RESULTS

OMC AC EIR

4.5 RESULTS

4.5.1 Vinual Arrival Traffic under negative exponential channel holding times In our cell traffic characterising model, we have shown that the vinual arrival traffic is fundamental to calculating hand-off traffic. Equations (4.19) and (4.20) are of such a nature that it is sufficient to show that the virtual arrival traffic (MoEN; VGEN] is smooth (or peak), to prove that the respective hand-off arrival traffic [M,en ; V,en] is smooth (or peak). Simple rearrangement of equations (4.19) and (4.20) show that the peakedness, where peakedness

=variance/mean, of the hand-off arrival traffic is less than one (or greater than one), if and only if, the peakedness of the virtual arrival traffic is less than one (or greater than one). In this section, we empirically show that the virtual arrival traffic under negative exponential channel holding times has a peakedness

Zo.

where

Zo

= V N"Ec/MN"EG , in the range (0,1).

For this purpose we chose the channel size in cell Ll to be one of the following values [2,5,10,20,50,100]. We nonnalised the service rates J..I. in cell Ll and J..I. in cell Lv to be one call per unit of time. For each possible value of channel size C. we varied the arrival rate "- from 0 to 6C. The peakedness of the virtual arrival traffic was detennined using equations

Figure 4-5: PeakcdDtss of the VlrtUlll Arrival Traffic under neg-expCHf 1.02

0

12

¹

g

0.98

"~ ~ 0.96

0;

€

0.94

";;

"

^0.92

.c

-

....

_0.9

0

~

~

!!

0.88

J

^0.86 _0.84

0.82

0 C 2C 3C 4C SC 6C

Arrival rate in cell

r,

Page 26, Chapter 4

Chapter 4: Proposed eeU Trofflc Characterising Model Perj'onnance Analysis of Cellular Networks

(4.33-4.38) and plotted on figure 4-5 for the various channel sizes. The vertical axis represents the peakedness of the vinual arrival traffic and the horizontal axis represents the arrival rate in cell L, as a multiple of the channel size C. The following conclusions may be inferred from figure 4-5. The virtual arrival traffic is a smooth process for a large range of channel sizes and arrival rates for negative exponential channel holding times. A more rigorous statement would require the analytical proof that the virtual arrival traffic is a smooth process and this fonnal mathematical proof is beyond the scope of this work. The peakedness

Zo

of the virtual arrival traffic, for the different channel sizes examined. is a concave function with the following asymptotic behaviour:

Zo

tends to 1 as A tends to O.

and

Zo

tends to 1 as A tends to ^00. The peakedness Zo of the virtual arrival traffic, for the different channel sizes ex.amined, has a minimum approximately in the region where the offered traffic A=Alj..I.=C. The region AzC is the optimal operating region for a cellular network; it would be a waste of resources to operate a cellular network in the region A«C and it would be bad service to the customers to operate in the region A>>C.

4.5.2 Vinual Arrival Traffic under det-neg and gamma channel holding times In the first subsection, we show that the expressions that we derived for the variance of carried traffic in the infinite queues, DNIDN/oo and GAMlGAMloo, are accurate. In the second subsection. we illustrate the accuracy of the expressions for the variance of the virtual arrival traffic, under det-neg and gamma channel holding time distributions (equations 4.50 & 4.54), for finite arrival rates).. and different channel sizes C in our virtual cell scenario.

4.5.2.1 Variance of carried traffic in the DNIDN/-queue.

We simulated a DNIDN/- queue, having identical det-neg inter-arrival and service distributions. We simulated an infinite server system by having a system with a very large number of servers. The mean offered traffic. as can be expected under identical inter- arrival and service distributions, is 1 Erlang. From our simulation. we determined the variance of carried traffic in the "infinite" server system for different det-neg probability parameters. p. We then analytically evaluated the variance of carried traffic in the infinite server system for different det-neg probability parameters, p, using the linear

Page 27, Chapter 4

Chapter 4: Proposed Cell TraffIC Characterising Model Performance Analysis of Cellular Networks

