Modeling Channel Assignment of Small-Scale
Cellular Networks
Hui-Nien Hung, Pei-Chun Lee, Yi-Bing Lin, Fellow, IEEE, and Nan-Fu Peng
Abstract—In a cellular telecommunications network, the call
blocking, forced termination, and call incompletion probabilities are major output measures of system performance. Most previous analytic studies assumed that the handover traffic to a cell is a fixed-rate Poisson process. Such assumption may cause significant inaccuracy in modeling. This paper shows that the handover traffic to a cell depends on the workloads of the neighboring cells. Based on this observation, we derive the exact equation for the handover force-termination probability when the mobile station (MS) cell residence times are exponentially distributed. Then, we propose an approximate model with general MS cell residence time distributions. The results are compared with a previously proposed model. Our comparison study indicates that the new model can capture the handover behavior much better than the old one for small-scale cellular networks.
Index Terms—Call duration time, cellular network, channel
as-signment, handover.
I. INTRODUCTION
E
MERGING cellular telecommunications network tech-nologies have attracted considerable attention in academic research as well as commercial deployment. A cellular network supports telephony services when users are in movement [10]. The cellular phone service area is populated with base stations (BS’s). The radio coverage of a BS is referred to as a cell. Customers within a cell can connect to the corresponding BS via mobile stations (MS’s) or mobile phones. When a call for a customer occurs, one radio channel of the BS is used for connecting the MS and the BS. If all radio channels are in use when a new call is attempted, the call will be blocked and cleared from the system. If the call is accepted, a radio channel will be occupied until the call is completed, or until the MS moves out of the cell. When a communicating MS moves from one cell to another, the occupied channel in the old cell is released, and an idle channel is acquired in the new cell. During Manuscript received May 27, 2003; revised October 1, 2003, January 20, 2004; accepted January 27, 2004. The editor coordinating the review of this paper and approving it for publication is M. Zorzi. This work was supported in part by NSC Excellence project NSC93-2752-E-0090005-PAE, in part by NSC 93-2213-E-009-100, in part by NTP VoIP Project under Grant NSC 92-2219-E-009-032, in part by IIS/Academia Sinica, and in part by ITRI/NCTU Joint Re-search Center. The work of H.-N. Hung was supported in part by the National Science Council of Taiwan under Grant NSC-91-2118-M-009-006. The work of N.-F. Peng was supported in part by the National Science Council of Taiwan under Grant NSC-91-2118-M-009-008.H.-N. Hung and N.-F. Peng are with the Institute of Statistics, Na-tional Chiao Tung University, Hsinchu 30010, Taiwan, R.O.C. (e-mail: hhung@stat.nctu.edu.tw; nanfu@stat.nctu.edu.tw).
P.-C. Lee and Y.-B. Lin are with the Department of Computer Science and Information Engineering, 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/TWC.2004.842946
Fig. 1. Effect of network size on the call incompletion probability ( = 0:3).
this handover procedure, if no channel is available in the new cell, the call is forced to terminate before its completion. When the call is connected, the call may be completed after several successful handovers, or may be forced to terminate due to a failed handover. The duration of a call connection (if the call is completed) is referred to as the call duration time.
For billing and network planning purposes, the handover be-havior and the probability of call completion need to be ana-lyzed. Several analytic studies have contributed to cellular net-work performance evaluation [1], [3], [4], [6], [12], [13], [15], [17]. Most studies assume that the handover traffic to a cell is a fixed-rate Poisson process. This assumption is reasonable for large-scale cellular networks, or when the networks experience light load traffic [2]. In reality, the handover traffic to a cell de-pends on the workloads of the neighboring cells. This fact has significant impact on modeling of small-scale cellular networks. Fig. 1 plots the call incompletion probability against the user mobility and the call arrival rate where the number of radio channels in a cell is 9. The “ ” curve is generated from a previous analytic model that assumes fixed-rate handover traffic [12]. The “ ” curve is generated from simulation of a 64-cell mesh configuration, and the “ ” curve is generated from sim-ulation of a three-cell configuration (illustrated in Fig. 2). The simulation model [11] actually simulates the MS movement in mesh or hexagonal networks of cells. This simulation model is used throughout the paper. Fig. 1 indicates that the fixed-rate as-sumption is acceptable when the number of cells is reasonably large, but is inaccurate for small-scale cellular networks. In this paper, we derive the exact equation for the handover force-ter-mination probability when the MS cell residence times are expo-nentially distributed. Then, we propose an approximate model with general MS cell residence times. The results are compared with the previously proposed model [12]. Our comparison study 1536-1276/$20.00 © 2005 IEEE
Fig. 2. Three-cell cellular system.
