From Optimal Search Theory to Sequential Paging in Cellular Networks

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 15, NO. 7, SEPTEMBER 1997

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From Optimal Search Theory to Sequential Paging in Cellular Networks Ramin Rezaiifar and Armand M. Makowski, Senior Member, IEEE

Abstract— We propose a novel paging strategy based on the theory of optimal search with discrete efforts. When compared to conventional paging methods, the proposed scheme increases the mobile station discovery rate while decreasing the average number of times that a mobile station has to be paged in a location area. The proposal is fully compatible with existing cellular structures, and requires minimal computational power in the mobile switching centers. Index Terms—Optimal search, paging, search plan, sequential, stochastic approximation, wireless.

I. INTRODUCTION

A

more efficient use of limited wireless resources in emerging personal communication services (PCS’s) requires much smaller cells (microcells and picocells) than those used in conventional cellular networks. Tracking the mobile stations will become a challenging task as the cell sizes shrink and the number of cells increases. In order to ensure reliable communication, a mobile station has to send/receive signaling messages from/to the base station. Depending on the system design, signaling messages serve various purposes such as frame synchronization, base station identification, mobility management, handoff decision making, etc. With the expansion of both the number of services available to users and the number of mobile stations in service, the radio spectrum will become a scarce commodity. This calls for a reduction in the signaling load between the mobile station and the base station in order to make more bandwidth available for voice and data traffic. One way to reduce the signaling load associated with paging and location area updating is to deploy cellular networks with more intelligent mobile tracking and location management techniques. In this paper, we propose a novel paging architecture that reduces the signaling load while expediting the paging process. The method is based on the theory of optimal search with discrete efforts [11], and exploits the observation that paging a mobile station resident in one of several cells is analoguous to searching an object hidden in one of a finite number of boxes. Building on this analogy, we integrate some well-known optimal search strategies into a novel paging technique. The Manuscript received September 1, 1996; revised April 1, 1997. This work was supported in part by NSF Grant NSFD CDR-88-03012 and NASA Grant NAGW277S. R. Rezaiifar is with QUALCOMM Inc., San Diego, CA 92121 USA. A. M. Makowski is with the Department of Electrical Engineering and the Institute for Systems Research, University of Maryland, College Park, MD 20742 USA. Publisher Item Identifier S 0733-8716(97)05850-2.

proposed scheme requires minimal computational power in the base stations and switching centers, is fully compatible with standard mobility management techniques [4]–[6], [13], and does not require additional database resources to keep track of the movements of the mobile station within a location area. Moreover, through simulations, we show that the performance characteristics of this paging scheme are often superior to that of the conventional blanket paging algorithm. The paper is organized as follows. Section II describes the registration procedure in a GSM-type cellular network. Call delivery and paging form the subject of Section III. In Section IV, we review the relevant elements from the theory of optimal search. In Section V, we briefly explain how paging is done in current cellular systems and how it can be improved. In Section VI, the optimal search strategies described in Section IV are integrated into an intermediate class of paging strategies; these pave the way to the proposed paging scheme which is then discussed in Section VII. Finally, in Section VIII, through simulation results, we demonstrate the superiority of the proposed sequential algorithm over the conventional paging methods. The paper closes with Section IX where we summarize the obtained results, and discuss topics for future research.

II. REGISTRATION AND LOCATION AREA UPDATE In order to be able to deliver incoming calls to a mobile station (MS), a cellular system has to keep track of the whereabouts of the MS’s within the cellular system. In practice, the MS may inform the system of its new location periodically, on power-up, just before shut-down, and/or when it crosses the boundaries of certain zones known as location areas (LA’s). An LA is simply a geographic area covered by a group of cells. The size of an LA or the number of cells in it may vary depending on the rate at which cells receive calls, and on the intercell traffic characteristics. In fact, the size of an LA can be optimized to create a balance between the LA update rate and the expected paging rate within an LA [2], [3]. The BS’s within an LA are managed by a mobile switching center (MSC). Each BS periodically broadcasts a unique ID number that specifies the LA/MSC to which it belongs, and an MS stores the ID number of the current BS in its memory. When an MS switches to a new BS, it compares the ID number broadcast by the new BS with that in memory. If the two ID numbers are different, the MS sends a registration message to

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Fig. 1. Hierarchical structure of mobility management system.

