Modeling and analysis of hierarchical cellular ... - Semantic Scholar
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 51, NO. 6, NOVEMBER 2002
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Modeling and Analysis of Hierarchical Cellular Networks With General Distributions of Call and Cell Residence Times Kunmin Yeo and Chi-Hyuck Jun
New (handoff) call arrival rate to a microcell from slow MSs. Take-back occurrence rate to a microcell. Overall call arrival rate to a macrocell. New (handoff) call arrival rate to a macrocell from fast MSs. Overflow occurrence rate to a macrocell. Channel occupancy time of new (handoff) calls from slow MSs in a microcell. Channel occupancy time of new (handoff) calls from fast MSs in a macrocell. Channel occupancy time of overflow calls from slow MSs in a macrocell. Blocking probability of new calls from slow (fast) MSs in a microcell (macrocell). Handoff failure probability of handoff calls from slow (fast) MSs in a microcell (macrocell). Dropping probability of new (handoff) calls from slow MSs. Take-back failure probability from a macrocell to a microcell. Overflow failure probability from a microcell to a macrocell.
Abstract—We present an analytic model for the performance evaluation of hierarchical cellular systems, which can provide multiple routes for calls through overflow from one cell layer to another. Our model allows the case where both the call time and the cell residence time are generally distributed. Based on the characterization of the call time by a hyper-Erlang distribution, the Laplace transform of channel occupancy time distribution for each call type (new call, handoff call, and overflow call) is derived as a function of the Laplace transform of cell residence time. In particular, overflow calls are modeled by using a renewal process. Performance measures are derived based on the product form solution of a loss system with capacity limitation. Numerical results show that the distribution type of call time and/or cell residence time has influence on the performance measure and that the exponential case may underestimate the system performance. Index Terms—Call time, cell residence time, channel occupancy time, hierarchical cellular system, overflow call.
NOMENCLATURE Number of microcells per macrocell. Call time with mean , probability density , and cumulative density funcfunction (pdf) . tion (cdf) Residual call time from an arbitrary epoch with pdf and cdf . Cell residence time of slow mobile stations (MSs) in and cdf . a microcell with pdf Residual cell residence time of slow MSs in a miand crocell from an arbitrary epoch with pdf . cdf Cell residence time of fast MSs in a macrocell with and cdf . pdf Residual cell residence time of fast MSs in a macroand cdf cell from an arbitrary epoch with pdf . Cell residence time of slow MSs in a macrocell. Residual cell residence time of slow MSs in a macrocell from an arbitrary epoch. Overall call arrival rate to a microcell from slow MSs. Manuscript received February 28, 2000; revised August 8, 2001. This work was supported by KOSEF through the Statistical Research Center for Complex Systems, Seoul National University. K. Yeo is with the Mobile Telecommunication Research Laboratory, Electronics and Telecommunications Research Institute, 305-350 Daejeon, Korea. C.-H. Jun is with the Department of Industrial Engineering, Pohang University of Science and Technology, 790-784 Pohang, Korea. Digital Object Identifier 10.1109/TVT.2002.804847
I. INTRODUCTION
A
S ONE solution to the need to increase the capacity of cellular mobile networks, a network architecture with a hierarchical layer of cells has been considered [1]–[6]. In a hierarchical cellular network, cells of different sizes in a multilayered structure provide multiple routes for calls that may be otherwise dropped or rejected to seize traffic channels when a part of the network is congested. Hence, it can provide higher traffic capacity by maintaining alternate routing through overflow and, in consequence, may reduce the call-dropping probability. Another benefit of the layered network is to provide service coverage of mobile stations (MSs) of various mobility classes. Using a speed-sensitive cell-selection mechanism, cell selection is performed by directing MSs to an appropriate cell layer according to their speed [1]. For example, in a two-layer (e.g., microcell and macrocell) hierarchical system, slow MSs request traffic channels to target microcells, while macrocells can be the first candidate for fast MSs. In the case of a speed-insensitive selection mechanism, originating calls are assigned to a default cell layer, which is in most cases the lowest (e.g., microcell) layer [2], [3]. Traffic channels of macrocells may be
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used for calls from slow MSs when there are not available channels in microcells to serve the calls. That is, a new or handoff call is overflowed from a speed-dependent preferred cell layer to an upper cell layer if it cannot be served by an idle traffic channel in the preferred layer [4], [5]. A number of publications related to the performance evaluation of hierarchical cellular systems assume that, for the sake of convenience and tractability, call time and/or cell residence time are exponentially distributed and that, in consequence, the channel occupancy time is exponentially distributed [1]–[6]. However, some results have been presented showing that this assumption, especially concerning the cell residence time, is not realistic. Zonoozi and Dassanayake [7] show that the cell residence time can be described by a generalized gamma distribution. Moreover, as pointed by Guerin [8], for a very low rate of change of direction, the channel occupancy time distribution shows some disagreement with the exponential model. As related to the call time distribution, it is found by Bolotin [9] that channel throughput drops significantly more under an exponential call time distribution model than under the actual measured call time distribution. Some experiments with field data show that channel occupancy times for cellular mobile systems are not exponentially distributed [10], [11]. Barcelo and Jordan [10] conclude that channel occupancy times and related time variables are not exponentially distributed from a series of tests for mobile radio cellular systems. They also show that a lognormal distribution or a mixture of Erlang distributions gives a better statistical fitting to the field data. Jedrzycki and Leung [11] also observe that a lognormal distribution is a better candidate for the channel occupancy time. On the other hand, nonexponential models mainly focus on the approximation of the cell residence time distribution [12], [13]. Orlik and Rappaport [12] model the cell residence time as the sum of hype exponential (SOHYP) random variables, which has the advantage of preservation of the Markovian property and thus facilitates construction of Markov chain. As pointed out in [13], however, the SOHYP model contains too many parameters to be identified; hence the statistical fitting is very complicated. Fang and Chlamtac [13] give a mobility model where the cell residence time is characterized by a hyper-Erlang distribution. They obtain the channel occupancy time distribution assuming that the call time is still exponentially distributed. Moreover, nonhierarchical systems are considered in [12] and [13]. In hierarchical systems, overflow calls may be difficult to characterize in the nonexponential case of the call time and the cell residence time. Hence, a more complicated modeling approach should be needed in order to deal with the general case. This paper proposes a new analytical model for the performance evaluation of a hierarchical cellular system exploiting speed-sensitive cell selection. Our model allows the call time and the cell residence time to be generally distributed. The call time is characterized by a hyper-Erlang distribution, which is thought to be a good approximate distribution for many nonnegative random variables of continuous type [13], [14]. The Laplace transform of the channel occupancy time distribution for each type of call (i.e., new call, handoff call, and overflow call) is derived as a function of the Laplace transform of the cell residence time distribution. In particular, we model the channel occupancy
Fig. 1. A hierarchical cellular network and ongoing calls (dot arrow: overflow calls from slow MS; solid arrow: calls from slow MS that are first directed to microcell; bold solid arrow: calls from fast MS that are first directed to macrocell).
time of overflow calls using a renewal process. Moreover, a call-admission scheme is proposed, which can effectively deal with heterogeneous and nonexponential channel occupancy time. A multidimensional birth–death process proposed by Orlik and Rappaport [12] may be applied to hierarchical systems, but it will require tremendous state space, particularly when the number of channels in each macrocell is large. Since our model gives the Laplace transform of the channel occupancy time distribution for each call type, this approach also would be useful in some other queueing models where new or handoff calls currently not served should wait in a buffer. The rest of this paper is organized as follows. Section II provides the model description and gives the stationary probability of the number of calls in each cell layer. In Section III, we derive the Laplace transform of channel occupancy time distribution for each call type. In Section IV, the arrival rates of handoff calls, take-back calls, and overflow calls are obtained assuming that all the cells in each layer are statistically homogeneous. The quantities obtained in Sections III and IV will be the key input parameters in the stationary probability in Section II. Section V contains the performance measures such as the blocking probability and the handoff failure probability. Numerical results and conclusions are presented in Sections VI and VII, respectively. II. MODEL DESCRIPTION In this section, we describe the system model and present the stationary probability of the number of each ongoing call type in a microcell or in a macrocell. Let us consider a two-layer hierarchical cellular network where each macrocell overlays contiguous microcells, as shown in Fig. 1. These microcells constitute the lower layer of microcells are overlaid by the two-layer hierarchy. Every a macrocell, and these overlaying macrocells form the upper layer. We assume that a call from slow MSs is first directed to a microcell, while a macrocell is the initial target for a call from fast MSs. The speed estimation of MSs can be achieved using the method of [15]. We further assume that an MS does not change its speed during a call session. If a call from slow MSs is blocked in a target microcell, it will be overflowed to the macrocell overlaying the microcell when the macrocell can
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afford to serve the call—both new and handoff calls have the same opportunity for overflow. The overflow from macrocell to microcell is not allowed, because relatively frequent handoffs across microcells are expected, and in consequence, the load of signal processing may be significant. The successfully overflowed calls will try to take back to a microcell. Two policies may be possible [1]: 1) take-back as soon as one of traffic channels becomes available in the current microcell and 2) take-back when an ongoing call reaches a border of a microcell. While the first policy may be more desirable with respect to channel utilization, the second policy can reduce the load of signal processing of monitoring the state of channels and can simplify the modeling of channel occupancy time of overflow calls. Hence, we adopt the second policy. We assume that combined traffic to a microcell arrives according to a Poisson process with rate and that incoming calls will be new calls, handoff calls, and take-back calls with rates , , and , respectively ( ). For a macrocell, we also assume a Poisson process with rate as the arrival traffic comprises new calls, handoff calls, and overflow , , and , respectively ( calls with rates ). Strictly speaking, the arrival processes may not be exactly Poisson processes because some functional dependencies exist among the call types. However, the Poisson assumption is often adopted in the analysis of queueing networks, and it is reported that the performance is little degraded under this assumption. The functional relations among the arrival rates of these calls will be given in Section IV. The stochastic behavior of one cell (a microcell or a macrocell) can be described by a loss system (i.e., a system with no queue) with multiservers (channels). Since a sudden forced termination during a call session will be more upsetting than a failure to connect a new call, it is desirable to give priority to handoff calls. As one method, the maximum number of channels allowed for new calls or overflow calls can be limited to a certain capacity, giving handoff calls more of a chance to seize channels. Hence, our system model becomes a loss system with channel-capacity limitation with respect to the type of calls. Such a system is known to have an insensitivity property, i.e., the stationary probability depends on the service time (channel occupancy time) distribution only through its mean (see [14] and [16]). Assume that and channels are available in each microcell be the maximum number and macrocell, respectively. Let ) and of channels allowed for new calls in a microcell ( and be the maximum number of channels allowed let for new calls and overflow calls in a macrocell, respectively , ). Then, call-admission policies in each ( microcell and macrocell are as follows, respectively. Microcell ongoing 1) A new call is blocked when there are at least new calls or channels are busy. 2) A handoff request is denied when channels are busy. Macrocell ongoing 1) A new call is blocked when there are at least new calls or channels are busy. 2) An overflow request is denied when there are at least ongoing overflow calls or channels are busy. 3) A handoff request is denied when channels are busy.
