non-uniform traffic issues in dca wireless multimedia networks
TRANSCRIPT
Wireless Networks 9, 605–622, 2003 2003 Kluwer Academic Publishers. Manufactured in The Netherlands.
Non-Uniform Traffic Issues in DCA Wireless MultimediaNetworks
JELENA MIŠIC ∗Department of Computer Science, University of Manitoba, Winnipeg, MB R3T 2N2, Canada
YIK BUN TAMDepartment of Computer Science, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Abstract. Wireless networks that utilize dynamic channel allocation (DCA) are known to perform better than those with fixed channelallocation, in terms of the call level QoS measures such as the handoff dropping probability. On account of this, the DCA networks areusually designed without the call admission control (CAC). However, given the decrease of cell sizes, together with ever increasing mobilephone and terminal population, dynamic channel allocation policies (such as channel borrowing) may not be sufficient to cope with the hot-spot area size and its traffic intensity. This paper analyses the performance of the DCA networks, both with and without the call admissioncontrol, under the hot-spot traffic regime. In such cases, the pure DCA approach fails to ensure sufficiently low level of QoS in both thehot-spot area and the surrounding cells. We propose a CAC policy that can stabilize the QoS under non-uniform traffic, whilst being easy tointegrate in the distributed DCA policies.
Keywords: multimedia wireless networks, dynamic channel allocation, adaptive call admission control, quality-of-service
1. Introduction
Ensuring reliable transfer of multimedia traffic is an impor-tant function of the new generation of cellular networks. Thesmall cell size of such systems results in frequent call handoffevents during the call’s lifetime. The probability of an un-successful handoff (handoff dropping) due to the lack of freechannels is an important measure of call level service quality.Keeping the handoff dropping probability below some spec-ified threshold, while simultaneously maximizing bandwidthutilization, is a challenging task; and the high bandwidth re-quirements of multimedia applications, together with an in-creased frequency of handoffs, do not make it any easier. Thistask can be attacked in two related directions: through the se-lection of channel allocation policy, and through the selectionof call admission control (CAC) policy. Channel allocationpolicies are generally classified into Fixed Channel Alloca-tion (FCA) and Dynamic Channel Allocation (DCA) subcat-egories. Under a FCA policy, each cell is allocated a fixednumber of channels, whereas a DCA policy allows a cell to“borrow” unused channels from its neighbors. It has beenshown that DCA policies offer lower handoff dropping prob-ability than their FCA counterparts [4,7,12], at the expense ofincreased overhead for channel allocation management in mo-bile switching control offices (MSCO) or base stations. More-over, even though the interference averaging techniques, suchas those used in CDMA, perform better than FCA schemes interms of spectrum efficiency [17], the interference avoidanceschemes (which, in effect, include DCA) perform even bet-
∗ Corresponding author.E-mail: [email protected]
ter than CDMA. Consequently, DCA policies are still consid-ered important in the context of third generation of wirelessnetworks with orthogonal frequency division multiple access(OFDMA) [5,18], as well as in the satellite systems.
A number of dynamic channel allocation policies havebeen proposed so far [6,13], and the maximum packing (MP)policy has been shown to be optimal [8]. This type of policyallocates and releases channels in such a manner that maxi-mum channel reuse is achieved. At the same time, it allowsthe number of available channels to be computed easily. TheMP policy normally requires centralized coordination amongcells, which might lead to both computational and commu-nication bottleneck problems with the reduction in cell size.Therefore, some distributed approximation of the MP pol-icy should be used in future wireless networks, preferablywith low to moderate computational and communication load.This is the reason why we adopt the distributed approximationof MP policy, as suggested in [7].
Another problem that needs attention is the QoS deterio-ration in the presence of hot-spots. Under a nominal load,channel borrowing ensures good call level QoS in DCA net-works. However, in an environment with small cell sizes, withever increasing mobile phone and terminal population, trafficintensity in some areas may easily exceed the nominal load– this is called a hot-spot. Depending on the size of the hot-spot, its traffic intensity, and the load in the surrounding cells,channel borrowing may not be possible, and the QoS in thehot-spot and its environment may be seriously affected. Theperformance of DCA networks under non-uniform traffic dis-tribution was evaluated by simulations in [1]. In this paper,we develop the analytical model for a DCA network which
606 MIŠIC AND TAM
explicitly addresses the impact of the implementation of thedistributed MP policy. We analyze the performance of a pureDCA network (i.e., a DCA network without call admissioncontrol) in the presence of hot spots of 1-cell, 7-cell, and in-finite sizes. We show that the QoS is deteriorated not onlyin the hot-spot cells, but also in the whole interference re-gion of the hot-spot cells. To remedy this, we propose a calladmission control algorithm based on our previous work inFCA networks [16]. The algorithm uses event-based hand-off load estimation to reserve bandwidth for future handoffevents, and it can be easily integrated into the DCA networkswith a distributed MP policy. The proposed CAC algorithmstabilizes the call level QoS under the conditions of high loadand non-uniform traffic, at the expense of moderate compu-tational complexity at the base stations. Since the distributedMP policy already requires the base stations to exchange theinformation on available channel sets, the extra communica-tion required for adaptive bandwidth reservation will not sig-nificantly increase the communication load.
Although our framework supports multimedia traffic, thehot-spot traffic analysis considers narrow-band traffic only.Note that in a mixture of narrow-band and wide-band traf-fic, the wide-band calls will experience much higher proba-bility of new call blocking than the narrow-band traffic. Con-sequently, the cell capacity utilization of the wide-band callswill quickly drop to zero, thus effectively making wide-bandtraffic much less important.
The paper is organized as follows. Section 2 describes theMarkov chain model for DCA wireless cell without admis-sion control. In section 3, the admission control algorithmis presented. Performance of DCA network, with and with-out CAC, when traffic is changing uniformly in all the cells,is analyzed in section 4. One- and 7-cell hot spot analysesare given in sections 5 and 6 respectively. Finally, section 7concludes the paper.
2. The system model for a DCA network under uniformload
Consider a wireless cellular system shown in figure 1. Thecells are arranged in an infinite hexagonal mesh. They arelabeled with (x, y), with x standing for the distance fromthe (arbitrarily chosen) center of the mesh – the cell labeledas (0, 0). The set of all cells at the same distance from thecenter is called a ring, and the second index y serves to dis-tinguish individual cells within the same ring. (The meaningof letters will be explained below.) We assume that the proba-bility of a handoff to any of the six surrounding cells is equalto 1/6, and that the call duration and dwell time are expo-nentially distributed, with parameters µ and h, respectively.Furthermore, only narrow-band traffic is assumed, consum-ing one channel per user, under uniform traffic intensity in allthe cells.
Due to the interference problems, the whole set of S chan-nels allocated to the network is divided among g neighboringcells. Each cell is associated with a given set of N nominal
Figure 1. Cell group and cluster representation for g = 7.
channels, i.e., channels that are preferred to other channelsfor allocation in that cell. The set of g cells that collectivelyuse the complete set S = gN of available channels will be re-ferred to as the cluster, and S denotes the cluster capacity. Thecluster size depends on the required signal-to-interference ra-tio, and we will assume that g = 7 in most discussions inthis paper. However, our model is applicable to other clustersizes as well [20], and the impact of the cluster size on theperformance is discussed in section 2.2.2.