Figure 4-6: Variance of Traffic

0.9 ^o 0 0 0 0 0 Simulalion

0 .• linear Appro)limation

Quadratic Appn»;imation 0.7

0.5 0.5

8 0.4

<

.~ ~ 0.3

0.2 0.1

approximation VO .. DT_NG"" (l-p ) and ^the quadratic approximation

2

• (I-£' _L). Figure 4-6 illustrates these results. As can be seen, the quadratic 2 2

approximation is superior to the linear approximation when compared to the simulation results. However, the true value of this quadratic approximation is that it enables us to bypass the exact and probably intractable analysis of a DNIDN/oo queue by way of a very simple result that is easy to apply and remarkably accurate.

4.5.2.2 Variance of carried traffic in the GAMlGAMJoo queue.

We also simulated a GAMlGAMloo queue, having identical gamma inter-arrival and service distributions. The mean offered traffic, under identical inter-arrival and service distributions, is 1 Erlang. From our simulation. we detennined the variance of carried traffic in the infinite server system for different gamma shape parameters, c. We then analytically evaluated the variance of carried traffic in the infinite server system for different ganuna shape parameters. c, ^{using our} conjecture

2r(2c T 1) from earlier. Figure 4-7 illustrates these results. As can be 2t: 2t: ne T l)r(eT 1)

Page 28. Chapter 4

Chapter 4: Proposed Cell Traff~ Characterising Model Perfonnance Analysis" oJ Cellular Networks

Figure 4-7: Variance of Traffic

0.9

0.' _o 0 0 0 0 0 Simulation

0.7 ^{- -- Analysis}

gO.6

> ~ 0.5

0.'

0.3

02

0.1';;----~~~~~~_'_;_--~-~---'

10° 10' 102

Gamma Distribution Shape Parameter, c

seen, our conjecture is very accurate when compared to the simulation results. However, due to the difficulties associated with proving the a~ve result., especially for cases where the shape parameter c is non-integral, the proof seems far from feasible and our result remains a conjecture.

4.5.3 Variance of the virtual arrival traffic.

In our cell traffic characterising model, we have shown that the virtual arrival traffic is fundamental 10 calculating hand-off traffic. Provided, we know the mean and variance of the virtual arrival traffic, we are able to determine the mean and variance of any hand-off arrival traffic where hand-off probability QI1 51 (equations 4.19 & 4.20). In this section, we evaluate the accuracy of our expression for the variance of the vinual arrival traffic offered by the first cell to the virtual cell (in figure 4-2b) under det-neg as well as gamma channel holding times. In both cases, we set channel size C to be one of the following values: C E

{3. 3D}. For each channel size C, we varied the offered traffic in the first cell. A = AI,u, from A

=

0 Erlangs up to A = 2.666 x C Erlangs. We analytically detennined the peakedness. Z = Variance / Mean. of the various vinual arrival traffic for the above

Page 29, Chapter 4

Chapter 4: Proposed Cell Troffu: Characterising Model Perjonnance Analysis of Cellular Networks

parameters. For comparison purposes we also simulated the simple virtual cell scenario under the same conditions. To imitate the infinite sized virtual cell of figure 4-2b, we set the channel size in the virtual cell in our simulator to be very large, namely 400 channels.

4.5.3.1 Variance of the virtual arrival traffic under det-neg channel holding time distributions

We set the channel holding time distributions in both cells to be identical det-neg distributions with a nonnalised mean of EfTJ=l/)l=i unit time. We considered four del- neg distributions with det-neg probability parameter pe {O, 113,2/3, I}. We plotted the simulation and analytical results in figures 4-8 & 4-9.

For the two different channel sizes that we considered, the peakedness of the virtual arrival traffic as determined by our quadratic approximation (equation 4.50) ties up extremely well with s.imuIation results when compared to the linear approximation. Similar results were obtained for various other channel sizes that we considered. However, to avoid repetition,

1.2

- _~

(11 0.8

·0

~

"

fS •

~ 0.. 0.4 0.2

Figure 4·8: Peakedness of the Virtual Arrival Traffic for Channel size C '" 3

0 0 0 0 0 0 Simulation

--

Linear approx.

Quadratic approx.

