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When a new call is generated, the mobile can move at a constant speed in four ... handoff arrival rate as a function of the class k new call arrival rate under an ...... packet/call level QoS constraints,” IEEE Globecom 2000 Conference Record,.
Resource Allocation in Mobile Cellular Networks with QoS Constraints* T.C. Wong1, J.W. Mark2 and K.C. Chua3 1

Centre for Wireless Communications National University of Singapore 20 Science Park Road, Singapore 117674

2

Centre for Wireless Communications University of Waterloo Waterloo, Ontario, Canada N2L 3G1

Department of Electrical and Computer Engineering National University of Singapore 10 Kent Ridge Crescent, Singapore 119260

with guard channels for K classes in our analytical models. The transmission bandwidth of each class can be an integer multiple of those for other classes. The simulation model considers a two-dimensional mobility pattern. A 10×10 square array of cells with wrap-around edges is considered and the arrival processes for new calls are Poisson distributed. When a new call is generated, the mobile can move at a constant speed in four directions: namely, north, south, east and west. The performance of fast and slow mobile users is evaluated. Furthermore, the effect of cell size is studied and the effect of the gain in system utilization through joint call/packet level QoS optimization is also investigated. The agreements between the analytical and simulation results for the CP and CS cases are good for low blocking probabilities. The key to connecting the analytical results with the simulation results is equation (17) which estimates the class k handoff arrival rate as a function of the class k new call arrival rate under an environment of low blocking probabilities.

Abstract - An approximate analytical formulation of the resource allocation problem for handling multiclass services in a cellular system is presented. Complete Partitioning (CP) and Complete Sharing (CS) schemes with guard channels are used for the resource sharing policies at the call level. The CP model is solved explicitly using a one-dimensional Markov chain while the CS model is solved using a K-dimensional Markov chain. Call and packet level qualities of service (QoS) are considered. The performance of fast and slow mobile users and the effect of cell size are evaluated. The agreements between the analytical and simulation results for the CP and CS cases are good for low blocking probabilities. Numerical results illustrate that higher gain in system utilization is achieved through the joint optimization of call/packet levels.

1.0 INTRODUCTION Resource allocation for call admission is basically a calllevel problem, with new call and handoff call blocking probabilities, and forced termination probability of handoff calls as QoS measures. When traffic flows are admitted into the network proper, QoS is measured in terms of packet loss rate and/or packet delay. In this case, scheduling and statistical multiplexing gains play a crucial role in determining the amount of traffic that can be admitted into the network proper while still satisfying the packet level QoS. Satisfying QoS constraints at the call level alone may limit the traffic load admitted, while the packet level can still sustain a larger load. There is thus reason to believe that system utilization can be enhanced by making use of both the call level and packet level properties. To our knowledge, Beshai et al. [1] are the first to suggest using the interaction between the call level and the packet level QoS in ATM networks to improve system performance. However, ATM networks do not support user mobility. In the case of wireless networks, the ability to support user roaming is the key feature, and user mobility affects the attainable system throughput and the satisfaction of QoS requirements. Cheung and Mark [2] have recently proposed a resource allocation strategy subject to joint packet/call level QoS constraints. They found that there is a significant improvement in system utilization when the deployment of system resources is subject to simultaneous satisfaction of both packet level and call level QoS constraints. However, they consider only one traffic class. For third generation cellular systems and beyond, many types of connections are anticipated, not just voice and data traffic. The connections could be voice, video, data, multimedia, web browsing, etc. That is, we have multiclass traffic. In [3,4], Mark et al. consider a complete sharing scheme with two classes. In the simulation model, the handoff time for each call in a cell is assumed to be exponentially distributed. In this paper, we consider CP and CS schemes

2.0 SYSTEM MODEL We consider a typical generic radio cell with physical capacity C in a cellular arrangement. For easy reference we define the basic unit of capacity as a channel. A user of some traffic class may transmit at a rate equal to one channel, while other transmission rates may require multiple number of channels. The cell-site (base station) supports K classes of services that can originate from at most N mobile users. The generic cell is characterized by the following system parameters used throughout the paper. System level parameters C: total physical capacity in a cell K: total number of traffic classes rk: number of basic channels (units) required by each class k call Ck: nominal capacity for class k N = ∑ kK=1Ck > C : total nominal capacity in a cell (capitalizing on statistical multiplexing gain) C k* : physical capacity for class k ( ∑ kK=1 C k* = C for CP)