The interference region of x, IR(x), is the set of cellsaround x such that, for any cell y ∈ IR(x), a channel that is inuse in x cannot be used in y due to interference. The channelsavailable to x can either come from its own nominal channelset, or from other cells in IR(x). For g = 7, IR(x) consistsof two rings of cells around the target cell (shown shaded infigure 1, assuming that x = (0, 0)). Cells within the inter-ference region that have the same set of nominal channels aresaid to belong to a cell group, and these groups are denotedwith capital letters in figure 1. For g = 7, each group in IR(x)has γ = |IR(x)|/(g − 1) = 3 cells.
2.1. Markovian model for one traffic type
The Markov chain that represents the system for the arbitraryadmission algorithm and a single type of traffic in a DCAnetwork, is shown in figure 2. The length of the Markov chainis R states, which will be calculated in section 2.2.2. Thestates in the chain represent the number of ongoing calls inthe target cell. The new call arrival rate is denoted with λ.We assume that the sum of current utilizations of the targetcell and the cells from the surrounding ring is constant andequal to the sum of their average utilizations. In this case, forthe target cell with k ongoing calls, there are 7Nρav − k usersin the surrounding ring with the “arriving parameter” h, andthe handoff arrival rate is λh,k = (7ρavN − k)h/6. The valueM = ρmaxN is the state wherein the call admission algorithmbegins to reject new calls (in case of a DCA network withoutthe admission control,M = R); beyond stateM , only handoffcalls are accepted. The state probabilities of the Markov chain
NON-UNIFORM TRAFFIC ISSUES IN DCA 607
Figure 2. Markov chain for a single traffic type, with indicated thresholds for borrowing (d = h+ µ).
are equal to:
Pk =
(c6
)k(A1k
)∑M
i=0
(c6
)i(A1i
)+ (A1M )
(A2M )
∑Ri=M+1
(c6
)i(A2i
)for 0 � k � M ,(
c6
)k(A2k
) (A1M )
(A2M )∑M
i=0
(c6
)i(A1i
)+ (A1M )
(A2M )
∑Ri=M+1
(c6
)i(A2i
)for M < k � R,
(1)
where c = h/(h + µ), A1 = 7Nρav + 6λ/h and A2 =7Nρav. The average utilization and the probabilities of newcall blocking and handoff call dropping for this system are,then, given by:
ρav = 1
N
R∑k=0
kPk, Pb =R∑
k=MPk, Phd = PR. (2)
2.2. The distributed channel borrowing policy
Let r denote the number of channels not being used in any cellin IR(x) ∪ {x}, and, therefore, available to x (which belongsto the A group of cells). If the channel reuse is perfect, thisnumber can be obtained as
r = gN − |B| − |C| − |D| − · · · − |G|
where |Z| denotes the maximum number of channels used bythe γ cells of the group Z (Z ∈ [B,C,D,E,F,G]). TheMarkov chain length R is the sum of the average number ofchannels available to cell x, and the average number of chan-nels utilized in x. In order to calculate it, we should note thefollowing. Assume that the channel i from the cell group, say,C = {(1, 4), (2, 0), (2, 3)}, was not used in any cell from thatgroup, or in their interference regions. Assume, then, that thecell x = (0, 0) borrows this channel from that group (whichwill subsequently be referred to as the donor group). Afterborrowing, the capacity of cell x is increased by one channel,and the channel i can not be used in any of the γ = 3 cellsfrom the group C. In this case, we say that the channel i issingle locked in the donor group C. Moreover, the channel iis made unavailable throughout the entire interference regionIR(x). Due to the maximum packing implementation throughthe cost functions [7], cost of borrowing the channel i in cell
k ∈ IR(x) can be:
Cx(k, i) =
0, if i /∈ !(k) and i /∈ FD(k),
1, if i ∈ !(k) and i /∈ FD(k),
2, if i /∈ !(k) and i ∈ FD(k),
3, if i ∈ !(k) and i ∈ FD(k)
(3)
where !(k) denotes set of available channels for cell k, andFD(k) denotes the set of nominal channels for cell k. There-fore, if some cell y (say, (3, 4)) from ring 3 – which sur-rounds the IR(x) ∪ {x} – wants to borrow the channel i fromcell group C, it will find that the cost of that channel in cells(IR(x) ∪ {x})∩ (IR(y)∪ {y}) is 2, if cell belongs to group C,or 0, otherwise. Since the borrowing decision is made bysumming the costs from all the cells in IR(y), channel i hasbetter chances to be borrowed than other channels from !(y)
which are not yet borrowed to any cell. So, when the cell(3, 4) borrows channel i, it will become double locked in thecell (2, 3) from group C. In a similar way, next borrowing bya cell from the group A (say, (3, 7)) will find that the chan-nel i is among the channels with lowest cost. If it is bor-rowed, it will become triple locked in cell (2, 3). Under per-fect channel reuse, the issue whether the channel i given outfrom donor cell (2, 3) is:
1. single locked by borrowing in either of the cells (0, 0),(3, 4), or (3, 7); or
2. double locked by borrowings in any pair of cells [(0, 0),(3, 4)], [(0, 0), (3, 7)], or [(3, 4), (3, 7)]; or
3. triple locked by borrowings from cells (0, 0), (3, 4), (3, 7)
depends only on the traffic intensity.In order to facilitate further analysis we will introduce
another assumption. Consider again the cell (0, 0) whichneeds to borrow a channel i from the donor group C ={(1, 4), (2, 0), (2, 3)}. This channel may be in one of the fol-lowing states:
• double locked in each cell of the group,
• double locked in two cells and single locked (or non-locked) in the third one,
• double locked in one cell and single locked (or non-locked) in two others,
• single locked in each cell of the group,
• single locked in two cells and non-locked in one cell,
• single locked in one cell and non-locked in two other cells,or
• non-locked in all the cells from the group.
608 MIŠIC AND TAM
Each of the previous cases will give a different cost to thechannel i in cell (0, 0). However, in order to reduce the di-mension of the problem, we will assume that the channel isdouble (single) locked in the entire donor group if it is double(single) locked in at least one cell of the group.
2.2.1. The relationship between the number of locked andborrowed channels
Assume that in each cell, under uniform traffic, the number ofborrowed channels in one cell is nb, and the number of lockedchannels is nl. The number of borrowed channels can be fur-ther broken into sum of borrowed channels which are single,double, and triple locked in their respective donor groups, i.e.,
nb = nb1 + nb2 + nb3. (4)
By the same token the number of locked channels in the targetcell can be broken in the sum of channels which are single,double, and triple locked, i.e.,
nl = nl1 + nl2 + nl3. (5)
The relationship between numbers of locked and borrowedchannels in one cell under uniform traffic conditions is:
nl1 = 3nb1, nl2 = 3
2nb2, nl3 = nb3. (6)
When channels are allocated according to maximum pack-ing (MP) policy, the first channels borrowed will be those thatare double locked in their group, since, according to the cost-based approximation of MP policy, their cost will be the low-est. Single locked channels will be borrowed only if there areno double locked channels available, and non-locked channelswill be borrowed only if there are no available single lockedchannels.