Fig. 2. The LA update process.

the new BS, which in turn forwards the message to the MSC managing this new BS. The MSC’s are connected to each other and to the PSTN through a land-line switching network (e.g., an ATM network). Each MS is bound to a home MSC. When the telephone number of an MS is dialed by another party, land-line switches first direct the call to the home MSC of the MS. The information about the whereabouts of the MS’s is stored in two types of databases—home location registers (HLR’s) and visitor location registers (VLR’s). A VLR database is usually smaller than an HLR, and isolates the small-scale movements of the MS’s by caching portions of the information stored in HLR’s. The territory of a VLR consists of the geographic area covered by one or more MSC’s. A unique HLR, called the home HLR, is assigned to each MS when it is put into service; it stores all the necessary information related to the MS such as the types of services to which the MS subscribes, the MS state (active/inactive), and the last VLR that informed the HLR of having the MS in its territory. Fig. 1 shows the hierarchical structure of the location tracking system described above. Different stages of an LA update procedure in a GSM-type system are depicted in Fig. 2; the numbers next to the dotted arrows represent the sequence of events. When an MS crosses the boundary into a new LA, the new MSC informs its VLR of the presence of the MS. If the information of this MS has

not been cached, a query is sent to the home HRL by the VLR. The home HLR is then responsible for informing the old VLR that the MS has moved out of its territory, and that the corresponding records in the old VLR can therefore be removed. As long as the MS travels among the LA’s covered by the new VLR, no query from the HLR is necessary because the profile of the mobile user and other related information already exist in the cache. If an MS leaves its current LA and enters another one that is supported by the same VLR, the new MSC will inform the VLR nonetheless. In this way, given the ID number of an MS, the VLR can specify the LA where the MS resides. III. CALL DELIVERY

AND

PAGING

In current cellular systems, the call delivery procedure uses the information stored in HLR’s and VLR’s to route the call to a given MS. The steps involved in the call setup procedure are shown in Fig. 3. The numbers next to the dotted arrows represent the sequence of events, and the solid arrows show the communication links after the call set-up is complete. When a mobile user’s telephone number is dialed, landline switches direct the call to the home MSC of the user. The home MSC then contacts the home HLR which has a pointer to the VLR currently serving the MS. Next, a query is sent to this VLR to find the MSC which currently serves

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Fig. 3. Call setup procedure in cellular networks.

the MS, and this MSC consequently sends a message back to the home MSC (via the VLR and the home HLR). Using the land-line network, the home MSC establishes a connection to the serving MSC. In the meantime, the serving MSC pages the MS within the LA to find the cell in which the MS resides, and establishes a wireline link between the BS and the MSC. In current cellular systems, the final step in the call setup procedure starts with the MSC sending a paging message to all of the BS’s under its control in order to find the MS. We refer to this scheme as the broadcast approach (this is the blanket polling of [8]). However, by monitoring the traffic pattern inside each cell in an LA, it is possible in principle to obtain a probability mass function (pmf) that quantifies the likelihood of discovering an MS in each cell. This pmf may be updated regularly to reflect changes in traffic pattern with the time of day. With this information available, it thus seems wasteful to page an MS in all of the BS’s that comprise an LA. For instance, a sequential search plan that starts the search with cells with a higher probability of discovery is expected to be more efficient. A sequential paging approach based on a user location pmf has been proposed by Rose and Yates in [7] and [8]. In a more recent paper [9], these authors have proposed a more efficient variation of their sequential paging strategy in which, instead of searching individual locations in decreasing order or probability (as in [7] and [8]), groups of locations are searched, where each group contains the most likely locations not already searched. Although these above-mentioned paging methods also seek to improve on the conventional broadcast paging strategy, they differ with the one proposed in this paper in several key aspects: Indeed, the latter: 1) accounts explicitly for the unavailability of paging channels in each cell, and 2) attempts to create a balance between the number of pages sent to cells with higher occupancy probability and those with lower occupancy probability. The objective is, of course, to minimize

the search effort, and cells with higher occupancy probability will experience a higher probability of blockage due to the unavailability of paging channels. In the remainder of this paper, we focus on optimizing the paging process, and show how an “intelligent” paging strategy can be devised with the help of ideas borrowed from the theory of optimal search. IV. OPTIMAL SEARCH

WITH

DISCRETE EFFORT

In this section, we summarize the requisite elements from the theory of optimal search with discrete efforts. After presenting a general mathematical model, we consider the problem of maximizing the probability of discovery for any given number of looks. A good and comprehensive treatment of the subject can be found in the monograph by Stone [11]. A. A Mathematical Model boxes where An object is placed randomly in one of denote the it remains throughout the search process. Let It is probability of the object being in box also assumed that a look at box may not necessarily lead to detection of the object even if the object resides in box Let denote the probability of failing to detect the object on the looks in box and of succeeding on the th try, first given that the object is in box A related parameter is the probability of not detecting the object on a single look in box given that the object is in box Under some natural independence assumptions, the and are clearly related by two probabilities

(4.1) A discrete search plan is any deterministic The search plan valued sequence