The dynamics of a radio cell is driven by new call requests, call terminations, and handoffs induced by user mobility. Maintaining an ongoing call is more important than admitting a new call. Hence, handoff calls should be given a higher access priority, or a lower blocking probability than new calls. Let Bn and Bh denote respectively the blocking probabilities of new and handoff calls, with Bh≤Bn. One way to facilitate this is to reserve capacity for admitting handoff calls, which is not accessible by new requests. The reserved capacity is sometimes referred to as guard capacity. Let CT, CG and Ci denote respectively the total capacity, the guard capacity and the instantaneous capacity occupancy. We have the following:

2 The work of this author has been supported by the Natural Sciences and Engineering Research Council of Canada under Grant no. RGPIN7779.

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3

713 717

the channel as a Markov process with the call level parameters in Section 3. A one-dimensional finite state Markov chain is used to solve for the complete partitioning scheme with class k. Let {P(nk)} be the steady state probabilities of the Markov chain for class k. Solving the Markov chain, we get

Admission Rule 1. Admit both new and handoff calls if CT − Ci > CG. 2. Admit handoff calls only if 0< CT−Ci ≤ CG, where (CT−Ci) is the free capacity for admitting new and handoff calls. The fraction of reserved (guard) capacity for handoff calls is then α = C G / CT

nk    C − CG  1   λk  0 < nk ≤  k P(0),  ,     µ k  nk !  rk    Ck −CG   C −C  P ( nk ) =    nk −  k G  rk     rk   C − CG   Ck  (λhk )  (λk ) P(0),  k  < nk ≤    (µ k )nk nk !  rk   rk 

(1)

with α conditioned such that CG =α CT is an integral multiple of the basic capacity unit. Deployment of the guard capacity policy has the following ramifications. • A handoff call is accepted as long as there is enough channels available. • A new call is accepted as long as the number of available channels (if it is admitted) is greater than CG.

(2)

where   Ck −CG     r nk 1   k   λk  + 1 ∑  µ  n ! k k 1 n =   k   C k − CG  P (0) =   C −CG   Ck    nk −  k    rk     rk   r  k  (λk ) (λhk ) + ∑ n  (µk ) k nk !  C −C   nk =  k G  +1  rk  

3.0 PROBLEM STATEMENT Consider a class k user. It’s QoS is specified by the new call blocking probability, Bnk, handoff call blocking probability, Bhk, and system utilization, Nuk, at the call level and the packet loss probability, Lk, at the packet level. The call level and packet level parameters used throughout the paper are listed below. Call level parameters Bnk: new call blocking probability for class k Bhk: handoff call blocking probability for class k Bn = ∑ kK=1 B nk : total new call blocking probability

           

−1

,

(3)

where x  denotes the greatest integer smaller than or equal to x. The class k new and handoff call blocking probabilities are given, respectively, by Ck / rk  bnk =

Bh = ∑ kK=1 B hk : total handoff call blocking probability

∑ P(nk ), nk = (Ck −CG ) / rk 

(4)

B=Bn+Bh: total call blocking probability nk: number of class k sources (users) in progress λnk: arrival rate of class k new calls λhk: arrival rate of class k handoff calls λ n = ∑ kK=1 λ nk : arrival rate of new calls

and

λ h = ∑ kK=1 λ hk

Letθhk denote the probability that the next arrival is a handoff call from class k. We have

(5) bhk = P (n k = C k / rk ). Let θnk denote the probability that the next arrival is a new call from class k. Then

: arrival rate of handoff calls

λ k = λ nk + λ hk : mean call arrival rate of class k calls λ = ∑ kK=1 λ k −1 µ ck : mean −1 µ hk : mean

(6)

θ hk = λhk ∑ kK=1λk .

(7)

θ k = λk ∑ kK=1λk .

(8)

Let θk denote the probability that the next arrival is from class k. Then

: total call arrival rate call holding time of a class k call dwell time (interhandoff time) of a class k call

The blocking probability for class k new call considering all classes is given by

µ k = µ ck + µ hk : mean equivalent rate of a class k call Packet level parameters Lk: packet loss probability for class k L = ∑ kK=1 L k : total packet loss probability

Bnk = bnk θ nk .