Further relationship between nl and nb can be deducedfrom figure 2. If a cell has given nl channels to the othercells, it has to start borrowing from its Markov chain stateN−nl +1, which means that there are R− (N−nl) channelsavailable for borrowing. Therefore, the number of borrowedchannels, nb, can be derived as:
nb =R∑
k=N−nl+1
Pk(k − (N − nl)
). (7)
The major problem in these calculations is to determine thenumbers of borrowed channels nb1, nb2, nb3 which are single,double and triple locked respectively in their donor groups.Since the MP packing policy first borrows channels that aredouble locked in their donor groups, then the channels thatare single locked, and finally those that are not locked at all,we introduce three groups of states in the Markov chain infigure 2. From the states (N − nl + 1) to (N − nl + nl2),only double locked channels are borrowed and turned intotriple locked channels. From the states (N − nl + nl2 + 1)to (N − nl + nl2 + nl1), single locked channels are borrowedand turned into double locked channels. Similarly, from thestates (N − nl + nl2 + nl1 + 1) to R, non-locked channels areborrowed and turned into single locked channels. Then, the
numbers nb1, nb2 and nb3 are obtained as:
nb3 =(N−nl+nl2)∑N−nl+1
Pk(k − (N − nl)
),
(8)
nb2 + nb3 =(N−nl+nl2+nl1)∑
(N−nl+1)
Pk(k − (N − nl)
).
2.2.2. Markov chain lengthThe number of locked channels nl in a cell is always greaterthan the number of borrowed channels nb (they approach eachother under extremely high loads). This limits the effectivenominal capacity of each cell to Nnom = N − (nl − nb), i.e.,to N − ( 2
3nl1 + 13nl2). As nl1 > nl2, the effective nominal
capacity is mostly influenced by the number of single lockedchannels.
Under ideal implementation of the MP policy, the sets ofsingle and double locked channels are identical within thecells of one group, and the length of the Markov chain in thatcase is:
R = N − (nl − nb)+ (g− 1)N(1 − ρav)− (g− 1)(nl − nb).
However, in the distributed, cost based implementation ofthe MP policy, the sets of single locked channels within thecell group may not be the same. Consider the situation inwhich
• There are simultaneous call arrivals in the γ = 3 cellsfrom the same group in IR(x). (To reduce the complex-ity of the problem, we did not take into account the caseswhen simultaneous new call arrivals occur in the cellswhich do not belong to the same cell group, and whichobey reuse distance.)
• Each cell from the group is in the situation to borrow thechannels from the set of non-locked channels.
• The sets of non-locked channels which are identical in allγ = 3 cells from the same group, and the number of chan-nels in the set is nl0.
• Due to the availability of information only about IR(x),the choice between two channels with the same cost is ran-dom.
Then, three possible cases may arise:
1. All three (γ = 3) cells borrow the same channel. Theprobability of this event is nl0/n
γ
l0.
2. Two (i.e., γ − 1) cells out of three borrow the same chan-nel, with the probability γ nl0(nl0 − 1)/nγl0.
3. Each of the cells borrows a different channel, with theprobability nl0(nl0 − 1)(nl0 − 2)/nγl0.
Note that the number of different combinations of borrowingincreases with γ .
Therefore, for each single locked channel in a donor cell,the total number of single locked channels in the donor cell
NON-UNIFORM TRAFFIC ISSUES IN DCA 609
Figure 3. Markov chain for two traffic types in DCA network.
group is:
φ =(
1
n2l0
+ 23(nl0 − 1)
n2l0
+ 3(nl0 − 1)(nl0 − 2)
n2l0
). (9)
The expression (9) gives only the approximate value for φ,since the mismatch can occur when single- and double-lockedchannels are borrowed. However, under loads about thenominal value, the sizes of these sets are much smaller thanthe number of non-locked channels. Under high loads, cellsborrow mostly single-locked and double-locked channels, andthe mismatch is minimal. These facts justify our approxima-tion of φ, which is further corroborated by the simulation re-sults, as will be shown later. Since single locked channels arethe dominant component among the locked channels, thereare approximately φ(nl − nb) channels that are not availablefor each cell group in the IR(x). In that case the size of theMarkov chain for a cell in the DCA network is:
R = N − (nl −nb)+ (g− 1)N(1 −ρav)−φ(g− 1)(nl −nb).
(10)
The entire DCA network without admission control, underuniform traffic conditions, may be described by the followingequations:
ρav = 1
N
R∑k=0
kPk, (11a)
R =N + N(g − 1)(1 − ρav),
− (φ(g − 1)+ 1
)(nl − nb), (11b)
nl = nl3 + nl2 + nl1, (11c)
nl3 + 2
3nl2 + 1
3nl1 =
R∑N−nl+1
Pk(k − (N − nl)
), (11d)
2
3nl2 + nl3 =
(N−nl+nl2+nl1)∑(N−nl+1)
Pk(k − (N − nl)
), (11e)
nl3 =(N−nl+nl2)∑N−nl+1
Pk(k − (N − nl)
)(11f)
which can be solved for variables R, ρav, nl, nl1, nl2 and nl3.The related numbers of borrowed channels can be found fromequation (6).
Note that the cluster size g appears in both second andthird terms of equation (11b). Assuming that the new callarrival rates are dimensioned according to the physical areacovered by the cluster, the third term will be the dominantone, meaning that the larger cluster sizes correspond to largernumber of cells per group, and consequently to larger valuesof the mismatch function φ.
2.2.3. Multiple traffic typesExact analysis of the performance with multiple traffic typesrequires the use of a multidimensional Markov chain. Wewill present the case of two traffic types, again without theadmission control, as shown in figure 3. Let bv = 1 denote
610 MIŠIC AND TAM
the bandwidth requirement of the voice traffic, and bd > 1 thebandwidth requirement of the wide-band traffic.
The states of the Markov chain are denoted with two in-dices, (k1, k2), where k1 denotes the number of ongoing voicecalls, and k2 denotes the number of ongoing wide-band flows.In the absence of wide-band flows, the Markov chain has thelength Rt,1, whereas in the absence of voice calls, the lengthof the chain is Rt,2.
Let us consider the set of states and edges between themat the boundary of the chain, between states (0, Rt,2) and(Rt,1, 0). These two states are connected by a sequence ofhorizontal segments interconnected by one-step vertical seg-ments. The length of horizontal segments can be differentfrom bd + 1 states, due to the mismatch function φ. Thearrangement of horizontal edge segments between (0, Rt,2)
and (Rt,1, 0) can be determined if the chain shown in figure 3is consideres as a set of columns. Each of the columns cor-responds to (0 � k1 � Rt,1) voice calls and has the heightof �Y (k1)�. The value of Y (k1) can be found from the set ofequations (11) under the assumption that the nominal capac-ity of the cell is (N − k1)/bd wide-band channels. Then, thenumber of states of this Markov chain is:
S =Rt,1∑k1=0
�Y (k1)�. (12)
The stationary probability distribution of the states in thisMarkov chain has the product form:
P(k1, k2) =1k1!(λ1+λh1µ1+h1
)k1 1k2!(λ2+λh2µ2+h2
)k2
G
where G is the normalization constant introduced in order toensure that P(k1, k2) is indeed a probability distribution [3],i.e.:
G =∑
(k1,k2)∈S
1
k1!(λ1 + λh1
µ1 + h1
)k1 1
k2!(λ2 + λh2
µ2 + h2
)k2
.
