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Modeling and Performance Evaluation of a Cellular Mobile Network Wei Li, Member, IEEE, and Xiuli Chao
Abstract—An analytic model of cellular mobile communications networks with instantaneous movement is investigated in this paper. This cellular mobile network is showed to be equivalent to a queueing network and furthermore the equilibrium distribution of this cellular mobile network is proved to have a product form. The explicit expressions for handoff rates of calls from one cell to another, the blocking probability of new calls and handoff calls are then obtained. Actual call connection time (ACCT) of a call in this cellular mobile network is characterized in detail, which is the total time a mobile user engages in communications over the network during a call connection and can be used to design appropriate charging schemes. The average ACCT for both complete call and incomplete call, as well as the probability for a call to be incomplete or complete, are derived. Our numerical results show how the above measures depend on the new call arrival process for some specific reserved channels numbers in each cell. The results presented in this paper are expected to be useful for the cost analysis for updating location and paging in cellular mobile network. Index Terms—Actual call connection time (ACCT), call blocking and dropping probability, cellular mobile network, mobility, queueing network.
I. INTRODUCTION
T
HE next generation of wireless mobile networks promises to provide a wide variety of services such as voice, data, and multimedia to users on the move. Mobile customers can make a phone call as in wired telephony, to make a connection to retrieve information messages such as email or stock information, or to make a connection to surf the Internet, or to do business over the Internet (electronic commerce). Due to the cellular nature of mobile communications, these networks have been modeled using queueing networks; a cell is modeled as a queue and channels as servers with limited numbers of channels representing the capacity restriction on the number of customers in the queue. The handoff concept in the wireless mobile communications is modeled as customers routing from one queue to another. A major difference between mobile networks and
Manuscript received September 25, 2001; revised May 5, 2002; approved by IEEE/ACM TRANSACTIONS ON NETWORKING Editor Z. Haas. The work of W. Li was supported in part by the Louisiana Board of Regents Support Fund under Grant LEQSF(2000-03)-RD-A-40 and by Wang Kuan Cheng Research Award in 2003 from Chinese Academy of Sciences. The work of X. Chao was supported in part by the National Science Foundation under Grants DMI-0196084 and DMI-0200306. W. Li is with the Department of Electrical Engineering and Computer Science, University of Toledo, Toledo, OH 43606 USA (e-mail:
[email protected]). X. Chao is with the Department of Industry Engineering, North Carolina State University, Raleigh, NC 27695 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TNET.2003.822641
queueing networks is that in cellular communications networks the requested call holding time of a call is related to the call, whereas the service time of customers in queueing networks is related to the queue [1]. In this paper, we formulate a cellular mobile communications network with a finite number of channels in each cell as a queueing network. The cellular network is characterized by the requested call holding time, cell residence time, and new call arrival process, as well as the capacity restrictions on the number of calls due to limited bandwidth. Because of the limited resources (time, frequency and code) of such networks, ongoing call connections may be disrupted or dropped, which has a detrimental effect on customer satisfaction. An attractive idea to address this issue is to apply different billing rates for complete calls and incomplete calls [6], [9]. Thus, the actual call connection time (ACCT), i.e., the time a mobile user actually uses the network service during a call, for both incomplete calls and complete calls are important quantities for the design of billing schemes [9], [11]. Wireless networks such as personal communication service (PCS) networks (see [7] and [13] and the references therein) cover service areas using base stations. The base stations are interconnected with either the public telephone network or a packet switching network, which forms the backbone network to provide services to mobile users. Each base station is responsible for providing service to mobile users in its area, called a cell. Within each cell, there are usually two classes of call traffic: new calls and handoff calls. A new call is the one which initiates in the current cell, while a handoff call is the one which initiates in another cell, and is handed over to the current cell. When a new call originates in the current cell, if an unused channel is available, it will be assigned for the communication between the mobile user and the base station and the new call is accepted for service. If no channel can be assigned to the new call, which depends on the channel allocation scheme, the new call may be blocked and may be cleared from the system, and in this case, the ACCT for this call is zero. If a new call is blocked in one cell, this does not mean that the new call is blocked in the network. From the concept of the mobile control handoff [12], the new call may be deemed as a handoff call in a neighboring cell and the ACCT of the new call depends on whether it can get a channel in the neighboring cell. When a new call is assigned a channel in the cell, it keeps the channel either until it is completed in the cell or until the mobile moves out of the cell. If the call is completed in the cell, the ACCT is equal to the requested call connection time (RCCT). If the mobile user does not finish its service in the originating cell, it moves out of the cell and attempts to gain service (a channel) in a neighboring cell. If a call
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possesses a simple solution structure. Our approach to solving this problem is to transform the wireless network problem into a queueing network problem with instantaneous movements. Furthermore, we study the handoff rates of calls (new calls and handoff calls) from one cell to another and obtain the blocking probability for new calls as well as handoff calls. The average ACCT for both complete call and incomplete call, and the probability for a call to be incomplete or complete, are discussed in detail. These measures have not been addressed under such a general framework in the existing literature. We expect these results are useful for the cost analysis for updating location and paging in cellular mobile networks. The rest of this paper is organized as follows. In Section II, we list the assumptions and notations used in the paper. In Section III, we show that the equilibrium distribution of this network has a product form. Some basic performance measures of the network are discussed in Section IV. The analytical characterization of the ACCT is presented in Section V, which also includes the analysis for the ACCT for incomplete calls and complete calls. A numerical analysis is provided in Section VI, where we show how the derived results from this paper depend on the new call arrival process for some specific reserved channel numbers in each cell. We conclude the paper with a discussion in Section VII. Fig. 1. Actual call connection time in the network.