(9)

The blocking probability for class k handoff call considering all classes is given by Bhk = bhk θ hk .

(10)

The blocking probabilities of new and handoff calls, denoted by Bn and Bh, respectively, are given by

4.0 ANALYTICAL MODELS 4.1 Complete Partitioning (CP) 4.1.1 Call Level Let us consider a single class k model for the CP case. The guard channels, CG, are reserved for handoff calls only. To facilitate analytical modeling, it is necessary to make certain assumptions about the traffic parameters. It is not unreasonable to assume that the holding time has a negative exponential distribution. Although a negative exponential distribution assumption may not be as reasonable for the cell dwell time, for analytical tractability, we will make the same assumption for cell dwell time (interhandoff time) and model

0-7803-7376-6/02/$17.00 (c) 2002 IEEE.

θ nk = λnk ∑ kK=1λk .

Bn = ∑ kK=1 B nk ,

(11)

Bh = ∑ kK=1 B hk .

(12)

and Summing Bn and Bh yields the total blocking probability (13) B = Bn + B h . The utilization for class k is given by Nuk =

714 718

λ (1 − bnk ) + λhk (1 − bhk ) × rk . nk P(nk ) × rk = nk µk nk =1

Ck / rk



(14)

where bk is the blocking probability considering class k only. The total system utilization is Nu = ∑ kK=1 Nuk .

4.2 Complete Sharing (CS) 4.2.1 Call Level Next, let us consider a K-class CS model. Assuming that the holding time for each call has a negative exponential distribution and the same assumption for cell dwell time (interhandoff time), we can model the channel occupancy as a K-dimension Markov chain with the call level parameters in Section 3. This Markov chain can be modeled and solved using the techniques in [5]. Let n=(n1,n2,…,nK) denote the state of the system with the number of users (nk) in each of the K classes, and let r=(r1,r2,…,rK) denote the number of basic channels (rk) required for each class k call with K classes. Let λk(n) denote the arrival rate and µk(n) the departure rate in the system. With N denoting the total number of nominal channels, the state space of the system, denoted by S, is given by S:={n:r⋅n≤N}. When the system is in state n and a class k call (new or handoff) arrives, an admission policy determines whether or not the call is admitted into the system. Here, the admission policy is a complete sharing scheme with guard channels. We can specify the admission policy by mapping f:=(f1,…,fK) for new and handoff calls, fG:=(fG1,…,fGK) for handoff calls, where fk and fGk: S→{0,1}, and fk(n) and fGk(n) each takes on the value 0 or 1 if a class k call is rejected or admitted, respectively, when the system state is n. They are defined by the following equations:

(15)

Equating the class k handoff call arrival rate to the product of the average handoff rate for a call and the average number of class k calls, we can get an approximate class k handoff call arrival rate as follows:

(1 − bnk )λ nk + (1 − bhk )λ hk N uk = µ hk × rk µ ck + µ hk . µ (1 − bnk )λ nk µ hk (1 − B nk θ nk )λ nk = = hk µ ck + bhk µ hk µ ck + ( Bhk θ hk ) µ hk

λ hk = µ hk ×

(16)

Thus, the class k handoff call arrival rate can be approximated under low blocking probabilities as follows:

(17) λ hk = µ hk λ nk µ ck , where µ hk = v k / s , v k is the speed of the class k mobile and

s is the cell length of a square cell.

4.1.2 Packet Level We assume that a class k call, after being admitted into the system, behaves according to an exponential on/off source. In the on state, a class k call generates packets at a rate such that rk channels are needed to transmit the packets. Assuming exponentially distributed on/off sources for each call, the fraction of time a class k call spends in the on state is given by

(

)

p k = t on,k t on,k + t off ,k ,

where ton,k and toff,k are the exponentially distributed on and off periods for a class k source, respectively. For a stationary admission control policy, the underlying process is Markovian. Let • rk be the number of basic channels a class k call needs to transmit its packets while it is in the on state. • nk be the number of class k calls in progress. nˆ k be the number of in-progress class k calls in the on • state. n k and nˆ k together characterize the state space, with state descriptor given by

