Random Number Generation for Excess Life of
Mobile User Residence Time
Hui-Nien Hung, Pei-Chun Lee, and Yi-Bing Lin, Fellow, IEEE
Abstract—In a mobile telecommunications network, the period when a mobile station (MS) resides in a cell (the radio coverage of a base station) is called the cell residence time of that MS. The period between when a call arrives at the MS and when the MS moves out the cell is called the excess life of the cell residence time for that MS. In performance evaluation of a mobile telecommunications network, it is important to derive the excess life distribution from the cell residence times. This distribution determines if a connected call will be handed over to a new cell, and therefore significantly affects the call dropping probability of the network. In mobile-telecommunications-network simulation, generating the excess-life random numbers is not a trivial task, which has not been addressed in the literature. This paper shows how to gen-erate the random numbers from the excess life distribution, and develop the excess-life random number generation procedures for cell residence times with gamma, Pareto, lognormal, and Weibull distributions. This paper indicates that the generated random numbers closely match the true excess-life distributions.
Index Terms—Cell residence time, excess life, handover, mobility management.
I. INTRODUCTION
A
MOBILE telecommunications network is populated with several base stations (BSs). Mobile users receive mobile telecommunications services by using mobile stations (MSs) connecting to the BSs. When an MS moves from the radio coverage (called cell) of a BS to the radio coverage of another BS, the MS is disconnected from the old BS and reconnected to the new BS. This process is called handover. Fig. 1 illustrates the relationship between movement of an MS and a call session to that MS. The MS moves to cell 1 at time t0, and then movesto cell i at time ti for i > 1. A call for the MS arrives at time
t1. If the call is not blocked or dropped, it completes at time
t5. At time t1, if cell 1 does not have enough radio resources
to accommodate this call (which can be a plain voice call or
Manuscript received April 3, 2005; revised September 3, 2005 and November 4, 2005. This work was supported in part by the National Science Council (NSC) Excellence project NSC 94-2752-E-009-005-Program for Aca-demic Excellence (PAE), NSC 94-2219-E-009-001, NSC 94-2213-E-009-104, National Telecommunication Development Program (NTP) Voice over Inter-net Protocol (IP) (VoIP) Project under Grant NSC 94-2219-E-009-002, NTP Service IOT Project under Grant NSC-94-2219-E-009-024, Intel, Chung Hwa Telecom, Institute of Information Science (IIS)/Academia Sinica, Industrial Technology Research Institute (ITRI)/National Chiao Tung University (NCTU) Joint Research Center, and MoE ATU. This work was also supported in part by the NSC of Taiwan under Grant NSC-93-2118-M-009-006. The review of this paper was coordinated by Prof. X. Shen.
H.-N. Hung is with the Institute of Statistics, National Chiao Tung Univer-sity, Hsinchu 30010, Taiwan, R.O.C. (e-mail: hhung@stat.nctu.edu.tw).
P.-C. Lee and Y.-B. Lin are with the Department of Computer Science, National Chiao Tung University, Hsinchu 30010, Taiwan, R.O.C. (e-mail: pjlee@csie.nctu.edu.tw; liny@csie.nctu.edu.tw).
Digital Object Identifier 10.1109/TVT.2006.874578
a multimedia call), the call is blocked. When the MS moves to cell i, the call is handed over from cell i− 1 to cell i. If no radio resources are available in cell i, the call is dropped or forced to terminate. Performance of a mobile telecommunications network is typically evaluated by the call blocking probability (a new call attempt is blocked), the call dropping probability or force-termination probability (a handover call is forced to terminate), and the call incompletion probability (a call is either blocked or dropped).
Many studies [3], [4], [6], [9], [15] have been devoted to eval-uate these probabilities for various radio resource-allocation strategies exercised in mobile telecommunications networks. Most of them utilized analytic approaches that provide use-ful insights to mobile-network modeling. However, analytic analysis has its limitations. For example, in Fig. 1, if the call holding time tc= t5− t1is nonexponential (which is probably
true for multimedia calls) [2], then it is difficult to derive the remaining call holding time τc= t5− ti after the MS moves
into cell i (for i > 1). Furthermore, most analytic studies made an approximate assumption that the handover traffic to a cell is a fixed Poisson process. This assumption is reasonable for large-scale mobile telecommunications networks, but may result in significant inaccuracy for small-scale networks [7], [16]. Also, if the resource-allocation policies under consideration are very complicate (which is probably true for wireless data sessions with QoS), it is impossible to find analytic solutions.