fails to acquire resource (either buffers or a channel), it is forced to terminate and may be cleared from the system. In this case, the ACCT is the time from the start of the call to the time that the call is terminated prematurely. If the handoff call is blocked in one cell, it again does not imply that the call is blocked in the network. The call may be deemed as a handoff call in a neighboring cell and the ACCT of this call will depend on whether it can get a channel in the neighboring cell. If the handoff is successful, then the call will continue in the new cell, and the ACCT will accumulate and the same procedure repeats. A diagram for the ACCT of a new call in this cellular mobile network is given means the cleared probability from in Fig. 1, in which the means the routing probability from current current cell and cell to the neighbor cell (see points 6 and 7 in Section II). To design a cellular mobile network, comparisons need to be made between the performance measures of different protocols. In a few special protocols, networks have a closed product equilibrium distribution [2], [5], [14] and this allows results to be obtained for blocking probabilities and other measures of interest. For the majority of networks, however, the equilibrium distribution for the number of calls cannot be expressed explicitly, and therefore, performance measures cannot be calculated exactly. In [1], the authors verified that a cellular mobile network with a general protocols is stochastically equivalent to a queueing network with age-dependent routing probability. From this result, they further showed three special cases of cellular networks in which the equilibrium distribution can be obtained in closed form. Here, we formulate a general cellular mobile communication network problem and define appropriate protocols for handoff calls, and under which we show that the network
II. DESCRIPTION OF THE WIRELESS MOBILE NETWORK A wireless mobile communications network (or part of it) connected cells. The other assumptions and noconsists of tations for this wireless mobile network are as follows. 1) Cell consists of channels. Among channels in cell , a number these are reserved for handoff calls; here is the number of channels, not individual channels. In addition there is for putting calls on a waiting buffer of size may be infinite. hold, where 2) New calls are generated in cell at rate . , of 3) The requested call connection time (RCCT), say a new call at cell is assumed to be exponentially dis. tributed with mean 4) The cell residence time, say , of each portable in cell (the interval that a portable stays in a cell) is exponentially . distributed with mean 5) The probability that a call moves from cell to a neighboring cell , given that it moves to a neighboring cell be, where . fore the call is completed, is and cell is a neighboring cell of if Clearly, and only if . 6) A new call at cell gains a channel for service when it arrives and finds that there are calls in the cell. If , the new call is blocked in cell and then will be cleared from the system with or will try to use a line from a neighboring probability and will be treated by cell, say , with probability cell as a handoff call, where , and again, is possible only when and are neighboring cells. This implies that the blocked call from cell will not be immediately cleared from the system but may still
LI AND CHAO: MODELING AND PERFORMANCE EVALUATION OF A CELLULAR MOBILE NETWORK
have the possibility to be deemed as a handoff call in a neighboring cell. 7) A handoff call to cell gains a channel for service when it arrives and finds calls in the cell. However, if a handoff call finds all channels busy it follows the following protocols. a) If there is a waiting space available in the buffer, i.e., , then the handoff call is placed on if hold in the buffer until it gains a channel for service, or until it departs from the current cell because of its mobility. If the handoff call moves out of cell before it gains a channel, it will either terminate and or be cleared from the system with probability deemed as a handoff call to a neighboring cell, say , with probability . b) If a handoff call to cell finds all buffers occupied, in the case of , it will be i.e., forced to terminate from cell and then will either or be cleared from the system with probability will attempt to use a channel from a neighboring . cell, say cell , with probability The introduction of enables us to model different cellular mobile network features in a unified manner. Let us specify some of these features. 1) If a call is blocked at one cell, it may not be blocked by the network. This is possible in practice, because cells often overlap to ensure complete coverage of the region and when a call is attempted, the mobile may be situated near the boundaries of two cells and it may be close to a third or fourth cell. A handoff attempt is possible to these neighboring cells when the first attempt is blocked. This protocol is called directed retry in [14]. 2) If a call arrives to a cell and finds all channels busy, it is possible to borrow a channel from a neighboring cell, provided this channel does not interfere with the existing calls. This is called simple borrowing strategy in [15]. Some related borrowing concepts can be found in the hybrid channel assignment strategy [8] and borrowing-with-channel-ordering strategy [4]. It is worthwhile to point out that the wireless network model described above can be extended to include impatient time. That is, the calls in the waiting buffer of cell are not patient and each reneges after an exponentially distributed amount of time with . A reneging call is terminated and cleared from the mean and attempts to gain a channel at a system with probability . The neighboring cell as a handoff call with probability analysis of such a model is the same except that the marginal of the next section is slightly different. distribution III. PRODUCT FORM ANALYSIS OF THE NETWORK In this section, we show that the cellular mobile network described above can be modeled as a queueing network with signals and instantaneous movements. First, we introduce a queueing network with signals and instantaneous movement. Then, we show it is equivalent to the cellular network model under certain conditions. Finally, by the results of the queueing network with signals and instantaneous movement, we show
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that the cellular network model has a product form equilibrium distribution. A. Product Form for a Queueing Network The basic description of the queueing network of interest is as follows. 1) The queueing network consists of nodes. 2) There are two classes of customers. 3) Class 1 customers arrive at node from the outside according to a Poisson process with rate . 4) Node has servers and there is a waiting space of size in node . customers at node 5) When there are the service rate is ; 6) After a class ( , 2) customer is served at node , it joins node as a class 2 customer with probability and it leaves the network with probability . 7) When a class 1 customer arrives at node and finds customers at node , it starts service immediately with , and otherwise it one free server if leaves node and joins node as a class 2 customer with and leaves the network with probability probability , where . 8) When a class 2 customer arrives at node and finds customers at node , it joins node if ; otherwise it leaves node and joins node as a class 2 and leaves the network customer with probability . with probability In terms of the concepts of a network with signals or instantaneous movements (or instantaneous triggering) in [3], we can prove that the above queueing network is quasi-reversible, and its equilibrium distribution is given by a product form. Using the terminology of [3], there are two classes of cusin the above queueing network. The state of tomers each node is the total number of customers at the node. Node is characterized by the following. The arrival effect of the two classes of customers are