The average utilizations per traffic class can be obtainedfrom:
ρav,1 = 1
N
Rt,1∑k1=0
�Y (k1)�∑k2=0
k1P(k1,k2), (13)
ρav,2 = bd
N
Rt,1∑k1=0
�Y (k1)�∑k2=0
k2P(k1,k2). (14)
The handoff dropping and new call blocking probabilitiesare:
Pb,2 = Phd,2 =Rt,1∑k1=0
P(k1, �Y (k1)�
),
Pb,1 = Phd,1 =Rt,2∑k2=0
P(�Z(k2)�, k2
)(15)
where Z(k2) is the solution of the system (11) for k2 wide-band sources. Note that Pb,2 � Pb,1, since the numberof components in the sum for Pb,2 is about kd times larger
than that for Pb,1. Under high arrival rates for narrow-bandand wide-band traffic, probabilities of the states �Y (k1)� willbe high, which means that the wide-band traffic will experi-ence high new call blocking and handoff dropping probabili-ties. Therefore, cell utilization under high arrival rates will bemostly contributed by narrow-band traffic.
3. Admission control
If QoS has to be tightly controlled, adaptive bandwidth reser-vation must be applied in the network. We use the conceptof the bandwidth reservation from the FCA framework [16],wherein a dedicated bandwidth reservation process is main-tained in each cell with the purpose of matching the hand-off call arrival process. In order to implement the bandwidthreservation to match the handoff arrival rate λh, we create thebandwidth reservation rate in the target cell using the entireuser population in the first surrounding ring. Namely, eachcall that arrives to a cell, either as a new call or a handoff call,will send some small number – called the bandwidth reserva-tion value – to all the surrounding cells. When the call leavesthe cell, that number is cleared from all the surrounding cells.Every base station maintains the sum of those numbers asso-ciated with the bandwidth requirement of the call. By analogyto the handoff call arrival rate, the sum of bandwidth reser-vation values corresponds to the average bandwidth reserva-tion rate of the bandwidth reservation process. Therefore, thebandwidth reservation rate in the target cell is
ν =∞∑m=0
6nmam
where nm denotes the number of calls with m executed hand-offs and am denotes the bandwidth reservation value sent af-ter the mth handoff. Although different bandwidth reserva-tion policies can be implemented by using various series am,in this work we focus on the feedback property of the band-width reservation under varying values of offered load, underthe assumption that am = const and ν = 6amNρav.
In order to achieve proper bandwidth reservation for hand-off calls, the call admission algorithm should consider boththe bandwidth reservation rate (which carries the informationabout potential handoff call arrivals) and the residual capacityof the cell (which has to accommodate the handoff calls). Theadmission criterion for new calls should be the capability ofthe residual cell capacity to accept arrivals of the bandwidthreservation process. For example, if the residual cell capacityis Q channels, then it can accommodate up to Q arrivals ofthe bandwidth reservation process. The probability that band-width reservation process will make more than Q arrivals isthe probability that residual capacity can not accommodatearrivals of the bandwidth reservation process; we will denoteit as the overload probability Pov.
To discuss this parameter let us consider the cell at time t ,with instantaneous values of cell capacity Rins, number of uti-lized channelsNρins, and bandwidth reservation rate νins. ThedifferenceRins −Nρins denotes the instantaneous residual ca-
NON-UNIFORM TRAFFIC ISSUES IN DCA 611
pacity of the cell. Then the instantaneous value of the over-load probability is equal to:
Pov = e−νins
∞∑q=�Rins−Nρins�
νq
ins
q!
= 1
-(Rins −Nρins)
∫ νins
0e−xxRins−Nρins dx. (16)
Figure 4 shows the overload probability for a range ofbandwidth reservation rates and residual cell capacities. Weobserve that Pov has low values if the bandwidth reservationrate is small and/or residual cell capacity is moderate to large.However, if the overload probability is high, this means thatthe reservation rate is large, and/or the residual capacity is toosmall to accommodate arrivals of the reservation process. Ifthe value of the overload probability is fixed to some thresholdvalue (denoted with PovT), we will get the new call admissioncondition that will keep the amount of residual capacity suffi-ciently high so as to match the bandwidth reservation rate.
Since we choose the bandwidth reservation values am suchthat ν � λh, we can choose PovT from the range (0..1) de-pending on the required handoff dropping probability. The
Figure 4. Overload probability as a function of bandwidth reservation rateand residual cell capacity.
smaller the value of PovT is, the smaller the handoff droppingprobability will be. The price to pay for this is an increase ofthe new call blocking probability – which necessitates a care-ful choice of PovT. The dependence of both handoff droppingprobability and new call blocking probability on tuning para-meter PovT is shown on figure 5. It shows handoff droppingand new call blocking probabilities for the values of the PovTequal to 0.5, 0.6, 0.7, and 0.8, respectively. We defer furtherdiscussion of this figure to section 4.
It can be shown that the relationship between the thresh-old PovT, the average cell capacity R at the new call arrivalrate λ, the average cell capacity utilization ρav, and the aver-age level of cell capacity utilization at which admission algo-rithm starts rejecting new calls ρmax (i.e., R − Nρmax is theaverage amount of residual capacity in the cell), is given by
PovT = 1
-(R −Nρmax)
∫ 6amρavN
0e−xxR−Nρmax dx. (17)
(The averages are taken over a very long time interval t ∈(0..∞).)
3.1. The admission algorithm based on Pov
We shall illustrate the algorithm using the cell (0, 0) as thetarget cell, as before. The main steps of the algorithm are:
1. Variables ν and ρcurr are set to zero.
2. When a new call arrives, the overload probability has tobe calculated using equation (16) in the cell (0, 0) and thesurrounding cells.Pov checking in cell (0, 0): ρcurr is incremented by 1/N ,and Pov(0, 0) is recalculated with the new value of ρcurr.If the updated value is larger than the predeterminedlimit, the new call is blocked.Pov checking in surrounding cells: In each cell (1, i) (0 �i � 5), the value of local ν is incremented by am, and thecorresponding Pov is recalculated. If the PovT in any ofthese cells is violated, the new call is blocked.
Figure 5. Impact of QoS tuning parameter PovT on handoff dropping and new call blocking probabilities under uniform load.
612 MIŠIC AND TAM
3. The calls which are admitted in cell (0, 0) can freelyhandoff to neighboring cells without invoking the calladmission algorithm again. The handoff call is droppedonly when there is insufficient bandwidth for serving thecall. When a call successfully executes its mth handofffrom cell (0, 0) to a neighbor cell (1, i), the followingactions must be performed:Update load info. in cell (1, i): ρcurr is incremented by1/N , and ν is decremented by the bandwidth reservationvalue am.Update load info. in the neighborhood of cells (0, 0)and (1, i): This action consists of the following steps:
(a) The new reservation values are distributed to cellssurrounding the cell (1, i), and the correspondingvalues of ν are updated. Note that some cells not pre-viously included in the neighborhood of cell (0, 0)will receive the call’s reservation factor for the firsttime. This reservation factor will be taken into ac-count in the execution of the admission algorithm forfuture new calls in these cells.
(b) The cells bordering the cell (0, 0), which was hostingthe call before the handoff, that do not belong to the1-ring neighborhood of the cell currently hosting thecall should delete the appropriate reservation factor.
4. When a call terminates, all corresponding bandwidthreservation values in the neighboring cells are deleted,and ρcurr in the hosting cell is decremented by 1/N .
3.2. Adaptivity of the admission algorithm
Due to the consideration of both the residual capacity and thebandwidth reservation rate this admission algorithm is adap-tive to the following factors:
1. The new call arrival rate in the target cell, since for fixedν residual cell capacity can be decreased as long as PovTis not exceeded.
2. The new call arrival rate in the cells belonging to thering which surrounds the target cell, since the bandwidthreservation rate is directly proportional to the number ofusers in the surrounding ring.