(n, nˆ ) = (n1 ,..., n K ; nˆ1 ,..., nˆ K ) .

n

k =1 K

= ∑ P (n − e k )λk (n − e k ){f k (n − e k ) + f Gk (n − e k )} k =1 K

+ ∑ P (n + e k ) µ (n + e k ), k =1

Assuming a priority structure with class 1 having the highest priority and no pa cket buffers, the class k packet loss probability for the CP scheme is given by

Lk =

K

∑ λi (1 − bi )

i =1



and

P(nk ) ∑ P(nˆk | nk )[nˆk rk − Ck ]+

nk =0

,

(21)

nk

P(nk ) ∑ P (nˆk | nk )[nˆk rk ] nˆk = 0

where C = ∑ kK=1Ck . The total packet loss probability is given by L = ∑ kK=1 L k .

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n ∈ S,

λ + λ hk , if f k (n) = 1 , λ k (n) =  nk if f Gk (n) = 1 λ hk ,

(26)

µk (n) = nk (µck + µhk ), 0 < r ⋅ n ≤ N .

(27)

The first condition in equation (26) allows both new and handoff calls to be admitted to (N-CG) channels, while the second condition allows only handoff calls to be admitted to CG channels. Equation (27) allows both new and handoff calls to be serviced with the total channel occupancy to be less than or equal to N channels. Equation (25) can be solved using LU decomposition together with the condition for the total probability of all states to obtain P(n). A new class k call is blocked from entering the system (and is assumed lost) if upon arrival it finds that it cannot be accommodated due to insufficient channels (excluding the guard channels) for an additional rk channels. Therefore the

nk

nˆk = 0

(25)

where ek is a K-dimensional vector of all zeros except for a one in the kth place,

(20)

 nk 



(24)

K

(19)

 k  nˆ ˆ P(nˆk | nk ) =   pk k (1 − pk )nk − nk . ˆ

nk = 0 C * / r   k k

 1, N − C G < r ⋅ n + rk ≤ N f Gk (n) =  0, otherwise

∑ [λk (n){f k (n) + fGk (n)}+ µ k (n)]P (n)

that there are nk calls in progress is given by

λk (1 − bk )

(23)

for which S (f ) + S (f G ) = S . Let P(n) denote the equilibrium probability that the system is in state n. The global balance equations for the Markov process under the policies f and fG are

The probability of nˆk in-progress calls in the on state given

C * / r   k k

 1, r ⋅ n + rk ≤ N − C G f k (n ) =  0, otherwise

(18)

(22)

715 719

blocking probability for a new class k call considering all classes is given by

Bnk =

N / r1  ( N −n1r1) / r2  ( N −n1r1 − n2r2 ) / r3  ∑



n1 =0



n2 =0

L sum =

  K −1   N − ∑ nk rk  / rK      k =1   



...

×

nK = 0

n3 = 0

K   × P(n1, n2 , n3 ,..., nK )θ nk , if  N − CG − ∑ nk rk  < rk . k =1  





n1 =0



n2 =0



n2



n3

n3 =0

.

P(n1 , n 2 , n3 ,..., n K )

(32)

nK

∑ ... ∑ P(nˆ1 | n1 ) P(nˆ 2 | n 2 )

nˆ1 =0 nˆ2 = 0 nˆ3 =0

nˆ K =0

K

× P(nˆ 3 | n3 )...P(nˆ K | n K ) ∑ nˆ i ri i =1

The total packet loss probability is given by equation (22). 4.3 Joint Call and Packet Levels QoS Optimization The optimization problem is then

  K −1   N − ∑ nk rk  / rK      k =1   

...

n3 = 0

n2 = 0

nK =0 n1

A class k handoff call is blocked from entering the system (and is assumed lost) if upon arrival it finds that it cannot be accommodated because the number of available channels is less than rk (including the guard channels). Therefore the blocking probability for a class k handoff call considering all classes is given by

Bhk =

n1 =0

  K −1   N − ∑ ni ri  / rK     i =1   

× ∑ (28)

N / r1  ( N −n1r1) / r2  ( N −n1r1 − n2r2 ) / r3 

N / r1 ( N −n1r1) / r2  ( N −n1r1 −n2r2 ) / r3  ... ∑ ∑ ∑

(33)

max{N u }

subject to



* Bnk ≤ B nk

nK = 0

* Bhk ≤ B hk

K   × P(n1, n2 , n3 ,..., nK )θ hk , if  N − ∑ nk rk  < rk . k =1  

Lk ≤ L*k (29)