indicates that the new model can capture the handover behavior much better than the old one for small-scale cellular networks.
II. EXACTANALYTICMODEL FOREXPONENTIALMS CELL RESIDENCETIMES
This section describes an exact analytic solution for exponen-tial MS cell residence times. Consider a cellular system with cells. For , let be the index set of cell ’s neighbors. That is, cell is a neighbor of cell if . Let be the number of cell ’s neighbors. For the illustration purpose, we consider a homogeneous system conforming to the following requirements:
1) Capacity: Each cell has channels.
2) Movement: The routing probabilities to the neigh-boring cells are the same. That is, for an MS at cell , it moves to each of cell ’s neighbors with probability
. We further assume that for
, . The MS cell residence time distributions are the same for all cells.
3) Call traffic: The new call arrival rates to all cells are the same. The call duration time distributions are the same for all calls.
The following input parameters are considered in our model. 1) : The new call arrival rate to a cell. The new call
ar-rivals are a Poisson stream [7] and the new call arar-rivals to each cell are independent.
2) : The mean call duration time. The call duration times have an exponential distribution.
3) : The mean MS cell residence time. The MS cell residence times are independent and identically dis-tributed (i.i.d.). This section assumes exponential MS cell residence times. In the next section, we will con-sider general MS cell residence time distributions. We assume that the call duration time and the MS cell residence time are independent of each other. Let random variable be the number of busy channels in cell . The fol-lowing output measures are evaluated in our study.
1) (the new call blocking probability): The number of new call blockings divided by the number of new calls. Since the system is homogeneous, for all cells are the same and for a cell , can be expressed as
A new call is blocked this new call occurs at cell
(1)
2) (the forced termination probability): The number of forced terminations divided by the number of han-dovers
3) (the call incompletion probability; i.e., the proba-bility that a call is either blocked or forced to termi-nate): The sum of the numbers of new call blockings and forced terminations divided by the number of new
calls. Note that .
To derive , we first define five events.
1) Event . A call is handed over into cell . For , are mutually exclusive events.
2) Event . A call is handed over out of a cell with busy channels. For , are mutually exclu-sive events.
3) Event . A handover occurs in the cellular system. The relationship among , , and is
(2)
4) Event . A call is handed over out of cell . For , are mutually exclusive events, and
(3)
5) Event . A call is handed over out of a specific cell
with busy channels. For and ,
are mutually exclusive events. Note that
(4)
Since the system is homogeneous, for all cells are the same, which can be expressed as
(5)
Since , (5) can be rewritten as
(6)
Equation (6) says that to compute , we need to consider how the flow-in handover traffic behaves. For example, there is no flow-in handover traffic into cell if for all cell ’s neigh-bors (i.e., there is no busy channel in any of cell ’s neighneigh-bors).
Since are mutually exclusive events, from [16], (6) can be expressed as
(7) In (7), let
(8) which is the probability that all channels in cell are busy given that a call is handed over from a cell with busy channels to cell . From (4), (8) can be rewritten as
(9)
Since are mutually exclusive events, from
[16], (9) can be expressed as
(10)
From (4), we have and (10) is rewritten as
(11)
In (11), represents the routing probability
that a call is handed over from cell with busy channels to cell , given that the call is handed over from a cell with busy channels to cell . Since we consider homogeneous topology and routing pattern, the MS can only move into cell from any one of cell ’s neighbors with probability . Therefore,
if
otherwise. (12)
Substitute (12) into (11) to yield
(13)
Due to homogeneous assumptions, (13) can be rewritten as
A call is handed over from cell to
cell (14)
where . It is clear that Event and Event {A
call is handed over from cell to cell } are conditionally
inde-pendent given Event , where . Therefore,
from [16], (14) can be expressed as
(15)
where . In (7), let
(16) Given that a call is handed over into cell , is the probability that the call is handed over out of a cell with busy channels.