Note that the first two equations imply that if the number of , then the arrival of customer in node is less than a class 1 customer increases the number of customer by 1 with , probability 1, and if the number of customer is at least the arrival of a class 1 customer does not change the state of the node (the call will not be accepted in cell ). A similar explanation applies to the other two equations. When a customer arrives at node , it may trigger instantaneous movements, specifically, we have
and other triggering probabilities are 0. The first equation implies that when a class 1 customer arrives at node and node has customers present, then the state of node
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does not change and it instantaneously triggers a class 2 departure with probability 1. Similarly, when a class 2 customer arcustomers present, then rives at node and node has the state of node does not change and it instantaneously triggers a class 2 departure with probability 1. The departure rates from node are . Note that there are only class 2 departures from each node. , 2) customers If node is in isolation with class ( and rearriving according to a Poisson process with rate spectively, then clearly it is a birth and death process with birth rate (1) and death rates distribution of node
for is thus given by
. The equilibrium
where . To check quasi-reversibility of node , we need to calculate the departure rate from node . To that end, note that for we have
and for
(4) where , is a matrix with element in the position , and is a -dimensional column vector with 1 in the th element and 0 otherwise, then the equilibrium distribution of the queueing network, denoted , is given by by (5) is the equilibrium distribution of node where and given by (4). given by (2) with
and is
B. Equivalence of the Wireless Mobile Network and the Queueing Network (2)
for
that is
, we have
, we have
Now, we show that the queueing network above with a special value of and is equivalent to the cellular mobile network described in . Section II if Let new calls be class 1 customers and handoff calls be class 2 customers. Then class 1 customers arrive at node according to a Poisson process with rate . Since the residence times of new calls and handoff calls have the same distribution, when there . Since are calls at node the service rate is each call is moving, the rate these calls move out of cell is . Thus in the queueing model the total service rate at node is . When a new call initiates at cell , it is equivalent to a class 1 customer arriving at node , if there are calls in cell then the new call is connected less than to the base station, and in the queueing model it is the same as a class 1 customer starts service. If, however, there are at least calls at cell then, according to our assumption on the cellular network, the new call is blocked and cleared from and it tries a line in the neighthe system with probability . In the queueing model this boring cell with probability is equivalent to the class 1 customer leaving the network with and switches to node with probability . probability The same argument applies to a handoff call. Consider a call occupying a channel in cell of the cellular network. The rate at which this call leaves cell is ; it is , and it is due to call completion with probability due to handoff to a neighboring cell with probability . There are such calls, where is the number of call in cell . If then there are calls in the buffer, and each moves out of cell as a handoff to a neighboring it is cleared cell at rate . As this happens with probability from the system and with probability it moves to cell as a handoff call. Therefore, the total rate at cell is
This shows that node in isolation is a quasi-reversible queue with signals. Thus it follows from Theorem 4.9 of [3] that, if and are the solution to traffic equations (3)
and each leaves the system with probability and attempts to gain a channel from cell with probability . Thus, if
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for then the wireless network is equivalent to a queueing network with service rate at cell being
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Thus the handoff rate of handoff calls from cell to cell can be calculated as
(6) and each departure leaves the network with probability and joins node as a class 2 customer with probability . Combining the above equivalent result and the product form result for the queueing network in Section III-A, we conclude , the equilibrium probability of that if calls in cell of having the cellular network is (7)
where
is given by (1) with is given by (6).
and
by (4), and 3) Handoff rates from cell to cell . From the result obtained above, the total handoff rate from cell to cell is
IV. HANDOFF RATES AND BLOCKING PROBABILITIES Once the network equilibrium distribution is calculated, we can compute many interesting measures of the system. In this section, we provide explicit expressions for various handoff rates and relevant blocking probabilities. Before we proceed, it be is convenient to note the following basic result first. Let the interval in which a handoff call arrives to cell and observes calls already in cell (where channels are being used and calls are in the buffer), to the moment that these calls are reduced to . It follows from the assumption is exponentially distributed of exponential distributions that . with rate
The total handoff rate to cell is the sum of the above expression over all . From the traffic equations (3), we can readily know that this sum is just . B. Blocking Probabilities 1) New call blocking probability in cell . The new call to cell is blocked and cleared from the system with probability
A. Handoff Rates 1) Handoff rates for a new call from cell to cell . Since and it is accepted the new call arrival rate to cell is , the rate at which a with probability new call to cell goes to cell as handoff call is
A new call to cell is blocked and moved to cell handoff call with probability
as a
and a new call to cell is blocked with probability
The overall blocking probability for a new call to the whole network is 2) Handoff rates for a handoff call from cell to cell . We need the Arrival Theorem in queueing networks (Theorem 10.21 and Corollary 5.7 of Chao et al. [3]), which states that any arrival to node , either from the outside or from another node, observes the network in equilibrium.