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Now, let denote the joint stationary probability and the number of handoff that the number of new calls is in a microcell. Since we calls (including take-back calls) is consider the take-back at each border crossing of microcells the channel occupancy time of take-back calls has the same probabilistic characteristic as that of handoff calls. In this scheme, we can treat take-back calls as handoff calls. Hence, from the theory of loss system [14], we have
(1) where
and
is the normalization constant as
Similarly, let denote the joint stationary probability that the numbers of new calls, handoff calls, and overflow calls are , , and , respectively, in a macrocell. Then
(2) where
and
is the normalization constant as
Note that the above system model slightly differs from the generally agreed concept of guard channel scheme (see [17]), in which one cell exclusively reserves a certain number of channels only for handoff calls. The difference lies in the call-admission policy (in particular, for new calls). For example, if we let be the number of guard channels out of a total of channels in a microcell, then in the guard channel scheme, blocking of a new ongoing calls. In fact, the call occurs if there are at least original motivation of this paper is not in the management of channels but in dealing with heterogeneous and nonexponential distributions in the model. Hence, if our scheme can be allowed by networks and implemented as a method of giving a priority to handoff calls, it has the following advantages in modeling aspect. 1) It gives a product-form solution concerning the joint stationary probability of the number of each call type. 2) It allows heterogeneous channel occupancy times to be nonexponentially distributed. 3) It may reduce the computational load—this is mainly due to the product-form solution. On the other hand, in the case of heterogeneous channel occupancy times, it is known that the guard channel scheme does not
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give a product-form solution and that multidimensional Markov chains (two- and three-dimensional Markov chains in each layer of microcells and macrocells, respectively) should be solved to compute the stationary probability, even provided that insensitivity property is assumed in relation to the channel occupancy time distribution. In hierarchical systems, heterogeneous channel occupancy times (for new calls, handoff calls, and overflow calls) should be managed, and moreover, the channel occupancy time of overflow calls does not follow an exponential distribution even when the call time and the microcell residence time are exponentially distributed. Sections III and IV are mainly devoted to the modeling of channel occupancy times and arrival rates for various call types, which will be used as the input parameters in (1) and (2). Also, based on (1) and (2), the performance measures are derived in Section V. III. DISTRIBUTIONS OF CHANNEL OCCUPANCY TIMES In this section, we obtain the Laplace transform of the channel occupancy time distribution for each type of calls: new, handoff, and overflow. The derived Laplace transform will be basically represented as a function of the Laplace transform of cell residence time distribution. As mentioned earlier, the generalization of the call time is based on the characterization by a hyper-Erlang distribution, which is represented as a mixture of Erlang distributions and thus is thought to be a relatively good approximate distribution for many nonnegative random variables of continuous type. Suppose that the call time follows a hyper-Erlang distribution with pdf as
Fig. 2. Channel occupancy times of new and handoff calls in a microcell or macrocell (bold solid line: channel occupancy time in a current cell; dot line: remaining channel occupancy time in adjacent cells; solid line: prior cell residence time at call initiation for new call and prior call time at handoff for handoff calls).
by the equilibrium distribution of ( ). From the relation , where is the pdf of , we can as easily obtain the tail distribution of
(6) and will facilitate The simple form of summation in the computation of complex integrals involved in the derivation of channel occupancy times. The Laplace transform of an arbitrary distribution with pdf is represented as
(3) (7) are positive integers and and are positive where and constants. The mean value of is easily calculated as (4) The hyper-Erlang distribution is also preferable from the computational point of view since its tail distribution has a simple form of summation as
(5) On the other hand, we can model the residual (or remaining) call time of handoff calls at an arbitrary handoff epoch as the excess of the call time. Note from renewal theory [18] that at an arbitrary epoch, the remaining renewal time follows the excess of the interarrival time of renewals, whose distribution can be represented by the equilibrium distribution of the interarrival times. Hence, the remaining call time ( ) at the point that an MS during call session enters a target cell is thought to have the same probabilistic property of the excess of the call time. This observation makes it possible to model the distribution of
In the rest of this paper, we will use an asterisk ( ) to represent the Laplace transform of the corresponding distribution. A. Distributions of Channel Occupancy Times of New and Handoff Calls Now, we derive the Laplace transform of channel occupancy time distribution for new and handoff calls. The method of Laplace transform provides a systemic approach to quantify the expected channel occupancy time. As illustrated in Fig. 2, for the new call, the channel occupancy time becomes either the call time (the bold solid arrow) if it is completed in the current cell where it is originated or the remaining cell residence time (the bold solid line) if it continues its call session to one of adjacent cells requesting a handoff. This case is due to the fact that at call origination, an MS has already consumed its part of potential cell residence time (the solid line for the new call). For the handoff call, the channel occupancy time becomes either the cell residence time (the bold solid line) if it continues its call session to one of adjacent cells requesting another handoff or the remaining call time (the bold solid arrow) if it is completed in the current cell because the handoff call has consumed some amount of its potential call time at the epoch of handoff (the solid line for the handoff call). Since a call from
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slow MSs is first assigned to a microcell, the channel occupancy times of new and handoff calls in microcell are represented as (8) (9) respectively. Similarly, since a call from fast MSs is initially directed to a macrocell, the channel occupancy times of new and handoff calls in macrocell are
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where the last equation follows since the Laplace transform of is and is defined as the th with alternating sign derivative of the Laplace transform of term. In (16), the interchange of integral and summation is justified since all terms are nonnegative (this property is satisfied in the rest of this paper). An easy computational formula for is given as (65) in Appendix A. Hence, by the Laplace can be obtained as transform,
(10) (11) and follow the equilibrium distrirespectively. Note that and with the following pdfs, respectively: butions of (17) and Since the pdf of
(12)
is (13)
then the Laplace transform is . Since
(14)
consists of two terms. The From (17), we can see that is just according to (4). Note first term can be directly derived from . that , which correHence, the second term will be sponds to the conditional expectation of the excess of the call time. This quantity perfectly matches the time duration depicted by the dotted line in Fig. 2. is Similarly, since the pdf of , we have the Laplace transform as . Note that
(15) then the Laplace transform of the channel occupancy time distribution for new calls is obtained by (16), shown at the bottom of the page, where (18) (19)
(16)
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Hence, the Laplace transform of the channel occupancy time distribution for handoff calls is obtained by
(20) where
successfully take back after a certain number of take-back failures; or it completes its potential call time without a take-back success using only a channel of a macrocell until the end of its (residual) lifetime—in this case the call is only served by the macrocell to which it is overflowed. On the other hand, the overflow call may continue its call session to another adjacent macrocell not being served by a microcell overlaid by the current macrocell: it fails to take back during the remaining cell residence time in the current macrocell. The various types of overflow calls are illustrated in Fig. 1. First, let us consider a new call that has been overflowed to a macrocell. Suppose that the call will successfully take back to a microcell at th trial after ( 1) consecutive failures. Let ( ) be independent and identically distributed with . with pdf as Now, let us define a random variable follows: .