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prescribes that the first look should be made in box and if the object is not detected, the second look should be made with this process continuing until the object is at box discovered. It is important to distinguish between a search plan and a search policy. The latter is an algorithm which, given the set of specifies the sequence parameters of boxes to be searched until the object is found. A search plan, however, is a realization of a search policy. Clearly, a good search policy is one that “learns” from the search history (i.e., number of times that each box has been visited). For a each search fixed set of parameters policy prescribes a unique search plan once the underlying randomness is generated. We find it convenient to identify a search policy by its corresponding search plan, a convention hereafter enforced. B. Optimal Search Plans Consider the situation when a fixed total number of looks is permitted. Let be the total number of looks permitted, denote the -step search plan With and let we associate a function each with representing the number of looks i.e., made into box under the plan (4.2) where i.e.,

is the indicator function for the condition if and

otherwise. The constraint on the total number of looks implies

The probability of detecting the object on or before the th is then given by look when using plan (4.3) where the term in the brackets is the probability of detecting th look in box given that the object on or before the A -step search plan is called the object is in box optimal if

A much stronger notion of optimality, according to which maximized for any number of search plans uniformly optimal if

where denotes the -step search plan corresponding to Uniformly optimal plans can be obtained by impleplan menting the recursive algorithm described below [11]; the proof of its optimality is available in [11]. according to Choose the first box (4.6) denotes the number of looks out of the first If looks that are placed in box under plan then the st is determined by look under plan (4.7) If more than one index maximizes the right-hand side of (4.7), then one should be selected according to some predetermined lexicographic order. fully As the set of parameters specifies the algorithm (4.6)–(4.7), we shall refer to it as the pq algorithm, and we shall call the resulting search plan the pq-search plan. requires only computing The determination of which is given by

Thus, depends only on and starting from the th element of the optimal plan can be obtained recursively. C. More Optimality algorithm generates an optimal search plan which The maximizes the probability of detection in any number of looks. Alternatively, if the search plan is required to minimize the expected number of looks necessary to find the object, then the following proposition shows that the algorithm will also be optimal under this new criterion. Proposition 4.1: A uniformly optimal search plan also minimizes the expected number of looks necessary to find the object. to be the probability of detection on Proof: Define represents the the th look under plan and recall that probability of detection on or before the th look. It is plain that (4.8)

(4.4)

denotes the expected number of looks necessary to find If then the object when using plan

optimality is that of uniform the probability of detection is looks. Let be the set of all A plan in is said to be

(4.9)

(4.5)

where we have used (4.8) to get the second equality, and by Writing in convention, we assume

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(4.9), and changing the order of summations, we get

(4.10) The inner summation is a telescopic sum, and can be replaced if we assume Therefore by (a)

(4.11) If the plan is such that we take by convention. is a uniformly optimal search plan, then by definition If

(b)

and the desired conclusion follows as we note from (4.11) that

Having reviewed the theory of optimal search with discrete efforts, we are now in a position to map the paging problem into the framework of search theory. We do so in the next two sections by first explaining how paging is done in current cellular systems, and by then describing the proposed paging algorithm. V. CONVENTIONAL VERSUS SEQUENTIAL PAGING To frame the discussion, we make the following operational assumptions. The paging requests (PR’s) for different MS’s are queued in the MSC to be processed in a first-in, first-out be the number of BS’s managed by the MSC, fashion. Let denote the number of paging channels available and let to each BS. In effect, paging channels per BS means that MS’s can be paged simultaneously by a given at most BS. We refer to time epochs at which a PR can be sent to a BS as paging cycles. In a first approximation, the paging mechanism is assumed to be perfect in that an MS is always paged successfully if the corresponding PR has been sent to the BS where the MS resides, and discovery takes place within that same paging cycle. paging channels per BS, the conventional With these paging scheme operates as follows. At the beginning of each PR’s are taken from the head of the main paging cycle, BS’s queue in the MSC, and are sent to each of the connected to the MSC. The set of PR’s sent to the BS’s is the same for all the BS’s. The paged MS’s reply by sending a message back to the MSC. In this way, exactly MS’s will be discovered per paging cycle. This brute force approach, although reliable, is clearly wasteful of system resources. In order to understand the

Fig. 4. (a) Conventional versus (b) ideal paging mechanisms.

inherent tradeoff between reliability and resource efficiency, we attempt to contrast the conventional paging method against an “ideal” sequential paging system. We do so in the simple and so that only one PR situation where and can be sent to a BS during a paging cycle. Let denote the three PR’s waiting at the head of the queue in that order at the beginning of a paging cycle. We assume that the and reside in BS1, corresponding MS’s, also denoted BS2, and BS3, respectively. Under the conventional algorithm, in the first paging cycle, is sent to all of the BS’s in the LA simultaneously, PR and a procedure which is then followed by the PR’s in the second and third paging cycles, respectively. This is illustrated in Fig. 4(a). We now try to imagine how an “ideal” sequential paging system would have handled the same situation: In such an ideal system, there would be an entity, hereafter referred to as a smart distributor (SD), which would know the locations of all mobile users within the LA [see Fig. 4(b)]. Acting on this information, at the beginning of the first paging cycle, the SD to BS1, PR to BS2, and PR to would then send PR BS3, so that all three MS’s will be discovered by the end of that first paging cycle. A comparison between the conventional and ideal sequential paging algorithms clearly reveals the superiority of the latter over the former. Indeed, in the conventional method, PR will take three paging cycles to get paged because it has to and PR to be serviced first. This is to be wait for PR compared to the delay of only one paging cycle experienced by PR in the ideal system. Also, fewer paging channels become occupied in the ideal system because each paging request has to be paged three times in the conventional system as opposed to only once in the ideal one.

