Call Completion Probability with Weibull Distributed Call ... - IEEE Xplore

Call Completion Probability with Weibull Distributed Call Holding Time and Cell Dwell Time Suwat Pattaramalai and Valentine A. Aalo

George P. Efthymoglou

Department of Electrical Engineering Florida Atlantic University Boca Raton, FL 33431, USA {spattara, aalo}@fau.edu

Department of Digital Systems University of Piraeus Piraeus 18534, Greece [email protected]

Abstract—In this paper, we derive a simple closed-form expression for the call completion probability in a wireless cellular network under Weibull distributed cell dwell time and call holding time. The derived expression is given in terms of the Meijer-G function which can be easily evaluated by commonly available computer software such as Mathematica. Numerical results are presented to show that the call completion probability is very sensitive to changes in the shape parameter of the Weibull distribution when its value is less than unity (heavy tail region). Keywords- call completion probability, Weibull distribution

I. INTRODUCTION In wireless broadband networks such as GPRS (general packet radio service), EDGE (enhanced data rates for GSM evolution), and 3G/UMTS (3rd generation universal mobile telecommunication system), multimedia data services are integrated with voice calls. The mobile users have access to services such as instant messaging, music and video download, and web services on their smart phones in the same way as using desktop PCs connected to fixed networks [1]. Moreover, the wireless network providers charge customers flat rate for unlimited data transferring to promote usage of these services. As a result, the call holding time for dial-up internet calls and World Wide Web calls follow heavy-tail distributions [2]. The call holding time is the time that a user stays connected to the network and has been studied for a variety of call types [2]-[4]. In [3], an overview of Enhanced UMTS deployment scenarios and supported services are presented. It shows that the Weibull distribution provides a good fit to multimedia-services type of calls. The Weibull distribution was also used to model call holding time in performance analysis of hierarchical cellular networks in [4]. Cell dwell time is another random parameter encountered in the performance analysis of wireless networks that may also be modeled with the Weibull distribution [5]-[6]. The performance analysis of homogeneous wireless networks under generalized cell dwell time and call holding time has been studied extensively by Fang et al. [7], [8]. Their approach, which is based on the residue theorem, requires that the Laplace transform of the distribution of the call holding time be a rational function. However, the Laplace transform of the Weibull distribution is not a rational function [2]-[4]. In fact, even when the call holding time has a gamma distribution with non-integer shape parameter, its Laplace transform does

not have a rational form and the approach based on the residue theorem is not applicable. In this paper, we use the notion of geometric random sum [9] to evaluate performance metrics under Weibull distributed call holding time and cell dwell time. Specifically, based on the Renyi limit theorem [9], we approximate the tail of the distribution of the sum of independent cell dwell times by the exponential function. The approximation is very accurate when the probability of handoff failure is very small. Consequently, the call completion probability can be easily derived and expressed in terms of the Meijer-G function which can be evaluated by such commonly used computer software as Mathematica and Maple. II.

WEIBULL DISTRIBUTION AND LIMITING CASES

A. Weibull Distribution Let Tc be a Weibull distributed random variable, with probability density function (pdf) given as [10]

v v −1 − (t / b )v t e t ≥ 0, , (1) bv where b is a positive scale parameter and v is the shape parameter. The moment generating function (MGF) of Tc is defined as [10] fTc ( t ) =

v ∞ − xt v −1 − (t / b )v e t e dt . (2) bv ∫ 0 We note that the exponential function in (2) can be expressed in terms of the inverse Mellin transform of the gamma function as [11]

MT ( − x ) = c

e

− (t b)

v

=

1 δ + i∞ t Γ(s)  ∫ 2π i δ − i∞ b

− vs

δ > 0,

ds ,

(3)

where Γ ( ⋅) is the gamma function [12] and i = −1 . Using (3) in (2), we have [12] MT ( − x ) = c

v

( xb )

v

1 2π i

δ +i∞

∫δ

− i∞

Γ ( s ) Γ ( v − vs )( xb ) ds, δ > 0. (4) vs

2634 1930-529X/07/$25.00 © 2007 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2007 proceedings.