From (3) and , (16) can be rewritten as
(17)
Since are mutually exclusive events, from [16], (17) can be expressed as
(18)
In (18), is the probability that under the condition that a call is handed over into cell , it came from cell . Because of homogeneous topology and routing pattern, the handover call is from any of cell ’s neighbors with the same probability. There-fore,
if
Substitute (19) into (18) to yield
(20)
where . Due to network homogeneity,
. Thus, (20) is rewritten as
(21)
where . In (21), Event represents the event
that a call is handed over from cell to cell , and Event means that a call is handed over from cell with busy channels to cell . Note that the mobility rate of an MS is . Because of homogeneous topology and routing pattern, if a busy channel in cell will be released due to MS movement to cell , then the channel is released at rate , and (21) can be rewritten as
(22) Substituting (15) and (22) into (7), we have
(23) The probability was derived in [12], which is expressed as follows:
(24) where is the Laplace transform of the MS cell residence time distribution. For exponential MS cell residence time
distri-butions, , and (24) can be expressed as
(25)
, and in (1) and (23) are
derived by solving an -dimensional Markov process, which are
Fig. 3. Timing diagram.
then used to compute . We note that our approach is similar to the one proposed in [14], where the model can be extended for heterogeneous network modeling (e.g., the numbers of channels in cells are different).
III. APPROXIMATE MODEL FOR GENERAL MS CELLRESIDENCETIMES
This section proposes an approximate solution for modeling general MS cell residence times. The idea is to adjust the exact analytic solution developed in the previous section. Specifically, we approximate the general MS cell residence time by an expo-nential distribution with the adjusted rate .
Consider the timing diagram in Fig. 3. In this figure, a call arrives when the MS resides in cell 1. The call duration time is . The MS cell residual time at cell 1 (i.e., the interval between when the call arrives and when the MS moves out of cell 1) is . For , if the call is successfully handed over to cell , then the remaining call duration time is . For , the MS cell residence time at cell (i.e., the interval between when the MS enters cell and when it moves out of cell ) is . Since the call duration times are exponentially distributed, and have the same exponential distribution. Let random variable be the number of call completions for a connected call. Note that the value of is either one or zero, depending on whether the call is eventually completed or forced to terminate. Let random variable be the number of handovers for a connected call. We derive and as follows. In Fig. 3, let
be the probability that a new call is not completed before the
MS moves out of the first cell, and be the
probability that a handover call is not completed before the MS moves out of cell , where From [8], we have
and (26)
With (26), is derived as follows:
A connected call is completed
A connected call is forced to terminate
In the right-hand side of (27), is the probability that a new call is completed before the MS moves out of the cell, and is the probability that a call is successfully handed over for times and is completed at the -st cell, where . From (27), we have
(28) Substitute (26) into (28) to yield
(29) Similarly, is derived as
(30) (31) In the right-hand side of (30),
is the probability that a call is successfully handed over for times before it is completed, or is successfully handed over for times and forced to terminate at the th handover. Substitute (26) into (31) to yield
(32) Define as
(33) Although and are dependent random variables, we have [5]
(34) From (34), (33) is rewritten as
(35) From (29) and (32), (35) can be expressed as
(36) For the exponential MS cell residence time distribution,
, and (36) can be simplified as
(37) To approximate the general MS cell residence time distribution by an exponential distribution, we define as the mobility rate
for the approximate exponential distribution. From (37), the ap-proximate mobility rate is
(38) By substituting (36) into (38), we have
(39)
Therefore, for MS cell residence time distribution with Laplace transform , this distribution can be approximated by an exponential distribution with mobility rate [expressed in (39)] for the channel assignment model. The probabilities ,
, and are computed as follows:
Input parameters: , , , , and . Output measures: , , , and . Step 1. Select an initial value for .