Hence,
means the duration from the overflow to the take-back Then, success at th trial. The Laplace transform of the distribution of is obtained by
can be obtained as
(26) (21) and , Using the same technique, we can derive and are derived, respectively. Let from which (22)
is formally interpreted as the Based on renewal theory [19], th renewal time of a renewal process, where the first interarand the next subsequent interarrival rival time follows . Particularly, this process is called a delayed times follow renewal process. Next, consider a handoff call that has failed in handoff and has been overflowed to a macrocell. Let us define a random variable with pdf as follows:
(23) (27)
Then, we obtain (24)
is interpreted as the time period from the initiation of Then, overflow to the time when the overflow handoff call will successfully take back to a microcell at th take-back trial after ( 1) consecutive failures. Now, the Laplace transform of the is given by distribution of (28)
(25) B. Distribution of Channel Occupancy Time of Overflow Calls Since slow MSs may traverse a number of microcells during overflow call session, the channel occupancy time of overflow calls ( ) is closely related to the cell residence times in microcell ( ) and macrocell ( ) and the take-back failure probability ( ). Hence, the derivation of its Laplace transform ) needs a more complicated approach. ( Suppose that a new or handoff call is overflowed to a macrocell. The call may be completed in the current microcell; it may
As illustrated in Fig. 1, there exist six types of overflow calls (types 1–6 in the figure): types 1–3 correspond to new calls and types 4–6 handoff calls. Type 1 (or 4) is an overflow new (or handoff) call that will complete its call time in a current microcell. Type 2 (or 5) is an overflow new (or handoff) call that will end its call time in another microcell after a certain number of consecutive take-back failures. Type 3 (or 6) is an overflow new (or handoff) call that will successfully take back to a microcell ) consecutive failures. at th take-back trial after ( First, consider overflow new calls (i.e., types 1–3). Let us construct the following probability intensity functions: (29)
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(30)
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is always less than . That is, , otherwise. Therefore
if
(39)
(31) is related to the channel occupancy time The first function of a type 1 call and is formally interpreted as . The second function is concerned with a type 2 call, representing the probability intensity that an overflow new call is eventually completed after a series of take-back is the failures. Note that the quantity and on condiprobability that the call time is between , i.e., . Finally, the tion that is related to the channel occupancy time of third function a type 3 call and is obtained from , which means the probability intensity that an overflow new call will successfully take back to a microcell after an arbitrary number of consecutive take-back failures. Hence, if the channel occupancy time of overflow new we denote by is represented by calls, then the pdf of
and
is represented as a series of s. If we can apNote that as a hyper-Erlang distribution and develop the proximate Laplace transform of the distribution of , we can easily deusing the methodology developed rive the th moment of follows a hyper-Erlang distribuearlier. Let us assume that tion with pdf (40) where and are positive integers and and are positive constants. Then, from (9) and (20), we obtain the Laplace transas form of the distribution of
(32) , which satisfies Note that the condition of the probability density function. Next, let be the channel occupancy time of overflow handoff calls (i.e., types 4–6). Similarly as in (29)–(31), let us construct the following intensity functions:
(41) where (42)
(33)
Hence
(34) (35) Then, the pdf of
is represented by (36)
. Also, it is satisfied that the virtual Before we proceed further, let us define by channel occupancy time of overflow calls with pdf (37) where represents the fraction of new calls to all the calls requesting overflows and is given by (38) The term “virtual” is used because is not truly . That is, the includes no consideration of , the cell residence quantity time of slow MSs in a macrocell. The quantity may be greater . This is the case when a slow MS with an ongoing than overflow call goes to an adjacent macrocell. However, note that
(43) To be complete, the above equation requires the th deriva[equivalently, in (42)], which is derived tive of in Appendix A embedded with a tractable algorithm. Further, Appendix B describes how to obtain the parameters of the disusing method of moments. Finally, the expected tribution of values obtained in this section are used as input parameters of (1) and (2) in Section II. IV. ARRIVAL TRAFFIC OF EACH TYPE OF CALLS In this section, we derive the arrival rates of handoff, overflow, and take-back calls in each layer of cells. It is assumed that all the cells in each cell layer have statistically the same characteristic of mobility in steady state. First, consider a microcell. The overall arrival rate into a microcell is represented as the sum of the arrival rates of new calls (which is given), handoff calls, and take-back calls ( ). Let us denote by the probability that a new call originated from slow MSs in a microcell will continue its call session ) and by the to an adjacent microcell (i.e., probability that a handoff call from slow MSs will continue its
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call session to an adjacent microcell (i.e., Then, taking a hyper-Erlang distribution for the call time in Section III gives
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). as
call will continue its call session to an adjacent macrocell (i.e., ). That is (51)
(44) (52) (45) Consider an arbitrary microcell surrounded by a number of adjacent microcells. Note that handoff requests into the microcell are from one of the following three call types: 1) new calls not blocked in adjacent microcells, continuing their call sessions to the microcell, 2) handoff calls successfully handed off into adjacent microcells and continuing their call sessions to the microcell, and 3) take-back calls that have successfully taken back into adjacent microcells and continue their call sessions to the microcell. Hence, the following equation should be satisfied in steady state: (46) Similarly, take-back requests into the target microcell are originated from the following three overflow call types in adjacent microcells: 1) overflow new calls, 2) overflow handoff calls, and 3) take-back calls that have not successfully taken back into adjacent microcells at the previous trial. Hence, in steady state, we have (47) Equation (47) follows since the take-back request occurs when an overflow call reaches a border of a microcell. Solving the system of (46) and (47), we obtain
Then, with a similar context as in (46) and (47), the following recursive equations should be satisfied in steady state: (53) Solving (53), we obtain (54) Finally, the arrival rates obtained in this section will be used as input parameters of (1) and (2) in Section II. V. PERFORMANCE MEASURES The blocking of a new call from slow MSs in a microcell occurs when the number of channels occupied by new calls in or there is not an available channel the microcell is equal to in the microcell. Hence (55) A handoff call from slow MSs cannot be served if all channels are occupied in a target microcell. The handoff failure probability of a handoff call being served is formally defined as the fraction of handoff attempts that are denied due to lack of channels. Hence (56)
(48)
(49) . where In a macrocell, the overall arrival rate comprises the arrival rates of new calls ( ), handoff calls ( ), and overflow calls and are from fast MSs, while is from ( ). Note that is independent from and slow MSs; and that the quantity . First, the arrival rate of overflow calls is simply obtained by (50) is the number of microcells in a macrocell. Next, where , let us denote by in order to derive the expression of the probability that a new call originated in a macrocell will continue its call session to an adjacent macrocell (i.e., ) and by the probability that a handoff
A take-back failure occurs if all the channels in a target microcell assigned to take-back requests are occupied. The take-back failure probability is formally defined as the fraction of take-back attempts that are denied due to lack of channels. If we give the same priority to both handoff and take-back . requests, then An overflow request is denied when the number of channels or there occupied by overflow calls in a target macrocell is is not an available channel in the macrocell. Hence (57) Because we assume that the arrival process is Poisson, by the property Poisson arrivals see time average (PASTA), the dropping probability of a new call from slow MSs is (58) Similarly, the dropping probability of a handoff call from slow MSs is (59)
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On the other hand, the blocking of a new call from fast MSs occurs when the number of channels occupied by new calls in a or there is not an available channel in macrocell is equal to the macrocell. Hence, the blocking probability (or equivalently dropping probability) is (60) Similarly, for a handoff call from fast MSs, the handoff failure probability (or equivalently dropping probability) is (61) Note that the performance measures (or equivalently, the joint stationary probabilities) are related to the arrival rates, which have also functional relationships with the performance measures (as shown in Section IV). Note also that the formula of the channel occupancy time of overflow calls has a functional form of the take-back failure probability (as shown in Section III). Hence, we need a recursive method to compute the performance measures. Basically, we use the iterative method proposed by [19]. If the size of or increases, the computation of and may be difficult. When this is the case, the method given in [20], which is devoted to the computation of asymptotic limiting distribution, may be used. All the numerical examples in this paper have convergent values of the performance measures. VI. NUMERICAL RESULTS In this section, we present numerical examples for a two-layer system with two populations of MSs with low and high mobilities. We assume that the total traffic of new calls follows a Poisson process with rate . Let be the fraction of new calls and . from slow MSs. Then Through the examples, we consider four identical microcells ). The mean cell resioverlaid by a single macrocell ( dence time of each type of MS is obtained as follows [21]: and
(62)
and are the radii of microcell and macrocell, rewhere spectively, and and are the average speeds of slow and fast m, m, MSs, respectively. We assume that km h, and km h. We consider the following , , , , configuration of channels: . Basically, we chose to be 0.5, but other values are and also considered to see the effect of the mobility ratio. The call , time ( ) is assumed to have the following parameters: , , , , and (mean min and variance ). The cell residence times ( and ) are assumed to have gamma distribution with shape and variance parameter and scale parameter (mean ). The associated parameters of are and (mean min and variance ) and the parameters of are and (mean min and variance ). The numerical examples consist of three parts: 1) the comparison of a nonexponential case of all the related random variables
Fig. 3. The performance measures in the exponential model versus the nonexponential model. Circle: exponential case; diamond: nonexponential case; horizontal axis: new call arrival rate ( calls per min); vertical axis: performance measures [p in (a), p in (b), p in (c), p in (d), and p in (e)].