3. The call handoff rate h and call departure rate µ throughindirect and direct way. In indirect way, whenevercall handoff rate increases and/or call departure rate de-creases, the average cell capacity will tend to increase dueto the larger number of handoff calls, which will result inthe increase of the bandwidth reservation rate. This in-crease will request larger residual capacity for handoffcalls and therefore new call blocking probability will in-crease. In direct way, the bandwidth reservation valuesam are made directly proportional to the number of ex-ecuted handoffs [15]. In such approach the bandwidthreservation rate will grow as the average number of exe-cuted handoffs grows.
3.3. Discussion about the performance of admissionalgorithm under general session time distributions
Although the exponential distribution was traditionally usedto model holding times of voice telephone calls, this modelmay not be appropriate for other services in cellular net-work. (Remember that modern cellular systems support ser-vices such as fax, video, electronic mail, and data.) Even forvoice services, the analysis of the collected data shows thatvoice call durations can be better modeled with log-normaland Erlang-j, k distributions than with exponential distribu-tion [2,11].
In addition to call duration modeling, performance analy-sis of cellular networks also requires modeling of the calldwell time. The distribution of call dwell times is affectedby the variable cell sizes and irregular cell shapes, especiallyin micro- and pico-cell environments, as well as the users’direction and speed. Although dwell times have been tradi-tionally modeled with exponential distribution, it is claimedthat better results have been obtained with hyper-exponentialmodeling [19] (and partly [9]) and sub-exponential modeling[9,10,14].
In the hyper-exponential approach [19], call (session)holding times and dwell times are modeled as sums of statis-tically independent hyper-exponential random variables (SO-HYP). The authors evaluate network performance for therange [1..4] of squared coefficient of variation κ . Their analy-sis shows that forced call termination probability decreasesvery slightly when the variations of call holding time anddwell time increase. Also, the increased variations of callholding time and dwell time lead to a decrease in the expectednumber of handoffs per call. This further means that the hand-off call arrival rate will be lower than in the case with expo-nential distribution. Consequently, the performance boundsderived for the exponential call holding times and dwell timescan be used.
Sub-exponential modeling of call holding times was dis-cussed in [9,10,14], under various distributions of dwell timeand various values of users’ mobility (which is representedas the ratio of the expected call holding time and cell dwelltime). Call holding times were modeled with Erlang distrib-ution [10,14] and with Erlang, Gamma, r-stage exponential,and hyper-Erlang distributions [9]. Dwell times were mod-eled with Gamma distribution where squared coefficient ofvariation κ was chosen to be 10, 1, and 0.1 in [14], and 2/3in [9]. The major conclusion of this work is that the call com-pletion probability (i.e., the probability that call is connectedand completed) decreases as the variance of the call holdingtimes decreases. This effect is negligible for low values ofusers’ mobility and/or large variance of dwell times κ � 1; itbecomes more pronounced when the variance of dwell timesis small and users’ mobility is high. (According to [9,10],“high users’ mobility” actually means values of 10 or higher.)The reason for an increase in call dropping is the higher av-erage number of handoffs, i.e., the larger handoff call arrivalrate compared to the case with exponential call holding timesand dwell times.
NON-UNIFORM TRAFFIC ISSUES IN DCA 613
Figure 6. Utilizations in the DCA network.
From this discussion we may conclude that performanceresults we have derived for exponential call holding timesand dwell times are quite satisfactory for hyper-exponentialcall holding and dwell times. Our results are also acceptablefor the sub-exponential call holding and dwell times with lowvalues of users’ mobility. Only in cases of sub-exponentialcall and dwell times with high users’ mobility, a more aggres-sive reservation is necessary. However, when enhanced withdirect mobility dependent bandwidth reservation, our schemehas the potential to cope with the increases of handoff call ar-rival rate which occurred due to the smaller variances of callholding times and dwell times and due to high mobility [15].
4. Results of the uniform load (infinite hot-spot) analysis
QoS in pure DCA network. The solutions of the system ofequations (11) for different values of the new call arrival rateare shown in figures 6–9. In all models we have assumedthat N = 50, h = 0.01, and µ = 0.002. Simulations wererun with the same parameters as the analytical model, and theresults obtained through simulation are shown on the samegraphs (except for the average Markov chain length).
Note that the length of the Markov chain approachesthe number of nominal channels N = 50 when the offeredload exceeds 150% of the nominal load λnom = 0.1(λnom/Nµ = 1), which means that network starts operatingclose to the FCA regime. This could be predicted by inspect-ing equation (11b), since under very high loads all channelstend to be triple locked, and the term nl − nb = 1
3nl2 + 23nl1
approaches zero. Figure 8 shows that the handoff droppingprobability exceeds 5% when λ exceeds 0.12 (i.e., when theoffered load exceeds the nominal load by as little as 20%).
QoS with admission control. The admission control will al-leviate the effect of shrinking of the Markov chain under in-creasing load, by inserting the threshold state where new callswill not be admitted and only handoff calls will be allowed.The performance of DCA network with admission control can
Figure 7. Average Markov chain length in the DCA network.
Figure 8. Handoff dropping probability in the DCA network with and with-out CAC (note that for pure DCA network handoff dropping and new callblocking probabilities are the same).
be assessed by simultaneous solution of the system (11) withthe equation (17) for a given value of the overload probabilitythreshold PovT. (In our calculations we used PovT = 0.5 andam = 0.0395.)
The results for DCA network with admission control areshown in figures 6–9. Due to the channel borrowing, thelength of the Markov chain is larger than the number of nom-inal channels for the cell. Therefore, the threshold state of theMarkov chain, M = Nρmax, can be larger than the number ofnominal channels N , depending on the new call arrival rateand PovT. As a consequence, we can obtain ρmax > 1.
The handoff dropping probability is much lower than inthe pure DCA case, on behalf of the moderate increase ofnew call blocking probability. The necessary value of Phdcan be tuned by choosing a suitable value for PovT. Figure 5shows new call blocking and handoff dropping probability forseveral values of tuning parameter PovT. We observe thatvalues of handoff dropping probability less than 1% can be
614 MIŠIC AND TAM
(a) (b)
Figure 9. Numbers of locked and borrowed channels in DCA network without CAC (a), and with Pov-based CAC (b).
achieved with new call blocking probabilities which are equalto several percents for nominal load and less than 30% forload that is three times higher than the nominal one. More-over, those blocking probabilities are only 20–25% higherthan their counterpart in the pure DCA case (without the ad-mission control) shown in figure 8.
4.1. Analysis of DCA parameters
In this subsection we will investigate the relationship betweenR, ρav, and ρmax under uniform load. When the new callarrival rate λ grows, the ρav will tend to grow in all the cells,and the Markov chain length will tend to decrease.
Let us first consider the case of pure DCA. Figure 9 showsthat the numbers of locked and borrowed channels changevery slowly with the new call arrival rate, so we can assumethat their first derivatives are approximately zero. Differenti-ating equation (11b), and taking into account that g = 7, weobtain:
dR
dλ≈ −6Ndρav
dλ(18)
which means that Markov chain length will decreaseN(g−1)times faster, compared to the increase of the average utiliza-tion. The decrease of Markov chain length when the new callarrival rate grows will lead to high handoff dropping proba-bility.