Assuming that the shape parameter v is a rational number, i.e., v = j k ( j and k are integers) and replacing s with ks, the MGF of Tc becomes

MT ( − x ) = c

j

c

k

1 2

j 1 + k 2

( xb )

j k 1− − 2 2

j k

( 2π )

1 2π i

1 n  k −1  m  ×∫ Γ  + − s ∏ Γ + s  ∏ δ −i∞  n=0 k j  m=0  k δ + i∞

4.5

( j ) 1 δ + i∞ js Γ ( ks ) Γ  j (1 k − s )  ( xb ) ds, j k ∫ δ −i ∞ i π 2 ( xb )

δ > 0 . (5) Using the multiplication theorem for the gamma function [12], we have j 1 j −1 1− j − js − 1 n Γ  j (1 k − s )  = ( 2π ) 2 j k 2 ∏ Γ  − s +  (6) j n=0 k and 1 1− k ks − k −1 m  Γ [ ks ] = ( 2π ) 2 k 2 ∏ Γ  s +  . (7) k  m=0 Substituting (6) and (7) in (5), we have

MT ( − x ) =

5

j −1

 j  =   xb 

j k

j ( 2π

)

s

 k  xb  j   k    ds   j  

2− j −k

Mean Variance

4 3.5 3 2.5 2 1.5 1 0.5 0 0

0.5

1

1.5

2 2.5 3 Shape parameter

B. Limiting Cases of the Shape Parameter The shape parameter v can take values between 0 and ∞ . In the case when v = 1, the Weibull distribution becomes the exponential distribution, whereas when v = 2, it reduces to the Rayleigh distribution. For v < 1 , the Weibull distribution belongs to a sub-exponential class of distributions which

4.5

5

exhibits a heavy tail property [2]. Next we investigate the Weibull distribution in the limiting cases when v tends to zero and infinity. From [10], the mean and variance of Tc are given by and Var [Tc ] = E [Tc ] = bΓ (1 + 1 v )

{

 k −1  1 2   xb  j 1,1 − k ,1 − k ,...,1 − k  , × Gkj,, kj k k   1 j − 1   j  1 1 1 1 2 + + + , , ,...,  k k j k j k j   (8) where we have used the Meijer-G function defined in [12, eq. (9.301)]. When v is an integer (i.e., k = 1 and j = v ), (8) reduces to  1  v v 1− v  v   xb   MTc ( − x ) =   v ( 2π ) G1,v v,1   1 v − 1 ,  1,1 + ,...,1 + xb v      v v  (9) which agrees with [13]. Note that for the special case when v = 1 (exponential distribution), the MGF reduces further to [12] 1 1,1  1 1 MTc ( − x ) = G1,1 , (10)  xb  = 1 xb + xb 1   as expected [10].

4

Figure 1. Mean and Variance of Weibull distribution versus shape parameter v with scale parameter b = 1.

b 2 Γ (1 + 2 v ) − Γ (1 + 1 v ) 

k

3.5

2

} , respectively.

Fig. 1 plots the

mean and variance versus the shape parameter v , when b = 1. Noting that lim Γ (1 + κ v ) = 1, for κ < ∞ , we observe that the v →∞

mean tends to the scale parameter b while the variance tends to zero, as the parameter v tends to infinity. This implies that when the shape parameter tends to infinity, Tc tends to a constant, i.e., fTc ( t ) = δ ( t − b ) . Consequently, the MGF of Tc becomes ∞

MT ( − x ) = ∫ e − xt δ ( t − b ) dt = e −bx . c

0

(11)

On the other hand, when the shape parameter tends to zero, all the gamma functions that appear in the expressions for the mean and variance tend to infinity. Therefore, both the mean and variance of Tc tend to infinity. Then the tail of the Weibull distribution becomes heavy. This implies that the Weibull distribution tends to a uniform distribution in the interval ( 0, b v ) , i.e., fTc (t ) = v b , 0 ≤ t ≤ b v and the MGF becomes

v b v − xt v e dt = (1 − e −bx v ) . (12) ∫ 0 b bx Note that as the value of the shape parameter v becomes smaller, the width of the uniform distribution gets wider.

MT ( − x ) = c

III. CALL COMPLETION PROBABILITY In this section, the Weibull distribution which was analyzed in the previous section is used to model the call holding time and the cell dwell time.


A. System Model Fig. 2 illustrates the timing diagram for the call holding time and the cell dwell time of a completed call over the wireless network. The call holding time Tc is the duration of time from the time a call is initiated and successfully connected to a cellular base station to the time when the call is ended by a mobile user. The duration of time a mobile unit resides in the i-th cell is defined as the i-th cell dwell time and is denoted by Ti ( i = 1, 2, 3,..., K − 1, K ,...) . We assume that the cell dwell times are independent and identically distributed (i.i.d.) random variables with common pdf f T ( t ) and the two moments E [T ] and E T 2  are finite. We also assume that the call initiates at an arbitrary time in the first cell and that the actual duration of the call in the first cell, R1 , is the time between the instant that a new call is initiated and the instant that the mobile user moves out of the first cell, given that the call has not ended.

which is a compound random sum. In a cellular network, the probability that the random variable K equals n is the probability that the call has completed ( n − 1) successful handoffs and the n-th handoff fails. Let p f be the probability that a handoff attempt fails and (1 − p f

