Numerical Results

We carried out analytic computation and event-based simulation to validate the analysis and capture the departure process behavior under various parameter settings and traffic arrivals. Analytical and simulation results of the queue length distribution and departure process distributions (inter-departure and burst size distributions) are shown in Figures 14, 15, 16, and 17 respectively. In the system setting, we adopt

ψ

= 25 or 100, and

τ

= 10, 20, 30, or ∞. In the MMBP, we adopt

0.225,

α

= β =0.025;

λ

_H =0.36 and

λ

_L =0.0933 at load 0.6; and

λ

_H =0.48 and

λ

_L =0.1244 at load 0.8. The batch size in any of states H and L was uniformly distributed between 1 and 9 (

b =

_H

b = 5). Accordingly, the burstiness of traffic is B =

_L 3 under both loads.

First, all analytical results are in profound agreement with simulation results. As shown in Figures 14(a) and 14(b), the queue length observed at burst departure time is small under lower load and lower

τ

value. It is caused by that most of the bursts are generated while the BATr is expired. If the

τ

value is large enough, the system accumulates

ψ

packets to assemble a burst. Since the large burst takes more transmission time than small burst, the number of packets arriving during burst transmission in large

τ

value is more than that in small

τ

value. The distribution of queue length in large

τ

value is to center on large queue size. As shown in Figure 14(c), the queue size distribution under large

ψ

value is expands to a greater scope.

In addition, we observe that the inter-departure time distribution is sensitive to

ψ

and

τ

. Under a medium load ( L =0.6) or high load ( L =0.8) condition, we observe the inter-departure time of zero (burst size=

ψ

=25 or 100) occurs with the larger probability under all

τ

values. It can be shown that during the burst transmission, there are enough packets (>

ψ

) accumulate in the queue under a high load. The other larger probability for different

τ

settings occurs at the inter-departure time being equal to the corresponding

τ

value. It is reasonable for that while system finish a burst transmission and queue is not empty, next burst wait at most

τ

slots. The results are shown by the spikes in Figures 15(a), 16(a) and 17(a).

The burst size distribution is also sensitive to

ψ

and

τ

. As shown in Figures 15(b), 16(b) and 17(b), if the

τ

value is small (=10), the system does not accumulate enough packets before BATr expired. The probability of burst size less than

ψ

is large under lower load (see Figure 15(b)) or under large

ψ

value. If the

τ

value is large enough (approached to limit), burst is always generated under the

ψ

^th packet arrived.

The probability of burst size equal to

ψ

is one. Deciding an appropriate

τ

value can make the burst almost complete and the delay could be controlled. For example, if we want to get 95% complete burst under

ψ

=25 and load=0.8, we must set

τ

=30. In the last, it has similar result with Figure 14 that there is a turning point at burst size 9 duo to the maximum batch size is 9

0 5 10 15 20 25 30 35 40 45 50 55 60 0.00

0.01 0.02 0.03 0.04 0.05

0.06 τ = 10 (P₀= 0.045)

ψ = 25

τ = (P₀= 0.001) τ = 30 (P₀= 0.004) τ = 20 (P₀= 0.010)

Load = 0.8

Probability density

Queue length Analysis Simulation

∞

0 5 10 15 20 25 30 35 40 45 50 55 60 0.00

0.01 0.02 0.03 0.04 0.05

0.06 τ = 10 (P₀= 0.204)

ψ = 25

τ = (P₀= 0.007) τ = 30 (P₀= 0.037) τ = 20 (P₀= 0.082)

Load = 0.6

Probability density

Queue length Analysis Simulation

∞

(a)

ψ

=25 under medium load (0.6)

(b)

ψ

=25 under high load (0.8)

0.01 0.02 0.03 0.04 0.05

0.06 τ = 10 (P

0= 0.033) ψ = 100

τ = (P

0= 4.2x10^-9) τ = 30 (P₀= 1.6x10^-4) τ = 20 (P

0= 0.002) Load = 0.8

Probability density

Analysis Simulation

∞

1 4 7 10 13 16 19 22 25 0.00

0.01 0.02 0.03 0.04 0.05 0.06 0.07

0.08 τ = 10 (P

₂₅

= 0.22)

ψ = 25 Load = 0.6 τ = (P

25

= 1.00) τ = 30 (P

25

= 0.76) τ = 20 (P

25

= 0.54)

Probabil it y dens it y

Burst size

Analysis Simulation

0 5 10 15 20 25 30 35 40

10

^-5

10

^-4

10

^-3

10

^-2

10

^-1

10

⁰

τ = 10

τ = τ = 30 τ = 20

Load = 0.6

Probabi lit y densit y

Slot

ψ = 25

Analysis Simulation

t

(a) Inter-departure time ( ) distribution

∞

(b) Burst size (s) distribution

∞

Figure 15. Departure process distributions (

ψ

=25 under medium load).

1 4 7 10 13 16 19 22 25 0.00

0.01 0.02 0.03 0.04 0.05

τ = 10 (P

₂₅

= 0.61)

ψ = 25 Load = 0.8 τ = (P

25

= 1.00) τ = 30 (P

₂₅

= 0.95) τ = 20 (P

25

= 0.86)

Probabil it y dens it y

Burst size

Analysis Simulation

0 5 10 15 20 25 30 35 40

10

^-5

10

^-4

10

^-3

10

^-2

10

^-1

10

⁰

τ = 10 (P

0

= 0.401)

τ = (P

0

= 0.571) τ = 30 (P

₀

= 0.553) τ = 20 (P

₀

= 0.518)

Load = 0.8

Probabi lit y densit y

Slot

ψ = 25 Analysis Simulation

(a) Inter-departure time (t) distribution

∞

∞

0 10 20 30 40 50 60 70 80 90 100 0.00

0.01 0.02 0.03 0.04

0.05 τ = 10 (P

100

= 0.022)