1
with an exponential one, 2) the effect of the maximum number of channels allowed for overflows calls ( ), and 3) the effect of the mobility ratio ( ). First, Fig. 3(a)–(e) shows the difference in various performance measures between our case and the exponential case having the same mean. To give a consistent comparison, the parameters of corresponding time variables (i.e., the call time and the cell residence time) are chosen to have the same mean level with each other. From these figures, we see that the performance may be affected by the distributions of the call and cell residence times, even when the same mean is assumed. and have different This is because the quantities values according to the corresponding distributions. That is, both quantities affect the expected channel occupancy times and the arrival rates, which are directly related to the performance measures. In all figures, the performance measures considered , , and ) are higher for the exponential here ( , , case than for the nonexponential case of hyper-Erlang call time and gamma cell residence time distributions. This phenomenon can be explained by the memoryless property of exponential . In distributions. Note that for new calls, the exponential case of , the residual cell residence time has the same probabilistic characteristic with . In other
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Fig. 4. The effect of the maximum number of channels allowed for overflow calls C . Diamond: C ; circle: C ; triangle: C ; horizon axis: new call arrival rate ( calls per min); vertical axis: performance measures [p in (a), p in (b), p in (c), p in (d), and p in (e)].
1
=3
=4
=5
words, the exponential distribution has no information on the time period of the solid line for new calls in Fig. 2. Hence, . In general, the exponential model gives has a probabilistically greater value than . however, Hence, the channel occupancy time has a tendency to increase in the exponential case. This fact gives rise to higher values of the performance measures. Handoff calls are also explained by the same reasoning. Therefore, the assumption of exponential distribution for both the call time and the cell residence time may underestimate the system performance as compared to the nonexponential case. The maximum number of channels allowed for overflow calls may influence the performance in both microcell and macrocell. increases, the performance in miFig. 4(a)–(e) shows that as crocell becomes better, while in macrocell the dropping probabilities of new and handoff calls increase. This observation corresponds to our intuition. Using this analysis, we can determine in order to balance the traffic load with the proper value of the desired quality of service. Fig. 5(a)–(e) shows the plots of the performance measures and respectively. This exin the cases of ample seems to show that as the ratio of MSs with low mobility increases, the performance in the layer of microcells becomes
= 04 1
= 05
Fig. 5. The effect of mobility ratio . Diamond: : ; circle: : ; triangle: : ; horizon axis: new call arrival rate ( calls per min); vertical axis: performance measures [p in (a), p in (b), p in (c), p in (d), and p in (e)].
=06
worse, while a better performance results in that of macrocells. However, this may not be always the case. For a given level of , as increases (equivalently, increases), more frequent blocking and handoff failure are expected—this increases the portion of the channels of a macrocell occupied by the overflow calls from the layer of microcells, degrading the performance of the layer of macrocell. Hence, there exists a tradeoff point with and . respect to VII. CONCLUSIONS We developed an analytic model for the performance evaluation of a hierarchical cellular system introducing general distribution for call duration and cell residence time. By means of the characterization of the call time by hyper-Erlang distribution, the Laplace transform of the channel occupancy time for each call type is obtained as a function of the Laplace transform of cell residence time. This methodology gives more flexibility in describing the cell residence times, which may have various probabilistic characteristics depending upon different mobility environments. In particular, overflow calls are modeled by exploiting a well-defined renewal process, an approach that captures the inherent overflow from a lower layer of cells to an upper layer of cells and quantifies the channel occupancy time
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of overflow calls in a tractable manner. Not addressed in this paper, the overflow from a macrocell layer to microcell layer can be treated by the proposed methodology. The developed Laplace transforms are thought to be also useful in handoff call queueing models where higher moments may be needed. The computational load is thought to be relatively low, and only a simple iterative method is sufficient to obtain the performance measures. Numerical results show that the distribution types of both the call time and the cell residence time have an obvious influence on the performance. This fact shows that the assumption of some specific distributions as the call and cell residence times may lead to over- or underestimation of the real performance. In particular, the exponential model has a worse performance than the nonexponential one. Although we have considered a two-layer hierarchical system, the approach can be extended to a multilayer hierarchical system with some modification.
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Hence
APPENDIX A DERIVATION OF If we let
for
, then
From (14) where
is defined as
(63)
With the interchange of integral and summation
where the term
An algorithm for Similarly
is given at the end of Appendix A.
is changed to The similar technique is applied in order to obtain and . First, from (19), we obtain
,
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With the interchange of integral and summation,
where the term
is changed to
Hence
respectively. The above . tation of where
is defined as A. Algorithm for the Computation of
(64) An algorithm for Finally
completes the compu-
and
Now, we give an algorithm to compute in (63) and in (64). For two arbitrary differentiable functions and for , it is satisfied that
is given at the end of Appendix A. where the subscript ( ) means the th derivative of the corresponding function. Hence
Furthermore, by a simple manipulation, we obtain the th as follows: derivative of
where the first derivative is computed as
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and the second one is computed recursively as
Similarly
Hence, for a sufficient large number and as follows:
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For a fixed value of , nonlinear programming may be . For needed to determine the values , , and a better approximation, more equations may be needed. Consider an Erlang distribution with shape parameter and scale pa. rameter Then, we obtain
, we can compute
where
means the closest integer to . From [21] and
Note that the quantities converge to zero for a given as in (16). used to compute
and increases. Finally, (65) is
APPENDIX B CHARACTERIZATION OF THE DISTRIBUTION OF If we define as the probability that a slow MS will traverse microcells overlaid by a macrocell ( ), then the can be represented as follows: pdf of
If we choose
then represents a Poisson probability when truncating with mean , representing the mean number of microcells that a slow MS traverses in a macrocell. Generally, we see have a tendency to increase according that the variance of to its mean level. Hence, the Poisson distribution can be a cansince the variance grows as the didate for the distribution of mean increases. Hence
Since
from (40) the first two moments are
and
where and denote the radii of a macrocell and a microcell, respectively. Hence, can be obtained from the following relation:
REFERENCES [1] B. Jabbari and W. F. Fuhrmann, “Teletraffic modeling and analysis of flexible hierarchical cellular networks with speed-sensitive handoff strategy,” IEEE J. Select. Areas Commun., vol. 15, pp. 1539–1548, Oct. 1997. [2] W. M. Jolley and R. E. Warfield, “Modeling and analysis of layered cellular mobile networks,” in Proc. 13th Int. Teletraffic Congr., Copenhagen, Denmark, June 1991, pp. 161–166. [3] X. Lagrange and P. Godlewski, “Teletraffic analysis of a hierarchical cellular network,” in Proc. IEEE Veh. Technol. Conf. (VTC ’95), 1995, pp. 882–886. [4] S. S. Rappaport and L. R. Hu, “Microcellular communications systems with hierarchical macro overlays: Traffic performance models and analysis,” Proc. IEEE, vol. 82, pp. 1383–1397, Sept. 1994. [5] L. R. Hu and S. S. Rappaport, “Personal communications systems using multiple hierarchical cellular overlays,” in Proc. IEEE Int. Conf. Universal Personal Commun. (ICUPC ’94), San Diego, CA, Sept. 1994, pp. 397–401. [6] P. Fitzpatrick, C. S. Lee, and B. Warfield, “Teletraffic performance of mobile radio networks with hierarchical cells and overflow,” IEEE J. Select. Areas Commun., vol. 15, pp. 1549–1557, Oct. 1997. [7] M. M. Zonoozi and P. Dassanayake, “User mobility modeling and characterization of mobility patterns,” IEEE J. Select. Areas Commun., vol. 15, pp. 1239–1252, Sept. 1997. [8] R. A. Guerin, “Channel occupancy time distribution in a cellular radio system,” IEEE Trans. Veh. Technol., vol. VT-35, pp. 89–99, Aug. 1987. [9] V. A. Bolotin, “Modeling call holding time distributions for CCS network design and performance analysis,” IEEE J. Select. Areas Commun., vol. 12, pp. 433–438, Apr. 1994. [10] F. Barcelo and J. Jordan, “Channel holding time distribution in cellular telephony,” in Proc. 9th Int. Conf. Wireless Commun. (Wireless ’97), vol. 1, Alta., Canada, July 1997, pp. 125–134. [11] C. Jedrzycki and V. C. M. Leung, “Probability distributions of channel holding time in cellular telephony systems,” in Proc. IEEE Veh. Technol. Conf. (VTC ’96), Atlanta, GA, May 1996, pp. 247–251. [12] P. V. Orlik and S. S. Rappaport, “A model for teletraffic performance and channel holding time characterization in wireless cellular communication with general session and dwell time distributions,” IEEE J. Select. Areas Commun., vol. 16, pp. 788–803, June 1998. [13] Y. Fang and I. Chlamtac, “Teletraffic analysis and mobility modeling of PCS networks,” IEEE Trans. Commun., vol. 47, pp. 1062–1072, July 1999. [14] F. P. Kelly, Reversibility and Stochastic Networks. New York: Wiley, 1979. [15] K. Kawabata, T. Nakamura, and E. Fukuda, “Estimating velocity using diversity reception,” in Proc. IEEE Veh. Technol. Conf. (VTC ’94), 1994, pp. 371–374.
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[16] R. W. Wolff, Stochastic Modeling and the Theory of Queues: Prentice Hall International, Inc., 1989. [17] S. Tekinay and B. Jabbari, “Handover and channel assignment in mobile cellular networks,” IEEE Commun. Mag., vol. 29, no. 11, pp. 42–46, 1991. [18] S. M. Ross, Stochastic Processes, 2nd ed. New York: Wiley, 1996. [19] Y. B. Lin, S. Mohan, and A. Noerpel, “Queueing priority channel assignment strategies for handoff and initial access for a PCS network,” IEEE Trans. Veh. Technol., vol. 43, pp. 704–712, Aug. 1994. [20] P. Gazdzicki, I. Lambadaris, and R. R. Mazumdar, “Blocking probabilities for large multirate Erlang loss systems,” Adv. Appl. Prob., vol. 25, pp. 997–1009, 1993. [21] B. Jabbari, “Teletraffic aspects of evolving and next generation wireless communication networks,” IEEE Personal Commun., vol. 3, pp. 4–9, Dec. 1996.