An alternative modeling technique to analytic analysis is discrete event simulation. There are two approaches to mobile-telecommunications-network simulation: the MS-based simula-tion and the call-based simulasimula-tion. In the MS-based simulasimula-tion, the number of MSs are defined in the simulation, and the MS objects are actually simulated for their movements (even if there are no calls destinated at these MSs). Examples of MS-based simulation can be found in [10]. In the call-based simulation [8], [12], the call arrival rate to the network is considered as the input that drives the simulation progress. In this approach, after a call arrival event is processed, the corresponding MS movement and the call termination events are generated follow-ing the timfollow-ing diagram illustrated in Fig. 1 (details of the call-based simulation is described in Appendix). When the number of MSs is small in a mobile telecommunications network, the MS-based simulation will produce more accurate results than the call-based simulation. When the number of MSs is large, both approaches produce results with similar accuracies. On the other hand, the execution time for the MS-based simulation is much longer than that for the call-based simulation (e.g., 100 times longer [10]). Since large MS population is expected in most third-generation systems such as Universal Mobile
Fig. 1. Timing diagram for MS movement and call arrival.
Telecommunications System (UMTS) [1], [11], the call-based simulation will become more important in advanced mobile telecommunications studies.
In mobile-telecommunications-network modeling, several random variables are defined. Two of them are elaborated here; others are described in Appendix. In Fig. 1, tm,1= t2− t0,
and tm,i= ti+1− ti (for i > 1) are the time intervals that the
MS resides in cell i. These cell residence times are typically modeled by a random variable with a specific distribution such as gamma and mixed Erlang [5], [8], [12]. The interval τm=
t2− t1 is the period between when a call arrives and when
the MS moves out of the first cell, which is referred to as the excess life of the cell residence time. In the call-based simu-lation, it is required to generate the random numbers for the excess life τm(see Appendix). Clearly, the τmdistribution must
be derived from the cell residence-time distribution. The call arrivals are typically assumed to be random observers of the cell residence times. If the cell residence times have the exponential distribution, then τmalso has the same exponential distribution
[14]. On the other hand, if the cell residence times have an arbitrary distribution, generation of the τmrandom numbers is
a nontrivial task. In this paper, we describe how to generate the
τm random numbers from the cell residence-time distribution.
For various cell residence-time distributions, generation of τm
random numbers need separate treatments. We show how to generate the excess-life random numbers for cell residence-time random variables with gamma, Pareto, lognormal, and Weibull distributions. Our study indicates that the generated random numbers closely match the true excess-life distributions.
II. DERIVATION OFEXCESSLIFEDISTRIBUTION
In Fig. 1, the cell residence times tm,i(i≥ 1) of an MS are
assumed to be independent identically distributed random vari-ables. Therefore, we use tmto represent an arbitrary cell
res-idence time with the density function fm(tm), the distribution
function Fm(tm) and the mean µ. Let τmbe the excess life of tm
with the density function rm(τm) and the distribution function
Rm(τm). Since the call arrivals form a Poisson process, a call
arrival is a random observer of the MS cell residence times. From the excess life theorem [14], we have
rm(τm) =
1− Fm(τm)
µ . (1)
It is difficult to generate the random numbers for the excess life of a cell residence-time random variable using (1) because
this equation involves the distribution function Fm(τm). To
efficiently generate the random numbers τm, we shall utilize a
variation of fm(τm). We will prove that rm(τm) can be derived
from the following function:
fT(t) = tfm(t) µ . (2) Since ∞ t=0 tfm(t) µ dt = 1 µ ∞ t=0 tfm(t)dt = µ µ = 1
it is obvious that fT(t) can be a density function. Let T be a
random variable with the density function fT(t). We have the
following theorem.
Theorem 1: Let τmbe the excess life of tm. Let random
vari-able U be uniformly distributed over the interval (0,1). Let T be random variable with the density function fT(t) = (tfm(t)/µ),
and U and T are independent. Then, the distribution of τmis the
same as the distribution of U × T .