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Fig. 5. The SD distributes paging requests to the BS’s.

As we will see shortly in the next section, the proposed sequential paging strategy is a natural application of the theory of optimal search to paging in wireless cellular networks. The key observation resides in the following natural connection between the abstract model of Section IV-A and the paging problem: the object to be found is the MS, and cells within an LA correspond to the boxes in the abstract model. In order to explore this connection further, we find it convenient to introduce a class of auxiliary paging policies; this will be useful in describing the proposed paging algorithm.

be presented later in the paper as a part of the proposed sequential algorithm. In principle, the assignment mapping may vary with each paging cycle and may be many-to-one. Paging signals are sent to BS’s via land lines, and are broadcast periodically over the paging channels. Each MS constantly monitors the paging channels to check if it is being paged. The SD redefines the distribution mapping for each paging cycle based on the information collected over previous paging cycles. As soon as an MS is discovered, the corresponding PR is purged from the SD buffer and a new request from the main queue replaces it. Throughout, we assume a perfect paging mechanism in that an MS will always respond to a paging signal destined to it, provided it receives the paging message. However, situations may develop that leave a MS undetected even though the SD elected to initiate a paging message for it to its resident cell. PR’s are Indeed, this will come about when more than Hence, as there are assigned to a BS by the SD through only paging channels available per BS, the PR’s in excess of (to be chosen by some prearranged mechanism) will be considered blocked, and will have to wait until the next paging cycle for another attempt at sending a paging signal to the selected BS. Hence, although an MS may be in the coverage area of the overloaded cell, the PR for it might be blocked and will not go through, with the net effect that the MS will not be discovered.

A. The Paging System

B. The

Of course, the ideal system requires additional (and essentially unavailable) information regarding the location of the MS’s within an LA, and the performance comparison is therefore not quite fair. However, this discussion already points to the possibility of improving the conventional paging algorithm. For instance, through traffic monitoring, it is not too difficult to collect statistics which are indicative of cell occupancy levels. Presumably, in an urban area, some cells are more crowded during a certain time of day than others, and this kind of information could be used to engineer a system that attempts to emulate the behavior of the ideal paging system. The details of how this can be done is the subject of the next section. VI. THE

-PAGING ALGORITHMS

Taking our cue from the discussion in Section V, we consider the system shown in Fig. 5. The buffer space at the MSC containing the PR’s is partitioned into a main queue and a queue associated with the SD; this SD queue is only deep. The PR’s arriving at the MSC are put first in the main queue, and over time are eventually moved to the SD buffer. This transfer is carried out by the SD whose function is to distribute PR’s to BS’s in accordance with some algorithm. In each paging cycle, the SD module processes each of the PR’s in its buffer, independently of each other. It does so by assigning a BS to each of the (no more than) requests in the SD buffer according to an assignment policy with or distribution mapping is the BS assigned to PR or the understanding that is sent to BS in order to be equivalently, that PR paged in the corresponding cell. There are many different ways to generate such assignment mappings; one method will

Start

-Search Plans as Paging Strategies with

a

given set and

of

parameter

values such

that (6.1) In the context of the system just described, we may think of any PR in the SD as specifying a search for the corresponding MS (i.e., object) in a cell (i.e., box) within the LA. As algorithm of Section IV-A is fully specified once the the and are provided, it is therefore possible to parameters algorithm on behalf of any of the PR’s buffered in run the the SD. This results in paging actions which are conducted -search plan (and, for each PR in accordance with the of course, the operational constraints of the system, e.g., algorithm run on blocking). In each paging cycle, the