(8)
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2) Handoff blocking probability in cell . A handoff call to cell is blocked in cell , i.e., leaves before starting service, and cleared from the system with probability
(this equation holds for
) by noting that
A handoff call to cell is blocked in cell and moved to cell as a handoff call with probability
V. ACTUAL CALL CONNECTION TIME A. ACCT for New Calls One performance measure that is of great interest is the ACCT per call, defined as the total time a call spends in the network. In the past, the ACCT has been studied for the cases when no queueing is allowed in the cell [7], [10]. In [10], this measure is called actual call holding time for the reserved channel scheme (RCS). We study the ACCT under the described network channel allocation scheme. In the model discussed in [10], the ACCT cannot be greater than RCCT because there is no buffer in the cell. However, in the current network scheme, we make the following observations: 1) the ACCT can be greater than the RCCT; 2) the call may be incomplete even if the ACCT is greater than the RCCT. This is possible because handoff calls can wait in the handoff area during the handoff process, and the ACCT contains not only the time in service but also the possible waiting time. We first consider the ACCT for a new call to cell and then consider the ACCT for an arbitrary new call to the network. We first introduce some notation. ACCT of a new call initially originated in cell . total time that a handoff call will spend on the network beginning from the time that the handoff call arrives to cell . distribution function of , i.e.,
distribution function of
in which the last equation holds for call to cell is blocked with probability
. A handoff
, i.e.,
under the condition . under the condition . under the condition . We note that and are both exponentially distributed . By conditional probability we obtain with mean
(10) where is an unit step function taking value 1 if 0 otherwise, and The overall blocking probability for a handoff call in the network is
(9)
and
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In the Appendix, we show that the second equation above can be simplified to
(13) Now, if we denote by
and (11) Since has exponential distribution, by memoryless property we have
then (12) and (13) can be written in matrix form as
Solving these equations yields Using (10), (11), and the relationship and we obtain the expectation of
, as follows:
(14) (15) (12)
Let be the average actual call connection time of an ar, then bitrary new call in the network, and by noting (3), we have
B. Analysis of Incomplete Calls 1) Average ACCT for Incomplete Calls: It is important to distinguish complete calls from incomplete calls. In [10], the authors introduced two different schemes of revenue structures based on the actual call holding time for a call, the actual call holding time for an incomplete call and the actual call holding time for a complete call. The first scheme implies that the revenue is assumed to be general but only an increasing function of the actual call holding time and the second revenue function is
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the superposition of the actual call holding time for an incomplete call and the actual call holding time for a complete call. In the following, we compute the call completion probability, the call dropping probability, and the ACCTs for complete calls and incomplete calls. In this section, we focus on incomplete calls, and complete calls are studied in the next section. We first consider the ACCT for an incomplete call to cell . Let the event that a new call to cell is an incomplete call; the event that a handoff call to cell is an incomplete call; the ACCT for an incomplete call initially originated in cell ; the ACCT for an incomplete call as a handoff call to cell ; the distribution of the ACCT for an incomplete call initially originated in cell , i.e., (17)
It follows from (16) and (17) that the distribution of the ACCT of an incomplete call as a handoff to cell ; i.e., (18) and
We conduct a similar analysis as in the last subsection for ACCT to obtain
(16)
and
In the Appendix, we show that the second equation above can be written as
(19)
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Thus if we denote by