Step 2. . Compute by using (39). Step 3. Use the standard -dimensional Markov process [14] to solve and with (1) and (23). Step 4. Let be a predefined value ( in our example). If , then go to Step 2. Oth-erwise, go to Step 5.
Step 5. The values for , , and converge. Compute by using (24).
IV. NUMERICALEXAMPLES
This section uses numerical examples to compare the analytic model in Sections II and III (called the “new model”) with the model we proposed in [12] (called the “old model”). For the demonstration purpose, we consider a three-cell cellular system (i.e., ; see Fig. 2), where each cell has nine channels (i.e., ) and two neighbors (i.e., ). Such systems have been manufactured and deployed in Asia.1For other small-scale cell
configurations, similar results are observed, which will not be presented in this paper.
Fig. 4 plots the blocking probabilities against the mobility rate for the exponential MS cell residence time model, where the call duration times are exponentially distributed with rate . The figure indicates that both and decrease as increases. Since the number of handovers increases as in-creases, increases as increases. The same phenomena were found in [1] and [9], and the reader is referred to these pervious studies for more details. We observe that the new analytic results almost match the simulation results, while the errors between the old analytic model and the simulation experiments can be up to 18%. The figure suggests that the new analytic model is more accurate than the old one. Fig. 4 also indicates that the higher the mobility, the more the inconsistency between the old analytic and the simulation results. Therefore, the advantage of
Fig. 4. Results for exponential MS cell residence time model( = 0:3). (a) p (%). (b) p (%). (c) p (%).
TABLE I
p VALUES ANDERRORS: ANALYTICMODELS VERSUSSIMULATION
( = 0:5). (a) p . (b) ERRORS OFp
the new analytic model becomes significant when the mobility is high. The old analytic model is not as accurate as the new one because it assumes , while in the new analytic model, we have proven that
(see (23)). When the mobility is low, the value of
is close to . Consequently, and the old analytic model works well.
For a general MS cell residence time distribution, its vari-ance may have significant impact on the output measures. For Gamma MS cell residence times, Table I shows the call in-completion probability values and their errors between an-alytic and simulation models, where the MS cell residence time
variances are , , and . In Table I, the
mobility rate is and the call completion rates are , , and , respectively. The results suggest that for various , the values of the new analytic model are much closer to the simulation results than that of the old ana-lytic model.
V. CONCLUSION
Most analytic modeling studies for cellular networks assume that the handover traffic to a cell is a fixed-rate Poisson process. This assumption may introduce significant inaccuracy for mod-eling small-scale cellular networks. This paper showed that the handover traffic to a cell depends on the workloads of the neigh-boring cells. We derived the exact equation for the handover force-termination probability when the MS cell residence times are exponentially distributed. Then we proposed an approximate model for general MS cell residence time distributions. The re-sults are compared with a previously proposed model, which indicate that the new model can capture the handover behavior much better than the old one for small-scale cellular networks.
ACKNOWLEDGMENT
The three anonymous reviewers have provided useful com-ments that significantly improve the quality of this paper.
REFERENCES
[1] I. Chlamtac, Y. Fang, and H. Zeng, “Call blocking analysis for PCS networks under general cell residence time,” in Proc. IEEE Wireless
Communications Networking Conf., New Orleans, LA, Sep. 1999, pp.
550–554.
[2] E. Chlebus and W. Ludwin, “Is handoff traffic really Poissonian?,” in
Proc. IEEE Int. Conf. Universal Personal Communications (ICUPC),
1995, pp. 348–353.
[3] Y. Fang and I. Chlamtac, “Teletraffic analysis and mobility modeling for PCS networks,” IEEE Trans. Commun., vol. 47, no. 7, pp. 1062–1072, Jul. 1999.