REZAIIFAR AND MAKOWSKI: FROM OPTIMAL SEARCH TO SEQUENTIAL PAGING

Fig. 6. The vector

P pBq

is affected by the choice of parameters

Q:

behalf of PR will determine the BS where paging to should be attempted in that cycle. Clearly, the BS be searched for PR changes from one paging cycle to the algorithm run on behalf of PR next in accordance with the This paging strategy (with some prearranged mechanism for selecting blocked PR’s) will be referred to as the -paging algorithm. The implementation details are given later in this section. Under weak statistical assumptions on the PR generation process, the system is expected to reach a steady state when denote operating under the -paging algorithm. Let the corresponding steady-state probability of an unsuccessful given that the mobile resides in paging attempt at cell These probabilities are organized that cell, into a probability vector Although depends on both and we do not display this explicit dependence in the notation. and were given We stress that the parameter vectors is a quantity derived from operating at the outset, while the system under the -paging algorithm driven by and in general, we have (6.2) However, as we go back to the abstract model of Section should be viewed as a surrogate for IV-A, we argue that Hence, given the blocking probability its optimality properties, the -paging algorithm would then be a reasonable paging strategy to use provided that, for all we have: 1) reflects the likelihood of an MS residing in cell ; is equal to . 2) In [10], Baser et al. have shown how occupancy probabilities can be estimated from each user’s calling data. If is fixed, then the -search plan the set of probabilities The diagram in Fig. 6 shows how depends only on the -paging algorithm, and are related. The arrows in this figure represent the cause and effect relationship. The is, in fact, a function of dotted arrow signifies that say so that A closedis unlikely to be available in most form expression for cases of interest, and there do not appear to be obvious ways to compute these blocking probabilities, short of resorting to simulations. However, in view of previous comments, it seems reasonsuch that able to seek a vector (6.3)

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and to process PR’s using the -paging algorithm associated Indeed, at the operating point defined by (6.3), the with paging algorithm would behave optimally on behalf of each of the PR’s in the SD with respect to a set of blocking probabilities which would coincide with the true resulting blocking probabilities. In that case, it is expected that the optimality of -search plans would then translate into a good behavior for the corresponding -paging algorithm. The major stumbling block to this approach lies in the computational difficulties raised earlier. This is addressed in Section VII, and leads to a novel sequential paging strategy. C. Implementation Aspects Having identified the different parameters involved in the model of the paging system, we are now in a position to explain the steps that need be taken in order to implement the paging scheme. To this end, for each PR, the SD must keep track of the number of times that a search has been attempted in each cell for this PR. For each PR, at the th paging cycle, is defined as an element of whose the vector th entry represents the number of times that the SD has chosen to send the PR to cell The SD updates and stores at the end of each paging cycle. -search plan, the SD Suppose that by following the decides to send a particular PR to cell One of the following three situations may then develop. 1) There are available paging channels in cell for that paging cycle, and the MS is within the coverage area of BS 2) Same as above, except that MS is not accessible by BS 3) The paging channels in BS are all occupied, and the PR is considered to be blocked for the current paging cycle. Of course, a well-defined procedure is needed to select the blocked PR’s from the population of more than PR’s which have been assigned to the same BS, say random selection for the sake of concreteness. In the first case, the MS responds to the PR, and the paging is successful. All of the records corresponding to a successful page can be removed from the database at SD, and a new PR is fetched from the main queue at the MSC. If the second case occurs, the SD must be forced to avoid trying cell for that particular PR in the following paging algorithm, we observe that cycles. Examining the recursive to a very large this can be accomplished by setting has the interpretation that number. A large value for cell has been searched several times, and the MS was not found there; thus, it is very unlikely for the object to be in that cell. by one to Finally, in the third case, we increment update the number of times a page has been attempted in cell and the information is stored in the vector to be used in the next paging cycle. The flowchart in Fig. 7 summarizes the update procedure of vector It only remains to provide a way of recursively computing the blocking probabilities associated with each cell, an issue which is addressed in the next section.

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Considering the complex probabilistic structure of the proposed paging scheme, a closed-form expression for is very hard, if not impossible, to obtain. Hence, to compute the fixed we have to resort to numerical methods. The point of such that when problem of finding a vector all we can observe is the random vector [and not naturally fits into the framework of stochastic approximation (SA) algorithms. In general, SA methods offer powerful schemes that can be utilized to find the root of a function whose exact analytic form is unknown or difficult to evaluate numerically. It is illustrative to compare an SA method with the Newton–Raphson method [12]. Suppose we want The to find the zeros of the mapping Newton–Raphson method provides a way of computing the zeros by the following recursive scheme: (7.2) Fig. 7. Updating the vector r (m):