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Therefore, if we use notation
and
then (18) and (19) can be written in matrix form as
then (20) and (21) can be written in matrix form as
Solving these equations, we obtain
Solving these equations yields (22)
If we denote by the average ACCT of an incomplete call in the network, then it follows from the results above that
(23) Let be the probability that a call in the network is incomplete, then it follows from the result above that
2) Probability of Incomplete Calls: In this subsection, we are concerned with the probability that a call is incomplete, a quantity of interest in the design of wireless networks. Denote (a new call to cell is an incomplete call); by (a handoff call to cell is an incomplete call). In the same way as we derived (16) and (17), we obtain
(24)
C. Analysis of Complete Calls
(20) and
(21)
1) Average ACCT for Complete Calls: Similar to the analysis above, we can also calculate the average ACCT for a complete call and the probability that a call is complete. We first consider the ACCT for a complete call to cell . Let the event that a new to cell is a complete call; the event that a handoff to cell is a complete call; the ACCT for a complete call initially originated in cell ; the ACCT for a complete call as a handoff to cell ; the distribution of the ACCT of a complete call initially originated in cell ; i.e.,
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the distribution of the ACCT of a complete call as a handoff to cell ; i.e.,
It follows from (25) and (26) that
(27) and
Similar to the analysis above, we have
(25) and
(28) In terms of the analysis in the Appendix, if we denote the item inside the big bracket of (28) by , it is easy to show that
Thus if we denote by
and (26)
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then (27) and (29) can be written as
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be the probability that a call to the network is a complete Let call, then it follows from the result above that
Solving these equations yields
Let be the average ACCT of a complete call in the network, then it follows from the result above that
(33) Compare the results in (22)–(24) and in (31)–(33), we have
and 2) Probability of Complete Calls: Denote by (a new call to cell is a complete call), and (a handoff call to cell is a complete call). Using the same argument as that of (25) and (26), we obtain
where is a -dimensional column vector of ones. This is intuitive because a new call to any cell , or to the network, will be either an incomplete call or a complete call. VI. NUMERICAL RESULTS AND DISCUSSION
(29) and
(30) Therefore, if we denote by
and
we then obtain the solution of (29) and (30) as (31) (32)
In this section, we present some numerical results to show how the performance measures obtained in the previous sections depend on the new call arrival process and the number of reserved channels in each cell. In the U.S., each cellular provider is allocated 25 MHz of spectrum, 12.5 MHz for transmitting and 12.5 MHz for receiving. Each phone conversation in the first generation cellular is allocated 30 kHz of spectrum. Therefore, each 12.5 MHz of bandwidth can handle 416 simultaneous phone calls. In the following examples, we consider 416 to be the maximum channel numbers in each cell. The other parameters are as follows. ; • The network consists of 20 cells channels. Among these 416 • Each cell has channels in cell , channels are reserved for handoff calls. In addition there is a waiting buffer of size 30 for putting the handoff calls on hold. per • New calls are generated in cell at the rate of minute. • The requested call connection time of a new call at cell is exponentially distributed with mean of 10/3 min . • The cell residence time of each portable in cell is expo. nentially distributed with mean of 5 min that a call moves from cell • The routing probability to cell is given by
In Fig. 2, we show how the new call blocking probability depends on the new call arrival process for the reserved channel numbers of 0, 10, 20, 30, and 40, respectively, in each cell. From this diagram, we observe that the new call blocking probability increases as new call arrival rate increases. In particularly, the new call blocking probability is almost zero if the
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Fig. 2.