Proof: The joint density function of U and T is f(U,T )(u, t) = tfm(t) µ , for 0 < u < 1 and t > 0 0, otherwise. Let W = U× T . Then Pr[W ≤ w] = Pr[U × T ≤ w] = 1 u=0 w u t=0
f(U,T )(u, t)dtdu
= 1 u=0 w u t=0 tfm(t) µ dtdu. (3)
From (3), the density function fW(w) of W can be derived as
fW(w) = d Pr[W ≤ w] dw = 1 u=0 w u f m(w/u) µ 1 u du = 1 µ 1 u=0 w u2 fm(w/u)du. (4)
Let y = (w/u). Then, (4) can be rewritten as fW(w) = 1 µ ∞ y=w fm(y)dy =1− Fm(w) µ = rm(w)
which means that W = U× T has the same distribution
as τm.
Theorem 1 allows us to generate a τmrandom number using
fm(·) as follows: We first generate a random number u for
the uniform random variable U in (0,1). Then, we generate a random number t for the random variable T with the density function fT(t) [see (2)]. Then, we multiply t by u to obtain the
random number for the excess life τm. Derivation of fT(t) is
not a trivial task, and some fT(t) functions cannot be derived
from the corresponding fm(t) functions. In the next section, we
show how to derive fT(t) for some popular distributions.
III. EXCESS-LIFERANDOMNUMBERGENERATION: SOMEEXAMPLES
This section derives the T distributions for cell residence times with distributions such as gamma, Pareto, lognormal, and Weibull. Then, we show how to generate the excess-life random numbers using Theorem 1 and the T distributions.
A. Gamma Distribution
Suppose that tm has a gamma distribution with the shape
parameter α and the scale parameter β. Then, the mean value is
µ = αβ and the density function fm(tm) is
fm(tm) =
e−tmβ tα−1
m
βαΓ(α) , for tm≥ 0. (5)
We have the following theorem.
Theorem 2: If tm has a gamma distribution with the
pa-rameters (α, β), then T has a gamma distribution with the parameters (α + 1, β).
Proof: From (2) and (5), we have fT(t) = te−βttα−1 µβαΓ(α), for t≥ 0 = βe −t βtα µβα+1Γ(α + 1)× Γ(α + 1) Γ(α) . (6)
Since Γ(α + 1) = αΓ(α) and µ = αβ, (6) is rewritten as
fT(t) =
e−βttα
βα+1Γ(α + 1), for t≥ 0. (7)
From (7), it is clear that T has the gamma distribution with
parameters (α + 1, β).
Fig. 2. rm(τm) function for gamma excess life.
Generation of an excess-life random number for gamma residence time with the parameters (α, β) includes the follow-ing steps: We first generate a uniform random number u in (0,1). Then, according to Theorem 2, we generate a random number t for the gamma random variable T with the parameters (α + 1, β). By multiplying u and t, we obtain a random number for the excess life τm. Fig. 2 plots the rm(τm) function for
gamma excess life.
In this figure, the symbols “” and “•” represent the values obtained from the random number generation. The solid and dashed curves are directly computed from (1). The figure indi-cates that our random number generation procedure accurately generates the excess-life random numbers for the gamma cell residence times.
B. Pareto Distribution
Suppose that tmhas the Pareto distribution with the
parame-ters (a, b), where a is the shape parameter and b is the scale parameter. Then, the mean is
µ =
ab
a−1, if a > 1
∞, if 0 < a≤ 1 (8) and the density function is
fm(tm) =
aba
ta+1m
(9) where tm≥ b, a > 0, and b > 0. We have the following
theorem.
Theorem 3: Suppose that tm has a Pareto distribution with
the parameters (a, b), where a > 1. Then, T has a Pareto distribution with the parameters (a− 1, b).
Proof: From (2), (8), and (9), we have fT(t) = taba ta+1 × a− 1 ab =(a− 1)b a−1 t(a−1)+1 . (10)
Fig. 3. rm(τm) function for Pareto excess life.
Equation (10) is a Pareto density function with the parameters
(a− 1, b).