---- _---

"

' " " ,

----

°0~---7---C2~----~3c---~4---=5---CS~--~~7c===~8

Offered Traffic in First Cell

Page 30, Chapter 4

Chapter 4: Proposed CeU TraffIC Characterising Model Performana 'Analysis of CeUukzr Networks

Figure 4·9: Peakedness of the Virtual Arrival Traffic for channel size C ^& 30

1.2

~

, f----<>--....

~ :>

-

B

1ij0.8

~

"

•

_{g: 0.6}

~ c

i

:0.4

0.2

o 0 0 0 0 0 Simulation Linear approx.

Quadratic approx.

p=o

--- --- ---

p

=

^1/3

- -- -- _{--- -}

p=l

Offered Traffic in First Cell

we have not presented those results here. The remarkable accuracy of the quadratic approximation for the variance of the virtual arrival traffic (equa~on 4.50) is a testament to the validity of the asymptotic analysis that we employed, where, we considered the behaviour of !he virtual cell system when !he arrival rate A. increased without bound and then borrowed the structure of the results to apply for finite arrival rates.

4.5.3.2 Variance of the virtual arrival traffic under gamma channel holding time distributions

In this section, we set the channel holding time distributions in both cells to be identical gamma distributions with a nonnalised mean of E[T}=l/p.=l unit time. We considered four gamma distributions with shape parameter c e {I, 3, 20, cc}. We ploned the simulation and analytical results in figures 4-10 & 4-11.

Page 31, Chapter 4

Chapter 4: Proposed CeU Traffu: Characterising Model Performance Analysis of Celluku Networks

Figure 4-10: Peakedness of the Virtual Arrival TraffIC for channel size C., 3

000000 Simulalion

1.2 Analysis

.:g § 0.8

;:

"

~O.6

]l

•

~0.4

0 . 2

o o

ooL---L---2L---3L---4L---5L---~6C----"=7c====<J8

1.2

I - _il

.:g 0.8

;:

"

~ GI 0.6

]

;

a.. 0.4

0.2

Offered Traffic in First Cell

Figure 4-11: Peakedneu of the Virtual Arrival TraffIc for Channel size C '" 30

000000 Simulation Analysis

c=-

00~----~10 C----C 2~0---"30C---40L--=::SO::==::~60d:::~7 ! 0 ====~ 80

Offered Traffic in First Cell

Page 32, Chapter 4

Chapter 4: Proposed Cell TraffIC Characterising Model ·Performance Analysis of Celluwr Networks

For the two different channel sizes that we considered, the peakedness of the vinual arrival traffic as determined by our analysis (equation 4.54) ties up reasonably well with simulation results. As can be seen. agreement between simulation and analysis was strongest for c=l (where the channel holding times are negative exponential) and c=oo (where channel holding times are deterministic). This is to be expected since we have exact results for these shape parameter values (equations 4.33-4.38 and equation 4.39). For other values of c, our expression (equation 4.54) is a reasonable approximation and one that we show in the next section to work well in practice. We obtained similar results for various other channel sizes that we considered.

4.6 Summary

Our proposed perfonnance analysis algorithm has three parts, a cell traffic characterising model, a cell traffic blocking model and a quality of service evaluation model. At the heart of our cell traffic characterising model is a simple two-cell scenario. In this chapter we analysed this two-cell scenario and presented results for the mean and variance of traffic offered by a cell to its neighbour in the two-cell scenario. We considered the following four different channel holding time distributions in our two-cell scenario:

• Negative exponential distribution

• Deterministic distribution

• Det-neg distribution

• Gamma distribution

The results that we presented for the neg-exp and deterministic distributions are exact whereas the results that we presented for the det-neg and gamma distribution are approximate, based on asymptotic analysis. We considered the behaviour of our two-cell!

virtual cell system when the arrival rate). increased without bound and then borrowed the structure of the results to apply for finite arrival rates. We have validated the accuracy of all our derivations using comparison with simulation results.

Page 33. Chapter 4

Chapter 5: Proposed Cell Traffic Blocking Models Performance AtuJ!ysis of Cellular Networks