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New call blocking probability versus new call arrival rate.
Fig. 4. Average ACCT of a new call versus new call arrival rate.
Fig. 3. Handoff call blocking probability versus new call arrival rate.
Fig. 5.
Average ACCT for a complete call versus new call arrival rate.
number of new call per unit time is less than 100 and it will significantly increase when the new call arrival rate is more than 100 calls per minute. Next, the new call blocking probability increases as the number of reserved channel increases. The less the reserved channel number is, the less the new call blocking probability. The new call blocking probability is almost zero no matter how many channel are reserved if is less than 100. In Fig. 3, we show how the handoff call blocking probability depends on the new call arrival process for the reserved channel numbers of 0, 10, 20, 30, and 40, respectively, in each cell. From this diagram, we observe that the handoff blocking probability increases as new call arrival rate increases. In particularly, the handoff call blocking probability is almost zero if the number of new call per unit time is less than 110 for zero reserved channels or if the number of new call per unit time is less than 170 but with a few reserved channels. This means that the reservation scheme significantly reduces the handoff blocking probability and is one of the reasons why the reserved channel scheme is better than nonpriority scheme from the viewpoint of reducing the handoff call blocking probability. Next, the handoff call blocking probability decreases as the number of reserved channel increases.
The more the reserved channel number is, the less the handoff call blocking probability. In Fig. 4, we show how the average ACCT of a new call depends on the new call arrival process for the reserved channel numbers of 0, 10, 20, 30, and 40, respectively, in each cell. From this diagram, we observe that the average ACCT is almost the same as the requested call holding time (10/3 min) if new call arrival rate is less than 100 no matter how many channels are reserved. The average ACCT decreases as new call arrival number increases from 100. When new call arrival rate is more than 100 calls per minute, the average ACCT significantly depends on the number of reserved channel and in this case it is a decreasing function of the number of reserved channels. In Fig. 5, we show how the average ACCT for a complete call depends on the new call arrival process for the reserved channel numbers of 0, 10, 20, 30, and 40, respectively, in each cell. From this diagram, we observe that the average ACCT for a complete call is almost the same as the requested call holding time (10/3 min) if new call arrival rate is less 100 no matter how many channels are reserved. The average ACCT for a complete call decreases as new call arrival rate increases from 100. When new
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Fig. 6.
Fig. 7.
Average ACCT for an incomplete call versus new call arrival rate.
Probability of incomplete call versus new call arrival rate.
call arrival rate is more than 100 calls per minute, the average ACCT for a complete call significantly depends on the reserved channels numbers and in this case it is a decreasing function of the number of reserved channels. In Fig. 6, we show how the average ACCT for an incomplete call depends on the new call arrival process for the reserved channel numbers of 0, 10, 20, 30, and 40, respectively, in each cell. From this diagram, we observe that the average ACCT for an incomplete call is almost the same no matter how many channel are reserved and it is much less than the requested call holding time (10/3 min), if new call arrival rate is less 100. The average ACCT for an incomplete call decreases as new call arrival rate increases from 100. When new call arrival rate is more than 100 calls per minute, the average ACCT for an incomplete call significantly depends on the reserved channel numbers and in this case it is a decreasing function of the number of reserved channels. In Fig. 7, we show how the probability of incomplete call depends on the new call arrival process for the reserved channel numbers of 0, 10, 20, 30, and 40, respectively, in each cell. From this diagram, we observe that the probability of incom-
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Fig. 8. Probability of complete call versus new call arrival rate.