)

be the probability

that a handoff attempt succeeds. Then, the random variable K follows a geometric distribution with probability mass function (pmf) given by n −1 (15) Pr ( K = n ) = p f (1 − p f ) , n = 1, 2, 3,...

and corresponding mean E [ K ] = 1 p f [10]. Substituting (15) in (14) and averaging over the distribution of the call holding time Tc , the call completion probability given in (13) may be evaluated as [4], [5] ∞  pc = (1 − po ) 1 − ∑ p f (1 − p f  n =1

)

n −1

B. Call Completion Probability We assume further that the call ends in the (random) K -th cell (as in Fig. 2). This implies that the call holding time is less than the sum of the first K cell dwell times. Let po denote the new call blocking probability, which is the probability that a new call is blocked before it can be connected to the wireless network. We also define the call completion probability pc as the probability that a new call that is not blocked is successfully connected to the cellular network and is not dropped until the call is ended only by the mobile user in any cell. It may be expressed as

×∫ Pr ( R1 + T2 + T3 + ... + Tn ≤ x Tc = x ) fTc ( x ) dx  , (16)  0 where fTc ( x ) is the pdf of the call holding time Tc .

pc = (1 − po ) ⋅ Pr (Tc < S K ) = (1 − po ) ⋅ 1 − Pr ( SK ≤ Tc ) ,

typically less than 5% [7], [8]. Furthermore, if the random variable K is independent of Ti , the conditional probability in (14) may be obtained as [9], [14], [15] lim Pr ( SK ≤ x Tc = x ) =

(13) where S K = R1 + T2 + T3 + ... + TK is a random sum of independent non-negative random variables. For a given call holding time Tc = x , the distribution of S K is given by

∞

The conditional distribution in (14) is called the geometric compound random sum when K follows the geometric distribution in (15) [9], [14], [15]. Based on the Renyi limit theorem [9], [14], [15], when p f is sufficiently small (rare events), the tail of the geometric compound sum converges in distribution to the exponential distribution. The statement and proof of Renyi theorem are given in the Appendix of [14]. In a practical cellular network, p f is usually very small,

p f →0

lim

Pr ( SK ≤ x Tc = x )

p f →0

∞

∑ p (1 − p ) f

n =1

∞

= ∑ Pr ( K = n ) ⋅ Pr ( R1 + T2 + T3 + ... + Tn ≤ x Tc = x ), n =1

(14)

where E [ S K ]

f

n −1

Pr ( R1 + T2 + T3 + ... + Tn ≤ x Tc = x )

 −x  , (17) 1 − exp   E [ S ]  K   is the mean of S K . Averaging over the

distribution of Tc , it follows that ∞ ∞ Pr ( S K ≤ Tc ) = ∫  lim ∑ p f (1 − p f 0  p f → 0 n =1

)

n −1

× Pr ( R1 + T2 + T3 + ... + Tn ≤ x Tc = x )  fTc ( x ) dx

∫

∞ 0

  − x  1 − exp    fTc ( x ) dx   E [ S K ]  

 −1  = 1 − MTc  .  E [ S ]  K   Figure 2. The time diagram for the call holding time and the cell dwell time.

(18)


Note that when the call originates at the beginning of the first cell (i.e., R1 = T1 ) [9], we have (19) E [ S K ] = E [ K ] ⋅ E [T ] = E [T ] p f . However, as is usually the case in practice, the call may not originate at the beginning of the first cell. Accordingly, the mean of S K is then given by E [ S K ] = E[ R1 ] + ( E [ K ] − 1) ⋅ E [T ] =

p f E [ R1 ] + (1 − p f ) E [T ] pf

,

(20) where E [ R1 ] is the mean of R1 . Substituting (20) in (18) and using the result in (13), the call completion probability can be accurately approximated as   − pf . (21) pc (1 − po ) MTc   p f E [ R1 ] + (1 − p f ) E [T ]   

C. Weibull Distributed Call Holding Time and Cell Dwell Time The call holding time is assumed to follow the Weibull distribution in (1), with the scale parameter b and shape parameter v. The dwell times Ti ’s ( i = 1, 2,...) are i.i.d. Weibull distributed random variables (with parameters β and γ ) with the pdf given by

γ γ −1 −(t / β )γ t e , βγ

t ≥ 0,

(22)

and corresponding m-th moment E T m  = β m Γ (1 + m / γ ) [10]. Using the MGF of the Weibull distribution (8) in (21), the call completion probability is j 1 +

pc

j k 1− − 2 2

(1 − po ) j k 2 ( 2π ) 1

k2 ×Gkj,, kj

 p f E [ R1 ] + (1 − p f ) E [T ]    bp f  

   bp f k k    j p f E [ R1 ] + (1 − p f ) E [T ]   