Kunmin Yeo was born in Daegu, Korea, in 1969. He received the B.S., M.S., and Ph.D. degrees in industrial engineering from the Pohang University of Science and Technology (POSTECH), Pohang, Korea, in 1995, 1997, and 2001, respectively. Since 2001, he has been a senior member of engineering staff in the Mobile Telecommunication Research Laboratory, Electronics and Telecommunications Research Institute, Daejeon, Korea. His research interests are performance evaluation of mobile telecommunications (in particular, traffic and probability modeling), wireless packet scheduling, and queueing theory.
Chi-Hyuck Jun was born in Seoul, Korea, in 1954. He received the B.S. degree in mineral and petroleum engineering from Seoul National University, Seoul, in 1977, the M.S. degree in industrial engineering from Korea Advanced Institute of Science and Technology in 1979, and the Ph.D. degree in operations research from the University of California, Berkeley, in 1986. Since 1987, he has been with the Department of Industrial Engineering, Pohang University of Science and Technology, Pohang, Korea, where he is now a Professor and the Department Head. He is interested in performance analysis of communication and production systems.
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Modeling and Analysis of Hierarchical Cellular Networks With General Distributions of Call and Cell Residence Times Kunmin Yeo and Chi-Hyuck Jun
New (handoff) call arrival rate to a microcell from slow MSs. Take-back occurrence rate to a microcell. Overall call arrival rate to a macrocell. New (handoff) call arrival rate to a macrocell from fast MSs. Overflow occurrence rate to a macrocell. Channel occupancy time of new (handoff) calls from slow MSs in a microcell. Channel occupancy time of new (handoff) calls from fast MSs in a macrocell. Channel occupancy time of overflow calls from slow MSs in a macrocell. Blocking probability of new calls from slow (fast) MSs in a microcell (macrocell). Handoff failure probability of handoff calls from slow (fast) MSs in a microcell (macrocell). Dropping probability of new (handoff) calls from slow MSs. Take-back failure probability from a macrocell to a microcell. Overflow failure probability from a microcell to a macrocell.
Abstract—We present an analytic model for the performance evaluation of hierarchical cellular systems, which can provide multiple routes for calls through overflow from one cell layer to another. Our model allows the case where both the call time and the cell residence time are generally distributed. Based on the characterization of the call time by a hyper-Erlang distribution, the Laplace transform of channel occupancy time distribution for each call type (new call, handoff call, and overflow call) is derived as a function of the Laplace transform of cell residence time. In particular, overflow calls are modeled by using a renewal process. Performance measures are derived based on the product form solution of a loss system with capacity limitation. Numerical results show that the distribution type of call time and/or cell residence time has influence on the performance measure and that the exponential case may underestimate the system performance. Index Terms—Call time, cell residence time, channel occupancy time, hierarchical cellular system, overflow call.
NOMENCLATURE Number of microcells per macrocell. Call time with mean , probability density , and cumulative density funcfunction (pdf) . tion (cdf) Residual call time from an arbitrary epoch with pdf and cdf . Cell residence time of slow mobile stations (MSs) in and cdf . a microcell with pdf Residual cell residence time of slow MSs in a miand crocell from an arbitrary epoch with pdf . cdf Cell residence time of fast MSs in a macrocell with and cdf . pdf Residual cell residence time of fast MSs in a macroand cdf cell from an arbitrary epoch with pdf . Cell residence time of slow MSs in a macrocell. Residual cell residence time of slow MSs in a macrocell from an arbitrary epoch. Overall call arrival rate to a microcell from slow MSs. Manuscript received February 28, 2000; revised August 8, 2001. This work was supported by KOSEF through the Statistical Research Center for Complex Systems, Seoul National University. K. Yeo is with the Mobile Telecommunication Research Laboratory, Electronics and Telecommunications Research Institute, 305-350 Daejeon, Korea. C.-H. Jun is with the Department of Industrial Engineering, Pohang University of Science and Technology, 790-784 Pohang, Korea. Digital Object Identifier 10.1109/TVT.2002.804847
I. INTRODUCTION
A
S ONE solution to the need to increase the capacity of cellular mobile networks, a network architecture with a hierarchical layer of cells has been considered [1]–[6]. In a hierarchical cellular network, cells of different sizes in a multilayered structure provide multiple routes for calls that may be otherwise dropped or rejected to seize traffic channels when a part of the network is congested. Hence, it can provide higher traffic capacity by maintaining alternate routing through overflow and, in consequence, may reduce the call-dropping probability. Another benefit of the layered network is to provide service coverage of mobile stations (MSs) of various mobility classes. Using a speed-sensitive cell-selection mechanism, cell selection is performed by directing MSs to an appropriate cell layer according to their speed [1]. For example, in a two-layer (e.g., microcell and macrocell) hierarchical system, slow MSs request traffic channels to target microcells, while macrocells can be the first candidate for fast MSs. In the case of a speed-insensitive selection mechanism, originating calls are assigned to a default cell layer, which is in most cases the lowest (e.g., microcell) layer [2], [3]. Traffic channels of macrocells may be
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used for calls from slow MSs when there are not available channels in microcells to serve the calls. That is, a new or handoff call is overflowed from a speed-dependent preferred cell layer to an upper cell layer if it cannot be served by an idle traffic channel in the preferred layer [4], [5]. A number of publications related to the performance evaluation of hierarchical cellular systems assume that, for the sake of convenience and tractability, call time and/or cell residence time are exponentially distributed and that, in consequence, the channel occupancy time is exponentially distributed [1]–[6]. However, some results have been presented showing that this assumption, especially concerning the cell residence time, is not realistic. Zonoozi and Dassanayake [7] show that the cell residence time can be described by a generalized gamma distribution. Moreover, as pointed by Guerin [8], for a very low rate of change of direction, the channel occupancy time distribution shows some disagreement with the exponential model. As related to the call time distribution, it is found by Bolotin [9] that channel throughput drops significantly more under an exponential call time distribution model than under the actual measured call time distribution. Some experiments with field data show that channel occupancy times for cellular mobile systems are not exponentially distributed [10], [11]. Barcelo and Jordan [10] conclude that channel occupancy times and related time variables are not exponentially distributed from a series of tests for mobile radio cellular systems. They also show that a lognormal distribution or a mixture of Erlang distributions gives a better statistical fitting to the field data. Jedrzycki and Leung [11] also observe that a lognormal distribution is a better candidate for the channel occupancy time. On the other hand, nonexponential models mainly focus on the approximation of the cell residence time distribution [12], [13]. Orlik and Rappaport [12] model the cell residence time as the sum of hype exponential (SOHYP) random variables, which has the advantage of preservation of the Markovian property and thus facilitates construction of Markov chain. As pointed out in [13], however, the SOHYP model contains too many parameters to be identified; hence the statistical fitting is very complicated. Fang and Chlamtac [13] give a mobility model where the cell residence time is characterized by a hyper-Erlang distribution. They obtain the channel occupancy time distribution assuming that the call time is still exponentially distributed. Moreover, nonhierarchical systems are considered in [12] and [13]. In hierarchical systems, overflow calls may be difficult to characterize in the nonexponential case of the call time and the cell residence time. Hence, a more complicated modeling approach should be needed in order to deal with the general case. This paper proposes a new analytical model for the performance evaluation of a hierarchical cellular system exploiting speed-sensitive cell selection. Our model allows the call time and the cell residence time to be generally distributed. The call time is characterized by a hyper-Erlang distribution, which is thought to be a good approximate distribution for many nonnegative random variables of continuous type [13], [14]. The Laplace transform of the channel occupancy time distribution for each type of call (i.e., new call, handoff call, and overflow call) is derived as a function of the Laplace transform of the cell residence time distribution. In particular, we model the channel occupancy
Fig. 1. A hierarchical cellular network and ongoing calls (dot arrow: overflow calls from slow MS; solid arrow: calls from slow MS that are first directed to microcell; bold solid arrow: calls from fast MS that are first directed to macrocell).