VII. THE SEQUENTIAL PAGING STRATEGY The sequential paging algorithm which we now introduce builds on the idea presented earlier that a -paging algorithm should be behaving well if the induced blocking probabilities The proposed algorithm can, constitute a fixed point for in fact, be interpreted as an adaptive version of the algorithm in the sense described below. As pointed out earlier, the difficulty with this approach lies in the complexity of evaluating the blocking probabilities. To gain some insight into the solution, we note that the blocking probabilities depend on the way the SD distributes PR’s among the BS’s. On the other hand, the uniformly optimal search plan given by (4.6)–(4.7) is directly affected by the set of parameters As pointed out earlier, it is important to make a distinction between the set of actual blocking probabilities that result from employing a certain paging scheme, and the set of parameters used by the optimal -search plan. An element in the latter set can be chosen to be any number in the interval [0, 1]. Of course, the proposed algorithm is designed fairly close to the actual blocking in such a way as to have in cell i.e., we need to find probability such that (6.3) holds. the vector and to arbitrary probability vectors and emFixing algorithm with these set of parameters, we can ploying the of the vector of blocking probabilities obtain an estimate in the form (7.1) where is a random vector representing the estimation error. is computed as follows. The vector and and let the algorithm run for a paging Fix paging cycles. Pick large window which consists of paging enough so that the system reaches steady state after let be the fraction of cycles. Then, for times that a PR has been denied access to a paging channel in cell within the paging window.

is the derivative of at Under where certain conditions, the sequence of iterates will converge to a root of To apply the Newton–Raphson and its derivative. What if method, we need to have both is available or computing the derivative only an estimate of is (computationally) too expensive? The Robbins–Monro of method is an SA algorithm which offers a relatively simple solution to this problem. The Robbins–Monro algorithm takes the form (7.3) is a sequence of positive numbers where and tending to zero with is the estimation error sequence. The similarity between the Robbins–Monro and Newton–Raphson algorithms is apparent with by comparing (7.2) and (7.3), and replacing in (7.2). As was the case for the Newton–Raphson algorithm, it can be shown [1], [12] that under certain conditions on the the sequence of iterates noise sequence converges to a solution for Assume, for the moment, that those conditions are met for and use the SA scheme to the function find the fixed point of the mapping By computing the fraction of times a PR fails to get paged within a finite number of paging cycles, we obtain only an estimate of the blocking probabilities, a fact which we denote by Starting from an arbitrary point in the iterate generated by sequence (7.4) can be expected to converge to a solution of almost surely. As part of the conditions for convergence it is satisfies required that the gain sequence and The usual candidate is

REZAIIFAR AND MAKOWSKI: FROM OPTIMAL SEARCH TO SEQUENTIAL PAGING

The proposed paging algorithm then operates as follows. 1) Pick a vector 2) Follow the recursive scheme (4.6)–(4.7) for a paging The length of the paging window window of length should be large enough so that the system reaches its steady state. 3) For each cell, compute over that paging window of the fraction of times it has blocked a PR length (because of the unavailability of paging channels). This —an estimate of will give 4) Update using

5) Increment

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TABLE I SIMULATION PARAMETERS

TABLE II CONVENTIONAL VERSUS SEQUENTIAL SCHEME

and goto step 2). VIII. NUMERICAL RESULTS

We compare the conventional and proposed methods on the basis of two performance measures which capture the efficiency of paging, namely: the expected number of times that an MS is paged before discovery; the expected number of MS’s discovered per paging • cycle. An efficient paging strategy is one that decreases the expected number of pages prior to discovery while increasing the rate of discovery. For the conventional paging method, these two measures are readily obtained. In each paging cycle, exactly distinct PR’s BS’s. This leads to the discovery are sent to each of the of exactly MS’s during a paging cycle. The reason is that only distinct PR’s are processed at each cycle, and since the paging messages are broadcast to all of the BS’s, the paged MS’s cannot miss the call setup requests. Thus, each mobile times (once in each cell) before it gets is paged exactly discovered, and exactly MS’s are found in each paging cycle (if we assume heavy traffic). Therefore •

and For the proposed paging method, no closed-form expresand even under the sions appear readily available for reasonable modeling assumption that PR’s arrive to the MSC according to a Poisson process. As a result, we resorted to simulations in order to evaluate these quantities. In what follows, we discuss several simulation experiments. All simulation runs assume that: 1) the PR’s are generated (PR/paging cycle); according to a Poisson process with rate BS’s in the LA under consideration; 3) the 2) there are size of the SD buffer (or, equivalently, the maximal number of PR’s being processed in a paging cycle) is set at the fixed (this value could have been tuned adaptively value in response to variations in the input rate in order to improve performance—this possibility will not be considered further is given in here); and 4) the occupancy probability vector Table I.