{

}

j k

of the Weibull distribution tends to infinity, we may use (11) in (21) to obtain the call completion probability as

{

pc (1 − po ) exp −b p f

( p E [ R ] + (1 − p ) E [T ])} . f

1

f

(24) Similarly, when the shape parameter v of the Weibull distribution tends to zero, we may use (12) in (21) to obtain

(

{

pc (1 − po ) 1 − exp − bp f

{

po = p f = 0.02 in all the numerical results. In Fig. 3, the call completion probability is plotted for different values of the shape parameter v of the call holding time distribution while shape parameter γ = 1 for the cell dwell time distribution. The results for the limiting cases of the shape parameter v are also plotted as the upper and lower bounds in Fig. 3. The roles of v and γ are reversed in Fig. 4. Both figures show similar trend for the variation of the call completion probability with the call-to-mobility factor of the user. Notice that, when shape parameter decreases, the call completion probability increases and is less sensitive to the mobility of the user. Also the effect of the shape parameter on system performance is negligibly small when the value is greater than two. However, in the sub-exponential region ( 0 < v < 1) , the call completion probability is very sensitive to changes in the value of the shape parameter. The same results are also observed in Fig. 5, that shows the call completion probabilities for different values of the shape parameters v and γ . From this graph we observe again that the call completion probability is greatly affected by the values of shape parameters that are less than one (heavy tail region) whereas the effect is much smaller for values that are greater than one (light tail region).

1 2 k −1  j  1,1 − ,1 − ,...,1 −  k k k  , 1 j − 1  1 1 1 1 2 , + , + ,..., +  k k j k j k j 

(23) where E [T ] = β Γ (1 + 1/ γ ) and E [ R1 ] = E T 2  ( 2 E [T ] ) = β Γ (1 + 2 / γ )  2Γ (1 + 1/ γ )  . When the shape parameter v

( v { p E [ R ] + (1 − p ) E [T ]})}) f

1

f

} ( bp ) .

× v p f E [ R1 ] + (1 − p f ) E [T ]

f

NUMERICAL RESULTS

In this section, we present some numerical results for the call completion probability based on the assumption of Weibull distributed call holding time and cell dwell time. All the figures are plotted against different values of the call-tomobility factor, defined as the ratio of the mean of call holding time and the mean of the cell dwell time, i.e., E [Tc ] bΓ (1 + 1 v ) ρ= Also we assume that = . E [T ] β Γ (1 + 1 γ )

v→0

1 0.95 0.9 Call Completion Probability

fT ( t ) =

IV.

0.85 0.8 0.75

0.65 0.6 0.55 0.5 0

v→∞

v= 4 v= 2 v= 1 v = 1/2 v = 1/3 v = 1/4 v = 1/5 v = 1/6

0.7

5

10

15 20 call-to-mobility factor ρ

( )

25

30

Figure 3. The call completion probability versus the call-to-mobility factor ρ for po = p f = 0.02 , γ = 1 and for different values of the parameter v.

(25)


can be easily computed using mathematical software such as Mathematica and Maple.

1

Numerical results show that light tail distributions do not affect the call completion probability significantly. On the other hand, the wireless network performance (measured with the call completion probability) is very sensitive to changes in the distribution model in the heavy tail region. Limiting cases of the shape parameter of the Weibull distribution were also investigated. These limiting case performance results may be useful in wireless network design.

0.95

Call Completion Probability

0.9 0.85 0.8 0.75 0.7 0.65

0

γ γ γ γ γ γ γ γ

=4 =2 =1 = 1/2 = 1/3 = 1/4 = 1/5 = 1/6

5

REFERENCES [1] 10

15 20 call-to-mobility factor( ρ )

25

30

Figure 4. The call completion probability versus the call-to-mobility factor ρ for po = p f = 0.02 , v = 1 and for different values of the parameter γ .

[2]

[3] [4]

1 0.95

Call Completion Probability

0.9

[5] 0.85 0.8

[6]

0.75 0.7

γ γ γ γ γ

0.65 0.6 0.55 0.5 0

5

= v= 4 = v= 1 = v = 1/2 = v = 1/3 = v = 1/4

10

15 20 call-to-mobility factor

[7]

[8] 25

30

Figure 5. The call completion probability versus the call-to-mobility factor ρ for po = p f = 0.02 and for different values of the parameters v and γ .

[9] [10] [11]

V. CONCLUSION The call completion probability for a wireless network under Weibull distributed call holding time and cell dwell time is derived. The Weibull distribution, which has the exponential and Rayleigh distributions as special cases, exhibits a heavy tail property when the positive shape parameter is less than one and is widely used to fit raw traffic data of internet networks. The derivation is based on the notion of random sums and results in easy to use expression in terms of the Meijer-G function, which

[12] [13]

[14] [15]

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