Without the admission control, the first derivative of aver-age utilization with respect to the new call arrival rate λ canbe found from equation (11a) by considering ρav as a func-tion, ρav(ρav(λ), R(λ), λ), and finding the corresponding par-tial derivatives. Note that the first term in the previous expres-sion comes from the dependency of average utilization on the
handoff call arrival rate, as shown in equation (1).
dρav
dλ= 1
N
R∑k=0
k∂Pk
∂ρav
dρav
dλ+ 1
N
∂(∑R
k=0 kPk)
∂R
dR
dλ
+ 1
N
R∑k=0
k∂Pk
∂λ. (19)
The partial derivative (∂/∂R)∑R
k=0 kPk can be calculatedusing its integral counterpart, i.e., the identity (d/dx)×∫ y
0 xf (x) = yf (y). After substituting expression (18) into(19), we obtain the expressions for the first derivative of aver-age utilization and handoff dropping probability for the pureDCA case:
dρav
dλ≈
1N
∑Rk=0 k
∂Pk∂λ
1 + 6RPhd − 1N
∑Rk=0 k
∂Pk∂ρav
, (20a)
dPhd
dλ≈
∂Phd∂λ
1 + 6RPhd − 1N
∑Rk=0 k
∂Pk∂ρav
. (20b)
In the presence of admission control, the increase of av-erage utilization 2ρav will be accompanied by the increaseof the bandwidth reservation rate, 2ν = 2ρavN6am. Thisincrease will cause the decrease of the threshold value M =ρmaxN , as shown in equation (16). This decrease will, in turn,diminish the total increase of ρav, thus effectively providing a(negative, or compensating) feedback action.
The action of the feedback can also be qualitatively ex-plained using figure 10. The Pov value corresponds to thearea below the integrand function f (x) = e−xx(R−Nρmax)/
(-(N(1 − ρmax))) in the segment x = [0, ν]. The functionf (x) has the maximum at x = (R −Nρmax). When λ grows,the average cell utilization tends to grow as well, and the endof the integrating segment ν tends to shift to the right, towards
NON-UNIFORM TRAFFIC ISSUES IN DCA 615
Figure 10. Representation of the regulation property of the admission algo-rithm.
the higher values of Pov. However, the Pov regulation main-tained by the admission algorithm has to keep the area belowthe function at a constant value PovT, so it will then shift theintegrand function f (x) to the right. (The shifted functiongets distorted in the process, but this is negligible for smallchanges of utilization.) Therefore,2ν ≈ 2(R−Nρmax), andwhen λ changes we have:
dν
dλ≈ dR
dλ− Ndρmax
dλ. (21)
Assuming that the bandwidth reservation is constant, ν =N6amρav, g = 7, we may combine equations (18) and (21) toobtain:
dρav
dλ≈ − 1
6(am + 1)
dρmax
dλ, (22)
dR
dλ≈ N
(am + 1)
dρmax
dλ≈ N
dρmax
dλ(23)
since 1 � am. Therefore, the increase in new call arrival rateswill cause the admission algorithm to move the clipping stateM = Nρmax of the Markov chain towards lower values. Thisfeature will control the QoS, since the rate of changes of ρav
and R are determined by the bandwidth reservation.Again, the first derivative of average utilization with re-
spect to new call arrival rate λ, can be found from equa-tion (11a) by considering ρav as a function, ρav(ρav(λ),ρmax(λ),R(λ), λ), and by finding corresponding partial deriv-atives:
dρav
dλ= 1
N
R∑k=0
k∂Pk
∂ρav
dρav
dλ+ 1
N
∂(∑R
k=0 kPk)
∂ρmax
dρmax
dλ
+ 1
N
∂(∑R
k=0 kPk)
∂R
dR
dλ+ 1
N
R∑k=0
k∂Pk
∂λ. (24)
The derivative (1/N)(∂/∂ρmax)∑R
k=0 kPk can be esti-mated (again, using its integral counterpart) as MPM −(M + 1)PM+1, where M = Nρmax is the clipping state inthe Markov chain after which the CAC algorithm ceases toadmit new calls. Since PM > PM+1 (see expression (1)),we may approximate this value with M(PM − PM+1). After
substituting expressions (22) and (18) into (24), we obtain thefirst derivatives of average utilization and Phd as:
dρav
dλ≈ 1
N
R∑k=0
k∂Pk
∂λ
(1+ 6RPhd + 6(am+1)M(PM−PM+1)
− 1
N
R∑k=0
k∂Pk
∂ρav
)−1
, (25a)
dPhd
dλ≈ ∂Phd
∂λ
(1+ 6RPhd + 6(am +1)M(PM−PM+1)
− 1
N
R∑k=0
k∂Pk
∂ρav
)−1
. (25b)
Under high loads, both expressions (20a) and (25a) willconverge towards zero, i.e., they will be bounded by the func-tion Kρ/λ, where Kρ is the proportion constant. The rea-son for this is as follows: the sum (1/N)
∑Rk=0 kPk grows
from 0 to 1 when either λ, ρav (via λh), or ρmax grow. Con-sequently, the partial derivatives (1/N)
∑Rk=0 k(∂Pk/∂λ),
(1/N)∑R
k=0 k(∂Pk/∂ρav), and (1/N)(∂/∂ρmax)∑R
k=0 kPkare positive and decreasing (actually, converging towardszero) functions of λ, ρav, and ρmax, respectively. Hence, thenumerators of both expressions (20a) and (25a) will convergetowards zero. However, since the average utilization is the lin-ear combination of all state probabilities of the Markov chain,ρav = (1/N)
∑Rk=0 kPk , the convergence of its derivative to
zero under high offered loads means that the derivatives ofother linear combinations of state probabilities (such as thehandoff dropping probability and new call blocking probabil-ity) will also converge to zero. Therefore, the numerators ofthe expressions (20b) and (25b) will also converge towardszero, i.e., they will be bounded by the functionKPhd/λ.
On the other hand, the denominators will have non-zerovalues. In the case of pure DCA, the main component of thedenominator is 6RPhd, so dρav/dλwill converge towards zeroon account of the large handoff dropping probability. Theupper bound for the handoff dropping probability can be ob-tained by solving the differential equation derived from (20b):
PhddPhd
dλ= KPhd
λ
and it is equal to P ∗hd = KPhd1 +KPhd2
√log(λ), where KPhd1
andKPhd2 are proportion constants determined from the initialconditions.
In the case of a DCA network with the CAC algorithm, themain part of the denominator in expressions (25a) and (25b)comes from the bandwidth reservation. In this case the upperbound for the handoff dropping probability can be obtained asP ∗
hd = KPhd3 +KPhd4 log(λ), whereKPhd3 and KPhd4 are againproportion constants determined from the initial conditions.
Since the length of the Markov chain is much larger in thecase with bandwidth reservation, the handoff dropping prob-ability under high loads will be several orders of magnitudesmaller than in the pure DCA case. For example, under the
616 MIŠIC AND TAM
load of twice the nominal value, the handoff dropping proba-bility in the pure DCA network is equal to 0.159, while in theDCA network with CAC and PovT = 0.5, the same probabil-ity is only Phd = 0.001.
5. Single cell hot-spot analysis
In this section we will analyze the case where the new callarrival rate λ in one target cell is different from the new callarrival rates in the rest of the network. (In other words, thereis a single-cell hot-spot in the network.) For simplicity, let thehot-spot be in the cell (0, 0). Again, we will compare the pureDCA network with the DCA network with admission control.We assume the following:
• The new call arrival rates in the non-hot-spot cells are con-stant and equal to λenv.
• There is one traffic type and directional probabilities areuniform.