ψ = 100 Load = 0.8 τ = (P

₁₀₀

= 1.000)

τ = 30 (P

₁₀₀

= 0.440) τ = 20 (P

₁₀₀

= 0.186)

Probability density

Burst size

Analysis Simulation

0 5 10 15 20 25 30 35 40

10

^-5

10

^-4

10

^-3

10

^-2

10

^-1

10

⁰

τ = 10 (P

₀

= 0.012)

τ = (P

₀

= 0.263) τ = 30 (P

₀

= 0.148) τ = 20 (P

0

= 0.075)

Load = 0.8

Probabili ty densi ty

Slot

ψ = 100 Analysis Simulation

(a) Inter-departure time (t) distribution

∞

(b) Burst size (s) distribution

∞

Figure 17. Departure process distributions (

ψ

=100 under high load).

To examine the effectiveness of shaping, we further compute the Coefficient of Variation (CoV) for the inter-departure time and burst size, under three

ψ

values (

ψ = 1, 10, and 100), three τ

values (

τ = 20, 40, and 80) and various MMBP

arrivals (B = 1, 3, and 5;

b =

_H

b = 5, 7, and 9) under

_L

α

=0.225, β =0.025. The CoV is a measure of dispersion of a probability distribution. It is defined as

Var[a]

Standard deviation of a CoV(a)

Mean of a E[a]

=  = 

   ,

where is a random variable. Distributions with CoV < 1 (such as an Erlang distribution) are considered low-variance, while those with CoV > 1 (such as a two-state MMBP distribution) are considered high-variance. Notice that the setting of

ψ = 1 corresponds to a FIFO system with no shaping. Numerical results are plotted in

Figures 18 and 19.

a

As shown in Figure 18, as expected, the CoV of the inter-departure time increases with the offered load. Crucially, under any MMBP arrival, we discover that CoV of inter-departure time is very large under

ψ

= 1. It means that departure process is of high variance. Then the CoV declines significantly with larger

ψ

values, yielding substantial reduction in burst loss probability. This fact will be again revealed in the network-wide simulation results presented in the Chapter 5.

Moreover, we observe from Figures 18 and 19 that the burstiness and batch

Figure 19, the CoV of the burst size declines with larger

τ

values under any MMBP arrival. Notice that greater

τ

values imply larger burst sizes, namely better shaping effect.

We then investigate the impact of (

ψ

,

τ

) setting on mean burst size. In this simulation, we adopted the batch size being uniformly distributed between 1 and 9 (i.e.,

b =

_H

b = 5) for MMBP arrivals. By the result of Figures 18 and 19, we have

_L learned that the larger

ψ

and

τ values results in better shaping. To satisfy a given

mean burst size, the results in Figure 20 can serve as a guideline for the selection of appropriate

τ

values under different traffic loads. For example, if

ψ = 200, to achieve

a mean burst size of 100 under load 0.6, the applicable values of

τ

are 40 and above.

The results of the mean burst size associated with a set of (

ψ

,

τ

) pairs are indicated in Figure 21. Under medium load (see Figure 21(a)),

τ

value must be set large enough to achieve large burst size. Under high load,

ψ has acted the main role to

decide the mean burst size. For example (see Figure 21(b)), if

τ > 20, mean burst size

is almost equal to the

ψ value.

0.80 0.82 0.84 0.86 0.88 0.90 0

2 4 6 8 10 12 14 16

b

H=b

L=5 B=1

B=3

ψ = 100 B=5

ψ = 10

C o ef fi cient of V ariation

^{τ = 80}

Load

ψ = 1

(b) Under different batch sizes and

ψ

values

0.80 0.82 0.84 0.86 0.88 0.90

0 2 4 6 8 10 12 14 16

B = 5

ψ = 100 ψ = 10

C o ef fi ci en t of V aria tio n

τ = 80

Load

ψ = 1

b_H=b_L=5 b_H=b_L=7 b_H=b_L=9 (a) Under different B and

ψ

values

0.80 0.82 0.84 0.86 0.88 0.90 0.0

0.1 0.2 0.3 0.4 0.5

b

_H=b_L=5

B=1 B=3 B=5

τ = 80

τ = 40 ψ = 100

Coef fi ci ent of Vari at io n

τ = 20

Load

(a) Under different B and

τ

values

0.80 0.82 0.84 0.86 0.88 0.90

0.0 0.1 0.2 0.3 0.4 0.5

ψ = 100

τ = 80 τ = 40

B = 5

C o ef fi cient of V ariation

^{τ = 20}

Load

bH=b

L=5 b_H=b_L=7 b_H=b_L=9

(b) Under different batch sizes and

τ

values Figure 19. CoV of the burst size.

0.3 0.4 0.5 0.6 0.7 0.8 0.9 0

50 100 150

200

τ = 50

τ = 40 τ = 30 τ = 20 τ = 10

ψ = 100

Mean burst size

Load 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0

40 80 120 160 200

ψ = 200

Mean burst size

Load

τ = 50 τ = 40 τ = 30 τ = 20 τ = 10

(a) Burst size under

ψ

= 200

(b) Burst size under

ψ

= 100

50 40 30

20

10 60 80 100 120 140 160 20

40 60 80 100 120 140 160

4

Load=0.5

Mean burst size (MBS)

τ ψ

50 40 30

20 10

60 80 100 120 140 160 20

40 60 80 100 120 140 160

4

Load=0.8

M e an Burst Size (MBS)

τ ψ

(a) Under medium load

(b) Under high load

Figure 21. Mean burst size associated with

ψ

and

τ

.

Chapter 5. Loss QoS Provision and Performance

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