TABLE III SIMULATION PARAMETERS

A. Simulation I The parameters and results for Simulation I are shown in Table II. It is plain from Table II that the proposed method greatly improves the performance of the system. • Each MS is paged 4.16 times in the new system as opposed to ten times (once in each cell) in the conventional GSM. • A larger number of MS’s are discovered in each paging cycle (14.63 MS’s in the new system as compared to nine MS’s in GSM). The occupancy probability vector and the blocking probare given in Table III. ability vector Through some additional simulation experiments, we exon the plore the influence of the incoming paging rate performance measures. Taking the system characterized by vary from 8 PR’s/paging Table II as a baseline, we let

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TABLE IV CONVENTIONAL VERSUS SEQUENTIAL (REDUCED PAGING CHANNELS)

(a)

(b)

TABLE V CONVENTIONAL VERSUS SEQUENTIAL (LIGHTLY LOADED)

Fig. 8. Efficiency of the sequential paging algorithm as a function of incoming paging rate.

cycle all the way to 16 PR’s/paging cycle. The results are depicted in Fig. 8. The top graph gives the relative decrease as a function of the signaling load of the incoming paging rate, while the bottom graph displays the relative increase the number of mobiles found per paging as a function of Both cycle graphs show improvement with respect to the conventional method (except for the slight decrease of discovery rate when the system is lightly loaded). It is clear from Fig. 8 that the efficiency of the sequential paging algorithm in terms of discovery rate increases with the paging load on the system. On the other hand, with respect to the signaling load, the sequential paging algorithm operates more efficiently when the paging system is lightly and loaded. As expected, the performance measures conflict with each other, in the sense that an increase in one is typically accompanied by a decrease in the other, and vice versa. B. Simulation II As the proposed algorithm can lead to a decrease in the signaling load due to paging, it is natural to inquire as to what happens when fewer channels are allocated for the purpose of paging. Requiring fewer paging channels leaves more radio spectrum available for transmission of voice/data, and therefore contributes to the overall improvement of the cellular system. Table IV summarizes the parameters and results of Simulation II; the only difference with the setup of Simulation I is that now the number of paging channels has been reduced to channels per BS. The 52.2% decrease in from signaling load and 58.3% increase is the rate of MS discovery once again reveals the advantages of using the sequential method for paging.

C. Simulation III In the first two simulations, the underlying assumption was that the system is heavily loaded, i.e., there is always a PR in the buffer to replace the PR’s getting through. In this heavy traffic regime, the proposed sequential paging algorithm exhibits a significant improvement over the conventional one. It is nevertheless of interest to examine the performance of the sequential algorithm in light PR traffic, i.e., when the expected number of incoming PR’s within a paging cycle is much less than the number of paging channels per BS (say, To this end, we choose the simulation parameters as listed paging channels per in Table V. As before, there are BS, so that the conventional method is capable of discovering 9 PR’s per paging cycle. The average number of new PR’s (as in per paging cycle has now been changed from and this can be easily handled by Simulation I) to be the conventional method with an SD queue size of The sequential paging algorithm still performs better than the conventional one in terms of the number of times that an MS has to be paged prior to discovery. Note that under the conventional algorithm, each MS is paged exactly once in each cell. This will add up to a total number of ten paging attempts

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relative to the case where it operated with the “true” occupancy Even in this case, the sequential paging probability vector scheme is still superior to the conventional one. Therefore, the fact that vector may not be estimated precisely does not affect drastically the performance of the algorithm, and the algorithm may thus be started with a rough estimate of the occupancy probability

TABLE VI TRUE AND ESTIMATED OCCUPANCY PROBABILITIES

IX. CONCLUSIONS

CONVENTIONAL

TABLE VII VERSUS SEQUENTIAL (SENSITIVITY TO

P)

per PR. The sequential algorithm reduces this figure to three, which is a 70% decrease, an improvement achieved at only a small price, namely, a 15% decrease in the discovery rate. D. Simulation IV A natural, and important, question in connection with the sequential algorithm is its performance sensitivity with respect Recall that quantifies the to the probability vector probability of finding an MS in the various cells within an LA. Clearly, not all of the MS’s follow the same mobility pattern, so that some of them may not conform with the chosen Ideally, we would like to use occupancy probability vector for each MS as the SD runs a specific probability vector algorithm on behalf of each PR in the SD buffer. Some a work has been done on deriving this probability distribution for each user based on the user’s calling data [10]. As such user profiles can only be known approximately, we have tested the sensitivity of the sequential algorithm with respect to in the following way. In Simulation IV, the newly arriving PR’s are all placed in the cells in accordance with the However, the same “true” occupancy probability vector underlying -search plan runs under some other “estimated” different from These two occupancy probability vector vectors are listed in Table VI. Table VII summarizes the other parameters, as well as the results of this simulation. Comparing Tables VII and II, we observe that the performance of the sequential scheme has degraded only slightly