By utilizing Theorems 1 and 3, the τm random number
generation procedure for Pareto cell residence times is sim-ilar to that for gamma cell residence times. Fig. 3 plots the
rm(τm) function for Pareto excess life. The figure indicates
that our random number generation procedure accurately gen-erates the excess-life random numbers for the Pareto cell resi-dence times.
C. Lognormal Distribution
Suppose that tm has a lognormal distribution with the
pa-rameters (θ, σ). Then, the mean value is µ = eθ+σ2/2 and the density function fm(tm) is fm(tm) = 1 σtm √ 2π e−(ln tm−θ) 2 2σ2 , for tm≥ 0. (11)
We have the following theorem.
Theorem 4: Suppose that tm has a lognormal distribution
with the parameters (θ, σ). Let Y = ln T . Then, Y has a normal distribution with the mean µ + σ2and the standard deviation σ.
Proof: From (2) and (11) fT(t) = 1 µσ√2π e−(ln t−θ)22σ2 , where t≥ 0 = 1 eθ+σ2 2 σ √ 2π e−(ln t−θ)22σ2 . (12)
Since Y = ln T , we have T = eY. According to the Jacobian
of the transformation [13], the density function of Y is ex-pressed as
fY(y) = fT(ey)
dydt = fT(ey)× ey (13)
where−∞ < y < ∞. Substitute (12) into (13) to yield
fY(y) = 1 eθ+σ2 2 σ√2π e−(y−θ)22σ2 × ey = 1 σ√2π e−[ y−(θ+σ2)]2 2σ2 (14)
where−∞ < y < ∞. From (14), Y is a normal random vari-able with the mean θ + σ2and the standard deviation σ.
Generation of an excess-life random number for lognormal cell residence time with the parameters (θ, σ) includes the following steps: We first generate a random number u from the uniform random variable U in (0,1). Then, according to Theorem 4, we generate a random number y for the normal random variable Y with the mean θ + σ2and the standard de-viation σ. By multiplying u and ey, we obtain a random number for the excess life τm. Details of the lognormal residence-time
curves will not be presented in this paper.
D. Weibull Distribution
Suppose that tm has a Weibull distribution with the scale
parameter θ and the shape parameter γ. Then, the mean value is µ = θ(1/γ)Γ(1 + (1/γ)) and the density function fm(tm) is
fm(tm) = γ θ tγ−1 m e− tγm θ , if tm≥ 0 0, if tm< 0. (15) We have the following theorem.
Theorem 5: Suppose that tmhas a Weibull distribution with
the parameters (γ, θ). Let Y = Tγ. Then, Y has a gamma
distribution with the parameters (1 + (1/γ), θ).
Proof: From (2) and (15), we have fT(t) = γ θ × tγe−tγθ θ1γΓ 1 + 1γ (16)
where t≥ 0. Let Y = Tγ. Then, T = Y(1/γ). According to
the Jacobian of the transformation [13], the density function of Y is fY(y) = fT yγ1 dt dy = fT yγ1 × yγ1−1 γ (17) where y≥ 0. Substitute (16) into (17) to yield
fY(y) = y1γe−yθ θ1+1γΓ 1 + 1 γ , where y≥ 0. (18)
From (18), Y has a gamma distribution with the parameters
(1 + (1/γ), θ).
Generation of an excess-life random number for Weibull cell residence time with the parameters (γ, θ) includes the following steps: We first generate a random number u of the uniform random variable U in (0,1). Then, according to Theorem 5, we generate a random number y for the gamma random variable
y(1/γ), we obtain a random number for the excess life τ m.
De-tails of the Weibull residence-time curves will not be presented in this paper.
IV. CONCLUSION
In performance evaluation of a mobile telecommunications network, it is important to derive the excess life distribution from the cell residence times. This distribution determines if a connected call will be handed over to a new cell, and therefore significantly affects the call dropping probability of the network. In mobile-telecommunications-network simula-tion, generating the excess-life random numbers is not a trivial task, which has not been addressed in the literature. This paper showed how to derive the excess life distribution and to generate the random numbers from the excess life distribution. We then developed the excess-life random number generation procedures for cell residence times with gamma, Pareto, log-normal, and Weibull distributions. Our study indicates that the generated random numbers closely match the true excess-life distribution [i.e., (1)]. Therefore, our procedures can be utilized to efficiently generate excess-life random numbers in mobile-telecommunications-network simulation.
