S α − ∑ j = 0 N j Q j S α = F − 1 F E L C C ( N , S , β )
u ( k , L ) = { 0 k = L 1 k ≠ L
Q i , j ( ( i + j ) μ + ( S − i − j ) λ ) =
Q i − 1 , j ( ( S − i − j + 1 ) λ ) · u ( N − i , N ) +
Q i + 1 , j ( i + 1 ) μ · u ( i , N ) +
Q i , j − 1 ( S − N − j + 1 ) λ · ( 1 − u ( i , N ) ) u ( S − N − j , S − N ) +
Q i , j + 1 ( j + 1 ) μ · u ( j , S − N )
( 0 ≤ i ≤ N ) ( 0 ≤ j ≤ S − N )
∑ i = 0 N ∑ j = 0 S − N Q i , j = 1
E O F L ( N , S , β ) = ∑ j = 0 S − N Q N , j
B O F L ( N , S , β ) = ∑ j = 0 S − N Q N , j ( S − N − j ) λ ∑ i = 0 N ∑ j = 0 S − N Q i , j ( S − i − j ) λ
C O F L ( N , S , β ) = A − Y A = S α − ∑ i = 0 N ∑ j = 0 S − N i Q i , j S α
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