• In the case when admission control is applied, all cellswill use the same value of PovT to regulate the utilization.Again, all numerical results were obtained with the valueof PovT set to 0.5, but other values may be used to fine tunethe QoS, and am = const = 0.0395.
Analysis now becomes more complex since the interactionbetween the hot-spot cell and the cells in its surrounding ringis non-symmetric. We modeled four rings around the hot-spotcell, assuming that the change of the utilization in the fifthring due to the increase of the new call arrival rate in the targetcell (0, 0) is negligible. (Obviously, adding more rings willincrease accuracy, but it will also increase the computationalcomplexity of our model.)
In the analytical model, we addressed the problem of non-equal utilizations in the cells of the same ring due to the dif-ferent number of the bordering edges towards the inner ring,as can be observed from figure 1. The cells from the sec-ond ring which have two bordering edges with the cells fromthe first ring are denoted as “odd” cells, and others with onebordering edge are modeled as “even” cells. In the third ring,cells with two bordering edges towards the second ring are de-noted as “odd” cells and the others with one bordering edgeare modeled as “even” cells. For simplicity, fourth ring wasmodeled with a single cell type. Altogether, we modeled 7different cell types t (t ∈ 0, 1, 2o, 2e, 3o, 3e, 4). For eachcell type t (except cell (0, 0), which belongs to cell groupA),we used the cell(s) from group C to model the interactionsamong the target cell and its neighbors. For target cell oftype t , we define the set of six neighboring cells as st (forparticular cell numbers, see figure 1). For example, the sumof average utilizations in six surrounding cells for cell type twill be denoted as
∑(st) ρ
(st)av . State probabilities of Markov
chains given in (1) for each cell type t , were modeled usingthe handoff load from the set (st).
The new call arrival rate in the hot-spot cell was variedfrom the nominal load λnom = Nµ = 0.1 to three times the
nominal load. The new call arrival rates in all other cells wereset to the constant value λenv.
Simulation results for average utilizations and handoffdropping probabilities are also included in the graphs. Simu-lations were run with the same parameters as in the analyticalmodel, except that they used 13 rings, while the analyticalmodel described four rings (five, if we count the hot-spot cellas a separate ring), and treated all other rings as if they hadidentical QoS as the ring no. 4.
5.1. Pure DCA case
All cells from the cell type t (t ∈ 0, 1, 2o, 2e, 3o, 3e, 4), hav-ing the set of six surrounding cells st were modeled with thefollowing system of equations:
ρ(t)av = 1
N
R(t)∑k=0
kP(t)k , (26a)
R(t) =Ng −∑st
ρ(st)av − (n(t)l − n
(t)b
)− φ
∑(st)
(n(st)l − n
(st)b
), (26b)
n(t)l = n
(t)l3 + n
(t)l2 + n
(t)l1 , (26c)
n(t)l3 + 2
3n(t)l2 + 1
3n(t)l1
= 1
6
∑(st)
R(st)∑N−n(st)l +1
P(st)k
(k −N + n
(st)l
), (26d)
2
3n(t)l2 + n
(t)l3
= 1
6
∑(st)
(N−n(st)l +n(st)l2 +n(st)l1 )∑(N−n(st)l +1)
P(st)k
(k −N + n
(st)l
), (26e)
n(t)l3 = 1
6
∑(st)
(N−n(st)l +n(st)l2 )∑N−n(st)l +1
P(st)k
(k −N + n
(st)l
). (26f)
Equation (26b) deserves more explanation. Namely, in thehot-spot environment, the numbers of locked and borrowedchannels within the cells in the same cell group are not thesame any more. Cells immediately surrounding the hot-spotwill have the largest number of locked channels. If that num-ber is used, we may underestimate the length of the Markovchain for the hot-spot cell(s). To cater for that, we used thecell at the largest distance from the hot-spot instead. Also,in order to reduce the computational complexity, we approx-imated the φ with a constant value derived for nominal loadunder uniform load conditions. (The accompanying loss ofaccuracy is negligible.)
The resulting system of 42 equations was solved numer-ically for variables R(t), ρ(t)av , n(t)l , n(t)l1 , n(t)l2 , and n(t)l3 . Fig-ures 11 and 12 show the call level QoS in the cells of the
NON-UNIFORM TRAFFIC ISSUES IN DCA 617
Figure 11. Average utilization and Phd (and also Pblock) for single cell hot-spot in pure DCA network.
Figure 12. Average Markov chain length and number of locked channels for single cell hot-spot in pure DCA network.
rings 0–4, under changing load in the hot-spot cell (ring 0),while the load in the surrounding cells was kept at λenv = 0.1.For rings 2 and 3, the average value of QoS between odd andeven cell types is shown as well.
Note that the hot-spot cell will borrow channels fromring 1, which will result in an increase of the average lengthof Markov chain in the hot-spot cell. However, since othercells are loaded with nominal load and the cell cluster capac-ity is limited, the growth of the Markov chain length in thehot-spot cell can not cope with the growth of new call arrivalrate. As a result, the handoff dropping probability in hot-spotcell will increase. Large utilization in the hot-spot cell willalso result in the increase of the number of locked channels indonor cells from ring 1. This will, in turn, cause the decreaseof the length of Markov chain in the donor ring 1. Further-more, since cells in the ring 2 belong to the interference re-gion of the hot-spot cell, channels borrowed by the hot-spotare locked in the ring 2 as well. This will decrease the lengthsof Markov chains of cells from ring 2, as well as their averageutilizations; it will also increase the handoff dropping proba-bilities in those cells. The number of totally locked channels
in ring 3 will decrease, due to the increased number of doubleand triple locked channels. Further, the decrease of averageutilization within ring 2 will cause the increase of the Markovchain length in ring 3 and the decrease of the total number oflocked channels and so on. Therefore, both rings 1 and 2 willexperience large handoff dropping probabilities, as a conse-quence of a hot-spot in cell (0, 0).
5.2. DCA with admission control
In this case the system of equations (26) is extended with theadmission equations:
PovT = 1
-(R(t) − Nρ(t)max)
∫ ν(t)
0e−xx(R(t)−Nρ
(t)max) dx (27)
where ν(t) = amN∑
(st) ρ(st)av . Figures 13–16 show call level
QoS in the cells of the rings 0–4 under changing load in hot-spot cell while the load in the surrounding cells was kept atλenv = 0.1. We observe that new call blocking probabilityin the hot-spot cell and first surrounding ring are around 40%
618 MIŠIC AND TAM
Figure 13. Average utilization and Phd for single cell hot-spot in DCA network with CAC.
Figure 14. New call blocking probabilities for single cell hot-spot in DCAnetwork with CAC.
larger than in the case without admission control. However,this is the price for achieved very low handoff dropping prob-abilities.