APPENDIX
CALL-BASEDSIMULATION
This Appendix describes the basic call-based discrete event simulation for mobile telecommunications network. Several random variables are defined: the intercall arrival time (the call arrivals are typically modeled as a Poisson process), the call holding time, the cell residence time, and the excess life of the cell residence time. Three basic event types are considered: the arrival event (a call arrival), the move event (an MS movement), and the complete event (a call completion). Every event is associated with a timestamp representing the time when the event occurs. All unprocessed events are inserted in an event list and are processed in the nondecreasing timestamp order. Details of the call-based simulation are described in the following steps.
Step 1) (Initialization) Generate the first arrival event and insert it in the event list.
Step 2) Remove the next event from the event list. If the event type is arrival then go to Step 3). If the type is move then go to Step 5). If the type is complete then go to Step 6).
Step 3) (Arrival) Check if the cell can accommodate this call based on some wireless resource-allocation policy. If not, reject the call, update the call statistics, and go to Step 4). Otherwise, generate the random numbers for the excess life τmof the cell residence time and
the call holding time tc.
Step 3.1) If τm> tc, generate a complete event
with timestamp “current time + tc.”
Step 3.2) If τm< tc, generate a move event with
timestamp “current time + τm.” Note that
when the next move event occurs, the
remaining call holding time is τc= tc−
τm.
Insert the generated event into the event list. Step 4) Generate the next arrival event according to the
Poisson process and insert it into the event list. Go to Step 2).
Step 5) (Move) The MS moves from the old cell to the new cell. Check if the new cell can accommodate this handover call. If not, drop the call, update the call statistics, and go to Step 2). Otherwise, generate the cell residence time tm. The remaining call holding
time is τc.
Step 5.1) If tm> τc, generate a complete event
with timestamp “current time + τc.”
Step 5.2) If tm< τc, generate the next move
event with timestamp “current time +
tm.” Note that when the next move event
occurs the remaining call holding time is τc = τc− tm.
Insert the generated event into the event list. Go to Step 2).
Step 6) (Complete) Reclaim the resources used by this call. Update the call statistics, and go to Step 2).
The simulation can be terminated based on various criteria. For example, at Step 3), we may check if some terminating conditions are satisfied (e.g., 1 000 000 call arrivals have been simulated). If so, the simulation terminates.
ACKNOWLEDGMENT
The authors would like to thank the three anonymous review-ers who have provided valuable comments that significantly improved the quality of this paper.
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Hui-Nien Hung received the B.S.Math. degree from National Taiwan University, Taipei, Taiwan, R.O.C., in 1989, the M.S.Math. degree from National Tsing-Hua University, Hsinchu, Taiwan, in 1991, and the Ph.D. degree in statistics from The University of Chicago, Chicago, IL, in 1996.
He is currently a Professor with the Institute of Statistics, National Chiao Tung University, Hsinchu. His research interests include applied probability, financial calculus, bioinformatics, statistical infer-ence, statistical computing, and industrial statistics.
Pei-Chun Lee received the B.S.C.S.I.E., M.S.C.S.I.E., and the Ph.D degrees in computer science from National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 1998, 2000, and 2006, respectively.
Her current research interests include design and analysis of a personal communications services net-work, computer telephony integration, mobile com-puting, and performance modeling.
Yi-Bing Lin (M’96–SM’96–F’04) received the B.S.E.E. degree from National Cheng Kung Uni-versity, Tainan, Taiwan, R.O.C., in 1983 and the Ph.D. degree in computer science from University of Washington, Seattle, in 1990.
He is currently the Chair Professor and Vice-President of Research and Development, National Chiao Tung University, Hsinchu, Taiwan. He has published over 200 journal articles and more than 200 conference papers. He is the coauthor of the books Wireless and Mobile Network Architecture with I. Chlamtac (New York: Wiley, 2001) and Wireless and Mobile All-IP Net-works with A.-C. Pang (New York: Wiley, 2005). His current research interests include wireless communications and mobile computing.