The blocking probabilities of new and handoff calls are given by equations (11) and (12), respectively. The total blocking probability is given by equation (13). The system utilization for class k is given by N uk =

×

5.0 NUMERICAL RESULTS The numerical results have been obtained by means of numerical analysis and simulation. Each simulation result is averaged over 2 different runs, each generating up to 100 million simulation minutes. A warm-up period of 10 million simulation minutes has also been used to minimize the effects of initial simulation transients. We consider two traffic classes (K=2). The parameters used in the numerical examples are as

N / r1  ( N − n1r1) / r2  ( N −n1r1 − n2r2 ) / r3  ... ∑ ∑ ∑ n1 =0

n2 =0

K −1   ( N − ∑ ni ri ) / rK    i =1



n K =0

n3 =0

(30)

n k P(n1 , n 2 , n3 ,..., n K )rk .

follows: N = 10 , C = 7 , C G = 2 , r1 = 1 , r2 = 2 , C1* = 4 ,

The total system utilization is given by equation (15).

C2* = 6 ,

4.2.2 Packet Level Assuming a priority structure with class 1 having the highest priority and no pa cket buffers, the class k packet loss probability for the CS scheme is given by   N / r1  ( N − n1r1 ) / r2  ( N − n1r1 − n2r2 ) / r3  ...  ∑ ∑  ∑ n2 = 0 n3 =0   n1 =0    K −1      N − ∑ ni ri  / rK     i =1   × P ( n1 , n 2 , n3 ,..., n K )  ∑   nK =0   nK   n1 n2 n3 ˆ ˆ × ∑ ∑ ∑ ... ∑ P (n1 | n1 ) P( n 2 | n 2 )    nˆ1 =0 nˆ2 =0 nˆ3 =0 nˆ K =0  × P ( nˆ 3 | n3 )...P ( nˆ K | n K )   +    k k −1      ∑ nˆ i ri  , if ∑ nˆ i ri ≤ C  ×  i =1 i =1     k −1 ˆ ˆ , if ∑ n i ri > C   n k rk i =1     Lk = L sum

t off ,2 = 1.5 s , 1 / µ c 2 = 2 minutes ,

t off ,1 = 0.650 s ,

θ n1 = θ n 2 ,

t on, 2 = 1.0 s ,

1 / µ c1 = 1 minute ,

1 / µ h1 = 1 / µ h 2 = 18 / 60 minute ,

v1 = v 2 = 4 km for slow mobiles, v1 = v 2 = 40 km

mobiles,

s = 200 m ,

Bn*1 = Bn*2 = 0.1 ,

for fast

Bh*1 = Bh*2 = 0.1

and L*1 = L*2 = 1× 10 −3 . Because of space limitation, in what follows we only present graphical results for the CS case. The trends of the results for the CP case are similar to those of the CS case. However, the performance of CS is generally better than that of CP as CP is quite rigid. Detailed numerical results for both CP and CS can be found in [6]. 5.1 Performance of Fast and Slow Mobiles 5.1.1Call Level Blocking Probabilities and System Utilizations The call level analytical and simulation results of new call and handoff call blocking probabilities for fast and slow mobiles are shown in Figs. 1 and 2, respectively. The analytical results are close to the simulation results under low blocking probabilities.

(31)

where

0-7803-7376-6/02/$17.00 (c) 2002 IEEE.

t on,1 = 0.352 s ,

716 720



New Call Blocking Probability

1.0E-01 Bn2-anal (slow)

CG=2

Bn1-anal (slow)

1.0E-02

























CG =2

































































Bn2-sim (slow) Bn1-sim (slow)

1.0E-03

Utilization

Bn2-anal (fast) Bn1-anal (fast)

1.0E-04





















Bn2-sim (fast) Bn1-sim (fast)

 







































1.0E-05

1.0E-06 0.2

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Total New Call Arrival rate, λn (/minutes)

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0.8

1

Total Ne w Call Arrival rate, λ n (/minutes)

Fig. 1. New call blocking probabilities Handoff Call Blocking Probability

0.4

Fig. 4. System Utilization for slow mobiles

1.0E-02

Bh2-anal (fast)

CG=2

Bh1-anal (fast)

1.0E-03

5.1.2 Joint Call and Packet Level QoS Optimization Figs. 5 and 6 show the effect of the gain in system utilization through joint call and packet level QoS optimization for fast mobiles and slow mobiles, respectively.