time of overflow calls using a renewal process. Moreover, a call-admission scheme is proposed, which can effectively deal with heterogeneous and nonexponential channel occupancy time. A multidimensional birth–death process proposed by Orlik and Rappaport [12] may be applied to hierarchical systems, but it will require tremendous state space, particularly when the number of channels in each macrocell is large. Since our model gives the Laplace transform of the channel occupancy time distribution for each call type, this approach also would be useful in some other queueing models where new or handoff calls currently not served should wait in a buffer. The rest of this paper is organized as follows. Section II provides the model description and gives the stationary probability of the number of calls in each cell layer. In Section III, we derive the Laplace transform of channel occupancy time distribution for each call type. In Section IV, the arrival rates of handoff calls, take-back calls, and overflow calls are obtained assuming that all the cells in each layer are statistically homogeneous. The quantities obtained in Sections III and IV will be the key input parameters in the stationary probability in Section II. Section V contains the performance measures such as the blocking probability and the handoff failure probability. Numerical results and conclusions are presented in Sections VI and VII, respectively. II. MODEL DESCRIPTION In this section, we describe the system model and present the stationary probability of the number of each ongoing call type in a microcell or in a macrocell. Let us consider a two-layer hierarchical cellular network where each macrocell overlays contiguous microcells, as shown in Fig. 1. These microcells constitute the lower layer of microcells are overlaid by the two-layer hierarchy. Every a macrocell, and these overlaying macrocells form the upper layer. We assume that a call from slow MSs is first directed to a microcell, while a macrocell is the initial target for a call from fast MSs. The speed estimation of MSs can be achieved using the method of [15]. We further assume that an MS does not change its speed during a call session. If a call from slow MSs is blocked in a target microcell, it will be overflowed to the macrocell overlaying the microcell when the macrocell can
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afford to serve the call—both new and handoff calls have the same opportunity for overflow. The overflow from macrocell to microcell is not allowed, because relatively frequent handoffs across microcells are expected, and in consequence, the load of signal processing may be significant. The successfully overflowed calls will try to take back to a microcell. Two policies may be possible [1]: 1) take-back as soon as one of traffic channels becomes available in the current microcell and 2) take-back when an ongoing call reaches a border of a microcell. While the first policy may be more desirable with respect to channel utilization, the second policy can reduce the load of signal processing of monitoring the state of channels and can simplify the modeling of channel occupancy time of overflow calls. Hence, we adopt the second policy. We assume that combined traffic to a microcell arrives according to a Poisson process with rate and that incoming calls will be new calls, handoff calls, and take-back calls with rates , , and , respectively ( ). For a macrocell, we also assume a Poisson process with rate as the arrival traffic comprises new calls, handoff calls, and overflow , , and , respectively ( calls with rates ). Strictly speaking, the arrival processes may not be exactly Poisson processes because some functional dependencies exist among the call types. However, the Poisson assumption is often adopted in the analysis of queueing networks, and it is reported that the performance is little degraded under this assumption. The functional relations among the arrival rates of these calls will be given in Section IV. The stochastic behavior of one cell (a microcell or a macrocell) can be described by a loss system (i.e., a system with no queue) with multiservers (channels). Since a sudden forced termination during a call session will be more upsetting than a failure to connect a new call, it is desirable to give priority to handoff calls. As one method, the maximum number of channels allowed for new calls or overflow calls can be limited to a certain capacity, giving handoff calls more of a chance to seize channels. Hence, our system model becomes a loss system with channel-capacity limitation with respect to the type of calls. Such a system is known to have an insensitivity property, i.e., the stationary probability depends on the service time (channel occupancy time) distribution only through its mean (see [14] and [16]). Assume that and channels are available in each microcell be the maximum number and macrocell, respectively. Let ) and of channels allowed for new calls in a microcell ( and be the maximum number of channels allowed let for new calls and overflow calls in a macrocell, respectively , ). Then, call-admission policies in each ( microcell and macrocell are as follows, respectively. Microcell ongoing 1) A new call is blocked when there are at least new calls or channels are busy. 2) A handoff request is denied when channels are busy. Macrocell ongoing 1) A new call is blocked when there are at least new calls or channels are busy. 2) An overflow request is denied when there are at least ongoing overflow calls or channels are busy. 3) A handoff request is denied when channels are busy.
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Now, let denote the joint stationary probability and the number of handoff that the number of new calls is in a microcell. Since we calls (including take-back calls) is consider the take-back at each border crossing of microcells the channel occupancy time of take-back calls has the same probabilistic characteristic as that of handoff calls. In this scheme, we can treat take-back calls as handoff calls. Hence, from the theory of loss system [14], we have
(1) where
and
is the normalization constant as
Similarly, let denote the joint stationary probability that the numbers of new calls, handoff calls, and overflow calls are , , and , respectively, in a macrocell. Then
(2) where
and
is the normalization constant as
Note that the above system model slightly differs from the generally agreed concept of guard channel scheme (see [17]), in which one cell exclusively reserves a certain number of channels only for handoff calls. The difference lies in the call-admission policy (in particular, for new calls). For example, if we let be the number of guard channels out of a total of channels in a microcell, then in the guard channel scheme, blocking of a new ongoing calls. In fact, the call occurs if there are at least original motivation of this paper is not in the management of channels but in dealing with heterogeneous and nonexponential distributions in the model. Hence, if our scheme can be allowed by networks and implemented as a method of giving a priority to handoff calls, it has the following advantages in modeling aspect. 1) It gives a product-form solution concerning the joint stationary probability of the number of each call type. 2) It allows heterogeneous channel occupancy times to be nonexponentially distributed. 3) It may reduce the computational load—this is mainly due to the product-form solution. On the other hand, in the case of heterogeneous channel occupancy times, it is known that the guard channel scheme does not
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give a product-form solution and that multidimensional Markov chains (two- and three-dimensional Markov chains in each layer of microcells and macrocells, respectively) should be solved to compute the stationary probability, even provided that insensitivity property is assumed in relation to the channel occupancy time distribution. In hierarchical systems, heterogeneous channel occupancy times (for new calls, handoff calls, and overflow calls) should be managed, and moreover, the channel occupancy time of overflow calls does not follow an exponential distribution even when the call time and the microcell residence time are exponentially distributed. Sections III and IV are mainly devoted to the modeling of channel occupancy times and arrival rates for various call types, which will be used as the input parameters in (1) and (2). Also, based on (1) and (2), the performance measures are derived in Section V. III. DISTRIBUTIONS OF CHANNEL OCCUPANCY TIMES In this section, we obtain the Laplace transform of the channel occupancy time distribution for each type of calls: new, handoff, and overflow. The derived Laplace transform will be basically represented as a function of the Laplace transform of cell residence time distribution. As mentioned earlier, the generalization of the call time is based on the characterization by a hyper-Erlang distribution, which is represented as a mixture of Erlang distributions and thus is thought to be a relatively good approximate distribution for many nonnegative random variables of continuous type. Suppose that the call time follows a hyper-Erlang distribution with pdf as
Fig. 2. Channel occupancy times of new and handoff calls in a microcell or macrocell (bold solid line: channel occupancy time in a current cell; dot line: remaining channel occupancy time in adjacent cells; solid line: prior cell residence time at call initiation for new call and prior call time at handoff for handoff calls).
by the equilibrium distribution of ( ). From the relation , where is the pdf of , we can as easily obtain the tail distribution of
(6) and will facilitate The simple form of summation in the computation of complex integrals involved in the derivation of channel occupancy times. The Laplace transform of an arbitrary distribution with pdf is represented as
(3) (7) are positive integers and and are positive where and constants. The mean value of is easily calculated as (4) The hyper-Erlang distribution is also preferable from the computational point of view since its tail distribution has a simple form of summation as
(5) On the other hand, we can model the residual (or remaining) call time of handoff calls at an arbitrary handoff epoch as the excess of the call time. Note from renewal theory [18] that at an arbitrary epoch, the remaining renewal time follows the excess of the interarrival time of renewals, whose distribution can be represented by the equilibrium distribution of the interarrival times. Hence, the remaining call time ( ) at the point that an MS during call session enters a target cell is thought to have the same probabilistic property of the excess of the call time. This observation makes it possible to model the distribution of
In the rest of this paper, we will use an asterisk ( ) to represent the Laplace transform of the corresponding distribution. A. Distributions of Channel Occupancy Times of New and Handoff Calls Now, we derive the Laplace transform of channel occupancy time distribution for new and handoff calls. The method of Laplace transform provides a systemic approach to quantify the expected channel occupancy time. As illustrated in Fig. 2, for the new call, the channel occupancy time becomes either the call time (the bold solid arrow) if it is completed in the current cell where it is originated or the remaining cell residence time (the bold solid line) if it continues its call session to one of adjacent cells requesting a handoff. This case is due to the fact that at call origination, an MS has already consumed its part of potential cell residence time (the solid line for the new call). For the handoff call, the channel occupancy time becomes either the cell residence time (the bold solid line) if it continues its call session to one of adjacent cells requesting another handoff or the remaining call time (the bold solid arrow) if it is completed in the current cell because the handoff call has consumed some amount of its potential call time at the epoch of handoff (the solid line for the handoff call). Since a call from
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slow MSs is first assigned to a microcell, the channel occupancy times of new and handoff calls in microcell are represented as (8) (9) respectively. Similarly, since a call from fast MSs is initially directed to a macrocell, the channel occupancy times of new and handoff calls in macrocell are
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where the last equation follows since the Laplace transform of is and is defined as the th with alternating sign derivative of the Laplace transform of term. In (16), the interchange of integral and summation is justified since all terms are nonnegative (this property is satisfied in the rest of this paper). An easy computational formula for is given as (65) in Appendix A. Hence, by the Laplace can be obtained as transform,
(10) (11) and follow the equilibrium distrirespectively. Note that and with the following pdfs, respectively: butions of (17) and Since the pdf of
(12)
is (13)
then the Laplace transform is . Since
(14)
consists of two terms. The From (17), we can see that is just according to (4). Note first term can be directly derived from . that , which correHence, the second term will be sponds to the conditional expectation of the excess of the call time. This quantity perfectly matches the time duration depicted by the dotted line in Fig. 2. is Similarly, since the pdf of , we have the Laplace transform as . Note that
(15) then the Laplace transform of the channel occupancy time distribution for new calls is obtained by (16), shown at the bottom of the page, where (18) (19)
(16)
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Hence, the Laplace transform of the channel occupancy time distribution for handoff calls is obtained by
(20) where
successfully take back after a certain number of take-back failures; or it completes its potential call time without a take-back success using only a channel of a macrocell until the end of its (residual) lifetime—in this case the call is only served by the macrocell to which it is overflowed. On the other hand, the overflow call may continue its call session to another adjacent macrocell not being served by a microcell overlaid by the current macrocell: it fails to take back during the remaining cell residence time in the current macrocell. The various types of overflow calls are illustrated in Fig. 1. First, let us consider a new call that has been overflowed to a macrocell. Suppose that the call will successfully take back to a microcell at th trial after ( 1) consecutive failures. Let ( ) be independent and identically distributed with . with pdf as Now, let us define a random variable follows: .