plete call increases as new call arrival rate increases. In particular, the probability of incomplete call is almost zero if the number of new call per unit time is less than 100 no matter how many channels are reserved. If the number of new calls per unit time is more than 100, the probability of incomplete call is increasing for both new call arrival rate and the reserved channels. This also means that the reserved scheme may enlarge the probability of incomplete call. In Fig. 8, we show how the probability of complete call depends on the new call arrival process for the reserved channel numbers of 0, 10, 20, 30, and 40, respectively, in each cell. From this diagram, we observe that the probability decreases as new call arrival rate increases. In particular, the probability of complete call is almost one if the number of new call per unit time is less than 100 no matter how many channels are reserved. If the number of new calls per unit time is more than 100, the probability of complete call is decreasing for both new call arrival rate and the reserved channels. This also means that the reserved scheme may reduce the probability of complete call. Next, the probability of complete call decreases as the number of reserved channel increases. The larger the reserved channel number is, the smaller the probability of complete call. VII. CONCLUSION In this paper, we analytically characterize the equilibrium probability and the ACCT and related performance metrics for a cellular mobile network under a general network allocation scheme. We obtain a product form solution for the equilibrium distribution of the network and explicit formulas for the expectation of the ACCT for any call in the network. The call completion probability, the call drop probability and the average ACCTs for both complete calls and incomplete calls are also obtained. With these analytical results for ACCTs, billing schemes for wireless network services can be evaluated and the billing rate can be determined. For example, the unit price of a call could be determined only by the ACCT, no matter if it is a complete call or an incomplete call. Alternatively, the billing rate of a call could also be reasonably designed as two different rates for incomplete call and complete call, respectively. These billing
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strategy has been investigated in [10]. Moreover, the analytical techniques developed in this paper may be useful for performance analysis of other telecommunications systems. We observe from our numerical analysis that there exists a threshold value, such that all the performance measures deteriorate rapidly beyond this threshold. For example, when each cell has 416 channels, the threshold value is 100 (see Figs. 2–8). This also means that if the new call arrival rate is about 100 in a specific population, it is better to arrange about 416 channels in each cell. Finally, we should point out that one of the limitation of the model is that the results hold only under restricted system has to take the form as specified. Without protocols, i.e., this condition, the network does not have a product form solution and the exact calculation of system performance measure will become extremely complicated.
To prove (19), we denote the item in the big bracket {} of by , in terms of the formula we just obtained above, we get
ACKNOWLEDGMENT APPENDIX To prove (13), we denote the item in the big bracket {} of by , then
The authors would like to express their sincere appreciation to the editor and the reviewers for their suggestions and comments which have greatly helped to improve the presentation of this paper. REFERENCES [1] R. J. Boucherie and N. M. Van Dijk, “On a queueing network model cellular mobile telecommunications networks,” Oper. Res., vol. 48, no. 1, pp. 38–49, 2000. [2] R. J. Boucherieand and M. Mandjies, “Estimation of performance measures for product form cellular mobile communications networks,” Telecommun. Syst., vol. 10, pp. 254–321, 1998. [3] X. Chao, M. Miyazawa, and M. Pinedo, Queueing Networks: Customers, Signals and Product Form Solutions. New York: Wiley, 1999. [4] S. M. Elnoubi, R. Singh, and S. C. Gupta, “A new frequency channel assignment algorithm in high capacity mobile communication system,” IEEE Trans. Veh. Technol., vol. 31, pp. 128–131, Aug. 1982. [5] D. Everitt and D. Manfield, “Performance analysis of cellular mobile communication systems with dynamic channel assignment,” IEEE J. Select. Areas Commun., vol. 7, pp. 1172–1180, Oct. 1989. [6] Y. Fang, I. Chlamtac, and Y. Lin, “Billing strategies and performance analysis for PCS networks,” IEEE Trans. Veh. Technol., vol. 48, pp. 638–651, Mar. 1999. [7] Y. Fang, I. Chlamtac, and Y. B. Lin, “Call performance for a PCS network,” IEEE J. Select. Areas