Bh2-sim (fast) 1.0E-04

Bh1-sim (fast) Bh2-anal (slow)

1.0E-05

Bh1-anal (slow) Bh2-sim (slow)

1.0E-06

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Figs. 3 and 4 show the call level analytical and simulation results of system utilization for fast and slow mobiles, respectively. The analytical and simulation results agree better for the slow mobile case than those for the fast mobile case.

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Fig. 5. System utilization with and without joint call/packet level QoS for fast mobiles

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Fig. 3. System Utilization for fast mobiles

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Fig. 6. System utilization with and without joint call/packet level QoS for slow mobiles

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The larger the load (total mean new call arrival rate), the larger the gain in system utilization is. This translates to more revenue for the network providers who in turn can lower the charges for the mobile users. Note that slow mobiles have better gain in system utilization than fast mobiles.

Joint optimization through call and packet level qualities of service (QoS) is investigated. The QoS performance metrics are call blocking probability, system utilization and packet loss probability. Numerical results illustrate that higher gain in system utilization is achieved through the joint optimization. This translates to more revenue for network providers who in turn can lower the charges for mobile users.

5.2 Effect of Cell Length The total new call arrival rate is set at 0.1 arrival per minute in a 100m cell length. The parameters are 1/µh1=1/µh2=(9,18,27,36)/60 minute with fast mobiles and 1/µh1=1/µh2=(90,180,270,360)/60 minute with slow mobiles, respectively, for cell size = (100m, 200m, 300m, 400m) with λn=0.1,0.4,0.9,1.6 arrival per minute.

7.0 REFERENCES [1] M. Beshai, R. Kositpaiboon, and J. Yan, “Interaction of call blocking and cell loss in an ATM network,” IEEE Journal on Selected Areas in Communications, vol. 12, pp. 1051-1058, August 1994. [2] M. Cheung and J.W. Mark, “Resource allocation in wireless networks based on joint packet/call level QoS constraints,” IEEE Globecom 2000 Conference Record, Vol. 1, pp. 271-275, November 2000. [3] J.W. Mark, T.C. Wong, M. Jin, B. Bensaou and K.C. Chua, “Resource Allocation for Multiclass Services in Cellular Systems,” IEEE International Conference on Communication Systems 2000 Conference Record, session T3-1 Cellular Systems, 2nd paper, November 2000. [4] T.C. Wong, J.W. Mark, M. Jin, B. Bensaou and K.C. Chua, “Resource Allocation in Mobile Multimedia Networks - Analytical and simulation results for cases 1 to 4 with fixed guard channels,” Technical Report RA001-07-2000, Centre for Wireless Communications, National University of Singapore, July 2000. [5] K.W. Ross, “Multiservice Loss Models for Broadband Telecommunication Networks,” Springer, 1995. [6] T.C. Wong, J.W. Mark and K.C. Chua, “Resource Allocation in Mobile Multimedia Networks with User Mobility - Case 1: Complete Partitioning and Complete Sharing,” Technical Report RA-001-06-2001, Centre for Wireless Communications, National University of Singapore, June 2001.

5.2.1 Call Level Blocking Probabilities and System Utilizations From numerical results of the call level analytical and simulation results of new and handoff call blocking probabilities for fast and slow mobiles [6], the analytical results are close to the simulation results under low blocking probabilities. From [6], the larger the cell length, the higher the blocking probabilities. From numerical results of the system utilization with different cell lengths for fast and slow mobiles [6], the analytical and simulation results agree better for the slow mobile case than those for the fast mobile case. 6.0 CONCLUDING REMARKS An approximate analytical formulation of the resource allocation problem for handling multiclass services with guard channels in a cellular system is presented in Section 4. Complete partitioning and complete sharing schemes are used for the resource sharing policies. The analytical models are solved using a one-dimensional Markov Chain for CP scheme and a K-dimensional Markov Chain for CS scheme.

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