Hence,
means the duration from the overflow to the take-back Then, success at th trial. The Laplace transform of the distribution of is obtained by
can be obtained as
(26) (21) and , Using the same technique, we can derive and are derived, respectively. Let from which (22)
is formally interpreted as the Based on renewal theory [19], th renewal time of a renewal process, where the first interarand the next subsequent interarrival rival time follows . Particularly, this process is called a delayed times follow renewal process. Next, consider a handoff call that has failed in handoff and has been overflowed to a macrocell. Let us define a random variable with pdf as follows:
(23) (27)
Then, we obtain (24)
is interpreted as the time period from the initiation of Then, overflow to the time when the overflow handoff call will successfully take back to a microcell at th take-back trial after ( 1) consecutive failures. Now, the Laplace transform of the is given by distribution of (28)
(25) B. Distribution of Channel Occupancy Time of Overflow Calls Since slow MSs may traverse a number of microcells during overflow call session, the channel occupancy time of overflow calls ( ) is closely related to the cell residence times in microcell ( ) and macrocell ( ) and the take-back failure probability ( ). Hence, the derivation of its Laplace transform ) needs a more complicated approach. ( Suppose that a new or handoff call is overflowed to a macrocell. The call may be completed in the current microcell; it may
As illustrated in Fig. 1, there exist six types of overflow calls (types 1–6 in the figure): types 1–3 correspond to new calls and types 4–6 handoff calls. Type 1 (or 4) is an overflow new (or handoff) call that will complete its call time in a current microcell. Type 2 (or 5) is an overflow new (or handoff) call that will end its call time in another microcell after a certain number of consecutive take-back failures. Type 3 (or 6) is an overflow new (or handoff) call that will successfully take back to a microcell ) consecutive failures. at th take-back trial after ( First, consider overflow new calls (i.e., types 1–3). Let us construct the following probability intensity functions: (29)
YEO AND JUN: MODELING AND ANALYSIS OF HIERARCHICAL CELLULAR NETWORKS
(30)
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is always less than . That is, , otherwise. Therefore
if
(39)
(31) is related to the channel occupancy time The first function of a type 1 call and is formally interpreted as . The second function is concerned with a type 2 call, representing the probability intensity that an overflow new call is eventually completed after a series of take-back is the failures. Note that the quantity and on condiprobability that the call time is between , i.e., . Finally, the tion that is related to the channel occupancy time of third function a type 3 call and is obtained from , which means the probability intensity that an overflow new call will successfully take back to a microcell after an arbitrary number of consecutive take-back failures. Hence, if the channel occupancy time of overflow new we denote by is represented by calls, then the pdf of
and
is represented as a series of s. If we can apNote that as a hyper-Erlang distribution and develop the proximate Laplace transform of the distribution of , we can easily deusing the methodology developed rive the th moment of follows a hyper-Erlang distribuearlier. Let us assume that tion with pdf (40) where and are positive integers and and are positive constants. Then, from (9) and (20), we obtain the Laplace transas form of the distribution of
(32) , which satisfies Note that the condition of the probability density function. Next, let be the channel occupancy time of overflow handoff calls (i.e., types 4–6). Similarly as in (29)–(31), let us construct the following intensity functions:
(41) where (42)
(33)
Hence
(34) (35) Then, the pdf of
is represented by (36)
. Also, it is satisfied that the virtual Before we proceed further, let us define by channel occupancy time of overflow calls with pdf (37) where represents the fraction of new calls to all the calls requesting overflows and is given by (38) The term “virtual” is used because is not truly . That is, the includes no consideration of , the cell residence quantity time of slow MSs in a macrocell. The quantity may be greater . This is the case when a slow MS with an ongoing than overflow call goes to an adjacent macrocell. However, note that
(43) To be complete, the above equation requires the th deriva[equivalently, in (42)], which is derived tive of in Appendix A embedded with a tractable algorithm. Further, Appendix B describes how to obtain the parameters of the disusing method of moments. Finally, the expected tribution of values obtained in this section are used as input parameters of (1) and (2) in Section II. IV. ARRIVAL TRAFFIC OF EACH TYPE OF CALLS In this section, we derive the arrival rates of handoff, overflow, and take-back calls in each layer of cells. It is assumed that all the cells in each cell layer have statistically the same characteristic of mobility in steady state. First, consider a microcell. The overall arrival rate into a microcell is represented as the sum of the arrival rates of new calls (which is given), handoff calls, and take-back calls ( ). Let us denote by the probability that a new call originated from slow MSs in a microcell will continue its call session ) and by the to an adjacent microcell (i.e., probability that a handoff call from slow MSs will continue its
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call session to an adjacent microcell (i.e., Then, taking a hyper-Erlang distribution for the call time in Section III gives
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). as
call will continue its call session to an adjacent macrocell (i.e., ). That is (51)
(44) (52) (45) Consider an arbitrary microcell surrounded by a number of adjacent microcells. Note that handoff requests into the microcell are from one of the following three call types: 1) new calls not blocked in adjacent microcells, continuing their call sessions to the microcell, 2) handoff calls successfully handed off into adjacent microcells and continuing their call sessions to the microcell, and 3) take-back calls that have successfully taken back into adjacent microcells and continue their call sessions to the microcell. Hence, the following equation should be satisfied in steady state: (46) Similarly, take-back requests into the target microcell are originated from the following three overflow call types in adjacent microcells: 1) overflow new calls, 2) overflow handoff calls, and 3) take-back calls that have not successfully taken back into adjacent microcells at the previous trial. Hence, in steady state, we have (47) Equation (47) follows since the take-back request occurs when an overflow call reaches a border of a microcell. Solving the system of (46) and (47), we obtain
Then, with a similar context as in (46) and (47), the following recursive equations should be satisfied in steady state: (53) Solving (53), we obtain (54) Finally, the arrival rates obtained in this section will be used as input parameters of (1) and (2) in Section II. V. PERFORMANCE MEASURES The blocking of a new call from slow MSs in a microcell occurs when the number of channels occupied by new calls in or there is not an available channel the microcell is equal to in the microcell. Hence (55) A handoff call from slow MSs cannot be served if all channels are occupied in a target microcell. The handoff failure probability of a handoff call being served is formally defined as the fraction of handoff attempts that are denied due to lack of channels. Hence (56)
(48)
(49) . where In a macrocell, the overall arrival rate comprises the arrival rates of new calls ( ), handoff calls ( ), and overflow calls and are from fast MSs, while is from ( ). Note that is independent from and slow MSs; and that the quantity . First, the arrival rate of overflow calls is simply obtained by (50) is the number of microcells in a macrocell. Next, where , let us denote by in order to derive the expression of the probability that a new call originated in a macrocell will continue its call session to an adjacent macrocell (i.e., ) and by the probability that a handoff
A take-back failure occurs if all the channels in a target microcell assigned to take-back requests are occupied. The take-back failure probability is formally defined as the fraction of take-back attempts that are denied due to lack of channels. If we give the same priority to both handoff and take-back . requests, then An overflow request is denied when the number of channels or there occupied by overflow calls in a target macrocell is is not an available channel in the macrocell. Hence (57) Because we assume that the arrival process is Poisson, by the property Poisson arrivals see time average (PASTA), the dropping probability of a new call from slow MSs is (58) Similarly, the dropping probability of a handoff call from slow MSs is (59)
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On the other hand, the blocking of a new call from fast MSs occurs when the number of channels occupied by new calls in a or there is not an available channel in macrocell is equal to the macrocell. Hence, the blocking probability (or equivalently dropping probability) is (60) Similarly, for a handoff call from fast MSs, the handoff failure probability (or equivalently dropping probability) is (61) Note that the performance measures (or equivalently, the joint stationary probabilities) are related to the arrival rates, which have also functional relationships with the performance measures (as shown in Section IV). Note also that the formula of the channel occupancy time of overflow calls has a functional form of the take-back failure probability (as shown in Section III). Hence, we need a recursive method to compute the performance measures. Basically, we use the iterative method proposed by [19]. If the size of or increases, the computation of and may be difficult. When this is the case, the method given in [20], which is devoted to the computation of asymptotic limiting distribution, may be used. All the numerical examples in this paper have convergent values of the performance measures. VI. NUMERICAL RESULTS In this section, we present numerical examples for a two-layer system with two populations of MSs with low and high mobilities. We assume that the total traffic of new calls follows a Poisson process with rate . Let be the fraction of new calls and . from slow MSs. Then Through the examples, we consider four identical microcells ). The mean cell resioverlaid by a single macrocell ( dence time of each type of MS is obtained as follows [21]: and
(62)
and are the radii of microcell and macrocell, rewhere spectively, and and are the average speeds of slow and fast m, m, MSs, respectively. We assume that km h, and km h. We consider the following , , , , configuration of channels: . Basically, we chose to be 0.5, but other values are and also considered to see the effect of the mobility ratio. The call , time ( ) is assumed to have the following parameters: , , , , and (mean min and variance ). The cell residence times ( and ) are assumed to have gamma distribution with shape and variance parameter and scale parameter (mean ). The associated parameters of are and (mean min and variance ) and the parameters of are and (mean min and variance ). The numerical examples consist of three parts: 1) the comparison of a nonexponential case of all the related random variables
Fig. 3. The performance measures in the exponential model versus the nonexponential model. Circle: exponential case; diamond: nonexponential case; horizontal axis: new call arrival rate ( calls per min); vertical axis: performance measures [p in (a), p in (b), p in (c), p in (d), and p in (e)].