Ideas from the theory of optimal search and stochastic approximations methods were combined into a novel paging algorithm suitable for locating mobile stations within a location area. The stochastic approximation technique is used to compensate for a lack of precise information concerning the complex stochastic structure of the system, and to provide the search algorithm with a required set of parameters that are almost impossible to obtain analytically. The proposed algorithm has two major advantages over the conventional paging method, namely, it reduces the signaling load and significantly expedites the paging process. The proposed paging method is computationally simple, and does not add to the complexity of the system. Through various simulation experiments, we have shown that: 1) the proposed sequential paging algorithm performs well both under heavy and light loads, and typically outperforms the conventional paging algorithm in several performance metrics; and 2) the algorithm is not sensitive to the This, of course, choice of the occupancy probability vector that has practical significance because obtaining a vector captures the occupancy probability of all MS’s represents a practical impossibility. Finally, we conclude with topics for future work concerning several interesting problems of practical significance which have not been addressed directly: 1) design the appropriate number of paging channels at each base station, given a certain traffic pattern for incoming paging requests; and 2) find rules to size up the depth of the SD queue in the mobile switching center and to determine the number of paging channels per base station given the size of the location area and the expected number of paging requests. ACKNOWLEDGMENT The authors thank the anonymous reviewers for their careful reading of the manuscript, and for their constructive comments. REFERENCES [1] H. J. Kushner and D. S. Clark, Stochastic Approximation Methods for Constrained Systems. New York: Springer-Verlag, 1978. [2] U. Madhow, M. L. Honig, and K. Steiglitz, “Optimization of wireless resources for personal communications mobility tracking,” IEEE/ACM Trans. Networking, vol. 3, p. 698, Dec. 1995. [3] J. Plehn, “The design of location areas in a GSM-network,” in Proc. 45th IEEE Vehicular Technology Conf., Chicago, IL, July 1995, pp. 871–877. [4] G. P. Pollini, K. Meier-Hellstren, and D. J. Goodman, “Signaling traffic volume generated by mobile and personal communications,” IEEE Commun. Mag., vol. 33, pp. 60–65, June 1995.

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[5] T. F. La Porta, M. Veeraraghavan, P. A. Treventi, and R. Ramjee, “Distributed call processing for personal communications services,” IEEE Commun. Mag., vol. 33, pp. 66–75, June 1995. [6] CCITT Recommendations, Q.1000 Series, International Telecommunication Union, 1988. [7] C. Rose, “Minimizing the average cost of paging and registration: A timer-based method,” ACM J. Wireless Networks, to be published. [8] C. Rose and R. Yates, “Minimizing the average cost of paging under delay constraints,” ACM J. Wireless Networks, vol. 1, no. 2, pp. 211–219, 1995. , “Ensemble polling strategies for increased paging capacity in [9] mobile communications network,” ACM J. Wireless Networks, 1996. [10] K. Basu, S. Madhavapeddy, and A. Roberts, “Adaptive paging algorithms for cellular systems,” presented at the 5th WINLAB Workshop Third Generation Wireless Inform. Networks, Apr. 1995. [11] L. D. Stone, Theory of Optimal Search, 2nd ed. Arlington, VA: ORSA Books, 1989. [12] M. T. Wasan, Stochastic Approximation. New York: Cambridge Univ. Press, 1969. [13] P. E. Writh, “Teletraffic implications of database architectures in mobile and personal communications,” IEEE Commun. Mag., vol. 33, pp. 54–59, June 1995.

Ramin Rezaiifar received the B.S. degree in electrical engineering from the Sharif University of Technology, Tehran, Iran, in 1988, and the M.S. and Ph.D. degrees in electrical engineering from the University of Maryland, College Park, in 1993 and 1996, respectively. He is currently a Senior Engineer at QUALCOMM Inc., San Diego, CA. He has held an internship position at the Naval Research Laboratory, Washington, DC. During the 1991–1993 academic years, he held an Institute for Systems Research Fellowship at the University of Maryland. His research interests are in the fields of handoff and paging in wireless communication systems and broad-band networks and services.

Armand M. Makowski (M’83–SM’94) received the Licence en Sciences Math´ematiques from the Universit´e Libre de Bruxelles in 1975, the M.S. degree in engineering–systems science from the University of California, Los Angeles, in 1976, and the Ph.D. degree in applied mathematics from the University of Kentucky, Lexington, 1981. He is presently Professor of Electrical Engineering at the University of Maryland, College Park. He has held a joint appointment with the Institute for Systems Research, one of the NSF Engineering Research Centers, since its establishment in 1985, and was its Associate Director for Research during 1995–1996. He is also a cofounder of and active participant in the Center for Satellite and Hybrid Communication Networks, a NASA center for the development and commercialization of space. Over the past few years, he has held visiting positions at the Technion (Israel), INRIA (France), the IBM T. J. Watson Research Center (Hawthorne, NY), and AT&T Bell Laboratories (Murray Hill, NJ). His current research interests include the performance evaluation of switching systems, traffic modeling in high-speed networks, and resource allocation in wireless networks. Dr. Makowski was a C.R.B. Graduate Fellow of the Belgian–American Eduational Foundation for the academic year 1975–1976; he is also a 1984 recipient of the NSF Presidential Young Investigator Award. He is currently serving as Associate Editor for Discrete Event Dynamic Systems—Theory & Applications.