We will model the feedback interaction between the hot-spot cell and the first surrounding ring by neglecting QoSchanges in the second ring. The motivation for this can befound in the small changes of utilization in the second andfurther rings shown in figure 13. On the basis of results fromsection 4.1, and taking into account size of the DCA cell clus-ter, g = 7, the derivatives of the Markov chain length in thehot-spot cell and the first surrounding ring are:
dR0
dλ≈ −N6dρ1
av
dλ, (28a)
dR1
dλ≈ −N
(dρ0
av
dλ+ 2
dρ1av
dλ
). (28b)
Furthermore, the approximate derivatives of the admissionequations are:
dν0
dλ≈ dR0
dλ− Ndρ0
max
dλ, (29a)
dν1
dλ≈ dR1
dλ− Ndρ1
max
dλ(29b)
which gives the feedback relations between the average andclipping utilizations as:
dρ1av
dλ≈ − 1
6(am + 1)
dρ0max
dλ, (30a)
dρ0av
dλ+ 2
dρ1av
dλ≈ − 1
am + 1
dρ1max
dλ. (30b)
The derivatives of the average utilizations for the hot-spotcell and its first surrounding ring are:
dρ0av
dλ= 1
N
R0∑k=0
k∂P 0
k
∂ρ1av
dρ1av
dλ+ 1
N
∂(∑R0
k=0 kP0k )
∂ρ0max
dρ0max
dλ
+ 1
N
∂(∑R0
k=0 kP0k )
∂R0
dR0
dλ+ 1
N
R0∑k=0
k∂P 0
k
∂λ, (31a)
dρ1av
dλ= 1
N
R1∑k=0
k
(∂P 1
k
∂ρ1av
dρ1av
dλ+ ∂P 1
k
∂ρ0av
dρ0av
dλ
)
+ 1
N
∂(∑R1
k=0 kP1k )
∂ρ1max
dρ1max
dλ+ 1
N
∂(∑R
k=0 kP1k )
∂R1
dR1
dλ(31b)
where the superscript 0 denotes the parameters (ρav, ρmax, R,M , Pk) from the hot-spot cell (0, 0), and superscript 1 denotesthe parameters from the first ring. Note the absence of the
NON-UNIFORM TRAFFIC ISSUES IN DCA 619
Figure 15. Average Markov chain length and clipping utilizations for single cell hot-spot in DCA network with CAC.
Figure 16. Number of locked channels for single cell hot-spot in DCA net-work with CAC.
term (1/N)∑R1
k=0 k(∂P1k /∂λ) in equation (31b), since λenv =
const.After substitution of expressions (28) and (30) into (31),
we obtain the utilizations in the hot-spot cell and the firstring as:
dρ0av
dλ≈
1N
∑R0
k=0 k∂P 0
k
∂λ
1 − D3D1D2
, (32a)
dρ1av
dλ≈ dρ0
av
dλ
D1
D2(32b)
where
D1 = 1
N
R1∑k=0
k∂P 1
k
∂ρ0av
− R1P 1hd
− (am + 1)M1(P 1M1 − P 1
M1+1
), (33)
D2 = 1 − 1
N
R1∑k=0
k∂P 1
k
∂ρ1av
+ 2R1P 1hd
+ 2(am + 1)M1(P 1M1 − P 1
M1+1
), (34)
D3 = 1
N
R0∑k=0
k∂P 0
k
∂ρ1av
− 6R0P 0hd
− 6(am + 1)M0(P 0M0 − P 0
M0+1
). (35)
According to the previous discussion (section 4.1), the nu-merator of the expression (32a) will converge to zero whenthe hot-spot new call arrival rate grows.
The first component in expression (33) represents the con-tribution of the handoff load, and we note that
1
N
R1∑k=0
k∂P 1
k
∂ρ0av<
h
6(h+ µ)
which means that the value of D1 is negative. Similar reason-ing shows that D3 < 0 and that |D2| � |D1|, |D2| � |D3|,which implies that dρ1
av/dλ will have small negative value.Hence, the derivatives of the average utilizations in the hot-spot cell and the first surrounding ring all converge to zero.This implies that the derivatives of the handoff dropping prob-abilities also converge towards zero under high load in thehot-spot cell.
6. Multiple cell hot-spot analysis
For reasons of simplicity, we will consider a seven-cell hot-spot placed in the cell (0, 0) together with its first surroundingring. The analysis is performed in the same environment asin the section 5, with the exception that the new call arrivalrate was varied synchronously in the cells (0, 0) and (1, i)(0 < i < 5), whereas the new call arrival rate in all othercell groups was kept constant at λenv. The results were ob-tained by numerically solving the same system of equationsas for the single- cell hot-spot case. However, the resultsdrawn from the feedback interactions among hot-spot cells, aswell as between the hot-spot group and cells with the lighterload, can be applied for the hot-spots of the arbitrary size andshape.
Figures 17 and 18 show the call level QoS in a pure DCAnetwork (new call blocking probability is equal to the hand-off dropping probability). The call level QoS is shown for
620 MIŠIC AND TAM
Figure 17. Average utilization and Phd (and also Pblock) for 7-cell hot-spot in pure DCA network.
Figure 18. Average Markov chain length and number of locked channels for 7-cell hot-spot in pure DCA network.
Figure 19. Average utilization and Phd for 7-cell hot-spot in DCA network with CAC.
NON-UNIFORM TRAFFIC ISSUES IN DCA 621
rings 0–4 under changing load in the hot-spot cells, while theload in the surrounding cells was kept at λenv = 0.1. It maybe noted that the Markov chain lengths in hot-spot cells andsurrounding ring 2 will decrease due to the interaction givenin equation (26b). This will result in high handoff droppingprobabilities in the hot-spot cells, as well as in ring 2. Thetotal number of locked channels will decrease because of theincrease in the numbers of double and triple locked channels,and decrease of the number of single locked channels. Asthe interference region of the hot-spot cells includes ring 3, itwill also experience the increased number of double and triplelocked channels.
Figures 19–22 show the call level QoS in a DCA networkwith the admission control algorithm, under the same condi-tions as in the previous case. The improvements in handoffdropping probability resulting from the use of the call ad-mission control are obvious. However, we note that new callblocking probability in the central cell and first surroundingring is around 30% higher than in the case without admis-sion control (the value of new call blocking probability in thecentral cell approaches the value of Pblock obtained for theuniform traffic case).
Figure 20. New call blocking probabilities for 7-cell hot-spot in DCA net-work with CAC.
7. Conclusion
In this paper we have analyzed the behavior of DCA networksin the presence of hot-spots. Our analytical model capturesthe impact of the cost-based distributed implementation of theMP policy on the number of borrowed and locked channels.We have shown that the handoff dropping probability dete-riorates in the hot-spot cells when nominal load is exceeded– even very slightly. Moreover, due to the channel borrow-ing from the surrounding cells, these will also experience anincrease in handoff dropping probability: the first as well asthe second surrounding ring. An adaptive admission controlapproach has been proposed in order to counteract the effectsof a hot-spot, thus maximizing channel utilization and main-taining tight control over the QoS. Furthermore, since basestations already need to communicate sets of available chan-nels for borrowing, the addition of the adaptive CAC will notsignificantly increase the communication overhead of the net-work. The improvements achieved through this algorithm areverified both with the analytical model and computer simula-tions.
Figure 22. Numbers of locked channels for 7-cell hot-spot in DCA networkwith CAC.
Figure 21. Average Markov chain length and clipping utilizations for 7-cell hot-spot in DCA network with CAC.
622 MIŠIC AND TAM
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Jelena Mišic received her Ph.D. degree in computerengineering 1993, from University of Belgrade, Yu-goslavia. She joined University of Manitoba in 2003where she is an Associate Professor. From 1995–2003, she was affiliated with Hong Kong Universityof Science and Technology. Her current researchinterest includes wirless cellular networks, wirelessad-hoc networks and mobile computing. She is themember of IEEE Computer Society.E-mail: [email protected]
Yik Bun Tam received his Bachelor degree in com-puter engineering from the Hong Kong Universityof Science and Technology (HKUST), Hong Kong,in 1998. He got MPhil in computer engineering in2000. Currently he is working in Hong Kong indus-try.