1
with an exponential one, 2) the effect of the maximum number of channels allowed for overflows calls ( ), and 3) the effect of the mobility ratio ( ). First, Fig. 3(a)–(e) shows the difference in various performance measures between our case and the exponential case having the same mean. To give a consistent comparison, the parameters of corresponding time variables (i.e., the call time and the cell residence time) are chosen to have the same mean level with each other. From these figures, we see that the performance may be affected by the distributions of the call and cell residence times, even when the same mean is assumed. and have different This is because the quantities values according to the corresponding distributions. That is, both quantities affect the expected channel occupancy times and the arrival rates, which are directly related to the performance measures. In all figures, the performance measures considered , , and ) are higher for the exponential here ( , , case than for the nonexponential case of hyper-Erlang call time and gamma cell residence time distributions. This phenomenon can be explained by the memoryless property of exponential . In distributions. Note that for new calls, the exponential case of , the residual cell residence time has the same probabilistic characteristic with . In other
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Fig. 4. The effect of the maximum number of channels allowed for overflow calls C . Diamond: C ; circle: C ; triangle: C ; horizon axis: new call arrival rate ( calls per min); vertical axis: performance measures [p in (a), p in (b), p in (c), p in (d), and p in (e)].
1
=3
=4
=5
words, the exponential distribution has no information on the time period of the solid line for new calls in Fig. 2. Hence, . In general, the exponential model gives has a probabilistically greater value than . however, Hence, the channel occupancy time has a tendency to increase in the exponential case. This fact gives rise to higher values of the performance measures. Handoff calls are also explained by the same reasoning. Therefore, the assumption of exponential distribution for both the call time and the cell residence time may underestimate the system performance as compared to the nonexponential case. The maximum number of channels allowed for overflow calls may influence the performance in both microcell and macrocell. increases, the performance in miFig. 4(a)–(e) shows that as crocell becomes better, while in macrocell the dropping probabilities of new and handoff calls increase. This observation corresponds to our intuition. Using this analysis, we can determine in order to balance the traffic load with the proper value of the desired quality of service. Fig. 5(a)–(e) shows the plots of the performance measures and respectively. This exin the cases of ample seems to show that as the ratio of MSs with low mobility increases, the performance in the layer of microcells becomes
= 04 1
= 05
Fig. 5. The effect of mobility ratio . Diamond: : ; circle: : ; triangle: : ; horizon axis: new call arrival rate ( calls per min); vertical axis: performance measures [p in (a), p in (b), p in (c), p in (d), and p in (e)].
=06
worse, while a better performance results in that of macrocells. However, this may not be always the case. For a given level of , as increases (equivalently, increases), more frequent blocking and handoff failure are expected—this increases the portion of the channels of a macrocell occupied by the overflow calls from the layer of microcells, degrading the performance of the layer of macrocell. Hence, there exists a tradeoff point with and . respect to VII. CONCLUSIONS We developed an analytic model for the performance evaluation of a hierarchical cellular system introducing general distribution for call duration and cell residence time. By means of the characterization of the call time by hyper-Erlang distribution, the Laplace transform of the channel occupancy time for each call type is obtained as a function of the Laplace transform of cell residence time. This methodology gives more flexibility in describing the cell residence times, which may have various probabilistic characteristics depending upon different mobility environments. In particular, overflow calls are modeled by exploiting a well-defined renewal process, an approach that captures the inherent overflow from a lower layer of cells to an upper layer of cells and quantifies the channel occupancy time
YEO AND JUN: MODELING AND ANALYSIS OF HIERARCHICAL CELLULAR NETWORKS
of overflow calls in a tractable manner. Not addressed in this paper, the overflow from a macrocell layer to microcell layer can be treated by the proposed methodology. The developed Laplace transforms are thought to be also useful in handoff call queueing models where higher moments may be needed. The computational load is thought to be relatively low, and only a simple iterative method is sufficient to obtain the performance measures. Numerical results show that the distribution types of both the call time and the cell residence time have an obvious influence on the performance. This fact shows that the assumption of some specific distributions as the call and cell residence times may lead to over- or underestimation of the real performance. In particular, the exponential model has a worse performance than the nonexponential one. Although we have considered a two-layer hierarchical system, the approach can be extended to a multilayer hierarchical system with some modification.
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Hence
APPENDIX A DERIVATION OF If we let
for
, then
From (14) where
is defined as
(63)
With the interchange of integral and summation
where the term
An algorithm for Similarly
is given at the end of Appendix A.
is changed to The similar technique is applied in order to obtain and . First, from (19), we obtain
,
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With the interchange of integral and summation,
where the term
is changed to
Hence
respectively. The above . tation of where
is defined as A. Algorithm for the Computation of
(64) An algorithm for Finally
completes the compu-
and
Now, we give an algorithm to compute in (63) and in (64). For two arbitrary differentiable functions and for , it is satisfied that
is given at the end of Appendix A. where the subscript ( ) means the th derivative of the corresponding function. Hence
Furthermore, by a simple manipulation, we obtain the th as follows: derivative of
where the first derivative is computed as
(65)
YEO AND JUN: MODELING AND ANALYSIS OF HIERARCHICAL CELLULAR NETWORKS
and the second one is computed recursively as
Similarly
Hence, for a sufficient large number and as follows:
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For a fixed value of , nonlinear programming may be . For needed to determine the values , , and a better approximation, more equations may be needed. Consider an Erlang distribution with shape parameter and scale pa. rameter Then, we obtain
, we can compute
where
means the closest integer to . From [21] and
Note that the quantities converge to zero for a given as in (16). used to compute
and increases. Finally, (65) is
APPENDIX B CHARACTERIZATION OF THE DISTRIBUTION OF If we define as the probability that a slow MS will traverse microcells overlaid by a macrocell ( ), then the can be represented as follows: pdf of
If we choose
then represents a Poisson probability when truncating with mean , representing the mean number of microcells that a slow MS traverses in a macrocell. Generally, we see have a tendency to increase according that the variance of to its mean level. Hence, the Poisson distribution can be a cansince the variance grows as the didate for the distribution of mean increases. Hence
Since
from (40) the first two moments are
and
where and denote the radii of a macrocell and a microcell, respectively. Hence, can be obtained from the following relation:
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[16] R. W. Wolff, Stochastic Modeling and the Theory of Queues: Prentice Hall International, Inc., 1989. [17] S. Tekinay and B. Jabbari, “Handover and channel assignment in mobile cellular networks,” IEEE Commun. Mag., vol. 29, no. 11, pp. 42–46, 1991. [18] S. M. Ross, Stochastic Processes, 2nd ed. New York: Wiley, 1996. [19] Y. B. Lin, S. Mohan, and A. Noerpel, “Queueing priority channel assignment strategies for handoff and initial access for a PCS network,” IEEE Trans. Veh. Technol., vol. 43, pp. 704–712, Aug. 1994. [20] P. Gazdzicki, I. Lambadaris, and R. R. Mazumdar, “Blocking probabilities for large multirate Erlang loss systems,” Adv. Appl. Prob., vol. 25, pp. 997–1009, 1993. [21] B. Jabbari, “Teletraffic aspects of evolving and next generation wireless communication networks,” IEEE Personal Commun., vol. 3, pp. 4–9, Dec. 1996.
Kunmin Yeo was born in Daegu, Korea, in 1969. He received the B.S., M.S., and Ph.D. degrees in industrial engineering from the Pohang University of Science and Technology (POSTECH), Pohang, Korea, in 1995, 1997, and 2001, respectively. Since 2001, he has been a senior member of engineering staff in the Mobile Telecommunication Research Laboratory, Electronics and Telecommunications Research Institute, Daejeon, Korea. His research interests are performance evaluation of mobile telecommunications (in particular, traffic and probability modeling), wireless packet scheduling, and queueing theory.
Chi-Hyuck Jun was born in Seoul, Korea, in 1954. He received the B.S. degree in mineral and petroleum engineering from Seoul National University, Seoul, in 1977, the M.S. degree in industrial engineering from Korea Advanced Institute of Science and Technology in 1979, and the Ph.D. degree in operations research from the University of California, Berkeley, in 1986. Since 1987, he has been with the Department of Industrial Engineering, Pohang University of Science and Technology, Pohang, Korea, where he is now a Professor and the Department Head. He is interested in performance analysis of communication and production systems.