Commun., vol. 15, pp. 1568–1581, Oct. 1997. [8] T. L. Kahwa and N. D. Georganas, “A hybrid channel assignment scheme in large-scale, cellular-structured mobile communication systems,” IEEE Trans. Commun., vol. 26, pp. 432–438, Apr. 1978. [9] W. Li and A. S. Alfa, “A PCS network with correlated arrival process and splitted-rate channels,” IEEE J. Select. Areas Commun., vol. 17, pp. 1318–1325, July 1999. [10] , “Channel reservation for handoff calls in a PCS network,” IEEE Trans. Veh. Technol., vol. 49, pp. 95–104, Jan. 2000. [11] W. Li, Y. Fang, and R. Henry, “Actual call connection time characterization for the wireless mobile networks under a general channel allocation scheme,” IEEE Trans. Wireless Commun., vol. 1, pp. 682–691, Oct. 2002. [12] Y. B. Lin and I. Chlamtac, Wireless and Mobile Network Architectures. New York: Wiley, 2001. [13] Y. B. Lin, S. Mohan, and A. Noerpel, “Queueing priority channel assignment strategies for PCS hand-off and initial access,” IEEE Trans. Veh. Technol., vol. 43, pp. 704–712, Aug. 1994. [14] D. L. Pallant and P. G. Taylor, “Modeling handovers in cellular mobile networks with dynamic channel allocation,” Oper. Res., vol. 43, no. 1, pp. 33–42, 1995. [15] S. Tekinay and B. Jabbari, “Handover and channel assignment in mobile cellular networks,” IEEE Commun. Mag., vol. 29, pp. 42–46, Nov. 1991.
LI AND CHAO: MODELING AND PERFORMANCE EVALUATION OF A CELLULAR MOBILE NETWORK
Wei Li (M’99) received the B.Sc. degree from Shannxi Normal University, Xian, China, in 1982, the M.Sc. degree from Hebei University of Technology, Tianjin, China, in 1987, and the Ph.D. degree from the Chinese Academy of Sciences, Beijing, China, in 1994. He is currently an Associate Professor in the Department of Electrical Engineering and Computer Science, University of Toledo, Toledo, OH. Before joining the University of Toledo, he was an Assistant Professor in the Department of Electrical and Computer Engineering, University of Louisiana, Lafayette, from 1999 to 2002, where he was named as the Bell South/BORSF Professor in Telecommunications from 2000 to 2002. He engaged in research and teaching at the University of Manitoba, Canada, and the University of Winnipeg, Canada, from 1995 to 1999. He has also been a Senior Researcher at the University of Newcastle, U.K., and an Associate Researcher at the Chinese Academy of Sciences. His research interests are in the evaluation, design and implementation of novel models for wireless and mobile networks, optimization of the radio resource allocations in wireless multimedia networks, traffic analysis, control and security in the wireless internet and mobile ad hoc networks, and queueing networks, reliability networks and their applications in telecommunications networks. Dr. Li is currently serving as an Editor for the EURASIP Journal on Wireless Communications and Networking. He is also serving or has served as a TPC Member/Session Chair for some IEEE professional conferences, including IEEE ICC’04, IEEE WCNC’04, IEEE VTC’03, IEEE GlobeCom03, IEEE ICC’02, and IEEE WCNC’00.
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Xiuli Chao received the B.S. degree from Shandong University, Jinan, China, in 1983, and the M.S. and Ph.D. degrees from Columbia University, New York, NY, in 1986 and 1989, respectively. He is currently a Professor of industrial engineering and the Co-Director of the Interdisciplinary Operations Research Programs at North Carolina State University (NCSU), Raleigh. Before joining NCSU, he was a Professor at the New Jersey Institute of Technology, Newark. He has also held visiting positions at the University of California at Irvine, the Tokyo University of Science, and Columbia University. He is a coauthor of Operations Scheduling with Applications in Manufacturing and Services (New York: Irwin/McGraw-Hill, 1998) and Queueing Networks: Customers, Signals, and Product Form Solutions (New York: Wiley, 1999). He is a department editor for IIE Transactions, an associate editor for Operations Research, Naval Research Logistics, Queueing Systems: Theory and Applications, and Operations Research Letters, and he is on the editorial board for Probability in the Engineering and Informational Sciences, Journal of Systems Science and Complexity, International Journal of Information Technology and Decision Making, and International Journal of Quality Technology and Quantitative Management. His research interest includes queueing theory and queueing networks, scheduling, optimal control, financial engineering, inventory theory, systems development, and supply chain management. Dr. Chao received the 1998 Erlang Prize from the Applied Probability Society of INFORMS and the Outstanding Overseas Young Chinese Scientist Award from the National Natural Science Foundation of China in 2002.