[4] R. Fantacci, “Performance evaluation of prioritized handoff schemes in mobile cellular networks,” IEEE Trans. Veh. Technol., vol. 49, no. 2, pp. 485–493, Mar. 2000.
[5] R. G. Gallager, Discrete Stochastic Processes. Norwell, MA: Kluwer, 1996.
[6] D. Hong and S. S. Rappaport, “Priority oriented channel access for cellular systems serving vehicular and portable radio telephones,” IEE
Proc.-I, vol. 136, no. 5, pp. 339–346, 1989.
[7] C. Jedrzycki and V. C. M. Leung, “Probability distribution of channel holding time in cellular telephony systems,” in Proc. IEEE Vehicular
Technology Conf., 1996, pp. 247–251.
[8] Y.-B. Lin, “Impact of PCS handoff response time,” IEEE Commun. Lett., vol. 1, no. 6, pp. 160–162, Nov. 1997.
[9] , “Performance modeling for mobile telephone networks,” IEEE
Network Mag., vol. 11, no. 6, pp. 63–68, Nov./Dec. 1997.
[10] Y.-B. Lin and I. Chlamtac, Wireless and Mobile Network
Architec-tures. New York: Wiley, 2001.
[11] Y.-B. Lin and V. W. Mak, “Eliminating the boundary effect of a large-scale personal communication service network simulation,” ACM Trans.
[12] Y.-B. Lin, S. Mohan, and A. Noerpel, “Queueing priority channel as-signment strategies for PCS hand-off and initial access,” IEEE Trans.
Veh. Technol., vol. 43, no. 3, pp. 704–712, Aug. 1994.
[13] P. V. Orlik and S. S. Rappaport, “On the handoff arrival process in cellular communications,” ACM/Baltzer Wireless Networks, vol. 7, pp. 147–157, Mar./Apr. 2001.
[14] , “Traffic performance and mobility modeling of cellular commu-nications with mixed platforms and highly variable mobilities,” Proc.
IEEE, vol. 86, no. 7, pp. 1464–1479, July 1998.
[15] J. M. Peha and A. Sutivong, “Admission control algorithms for cel-lular systems,” ACM/Baltzer Wireless Networks, vol. 7, pp. 117–125, Mar./Apr. 2001.
[16] S. Ross, A First Course in Probability. Englewood Cliffs, NJ: Prentice-Hall, 1998.
[17] A. Xhafa and O. K. Tonguz, “Dynamic priority queueing of handoff requests in PCS,” in Proc. IEEE Int. Conf. Communications, vol. 2, 2001, pp. 341–345.
Hui-Nien Hung received the B.S.Math. degree
from National Taiwan University, Taiwan, in 1989, the M.S.Math. degree from National Tsin-Hua University, 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, Taiwan. His research interests include applied prob-ability, financial calculus, bioinformatics, statistical inference, statistical computing, and industrial statis-tics.
Pei-Chun Lee received the B.S.CSIE and M.S.CSIE
degrees in 1998 and 2000, respectively, from Na-tional Chiao Tung University, Hsinchu, Taiwan, R.O.C., where she is currently working toward the Ph.D. degree.
Her research interests include the design and anal-ysis of personal communications services networks, computer telephony integration, mobile computing, and performance modeling.
Yi-Bing Lin (M’95–SM’95–F’03) received the
B.S.E.E. degree from National Cheng Kung Univer-sity, Tainan, Taiwan, R.O.C., in 1983, and the Ph.D. degree in computer science from the University of Washington, Seattle, in 1990.
He is currently a Chair Professor with National Chiao Tung University, Hsinchu, Taiwan, R.O.C.
Dr. Lin is an ACM Fellow.
Nan-Fu Peng received the B.S. degree in the applied
mathematics from National Chiao Tung University, Hsinchu, Taiwan, in 1981, and the Ph.D. degree in statistics from The Ohio State University, Columbus, in 1989.
He is currently an Associate Professor with the In-stitute of Statistics, National Chiao Tung University. His research interests include Markov chains, popu-lation dynamics, and queueing theory.
