Wireless Networks 1(1995) 147-160

147

Performance evaluation of a reservation random access scheme for packetized wireless systems with call control and hand-off loading * Izhak Rubin and Shervin Shambayati Department of Electrical Engineering, 56-125BENGR IV, Universityof California, Los Angeles, Los Angeles, CA 90024-1594, USA

ReceivedOctober 1994

Abstract. We consider a packet switched wireless network where each cell's communicationchannel is shared among packet voice sources. In this paper, we present a method for the design and analysis of wireless cells using a reservation random access (RRA) scheme for packet access control. This scheme is integrated with a call admission control procedure. We model the state process of a single cell as a vector Markov chain. We compute the steady state distribution of the Markov chain. This result is used to calculate the packet dropping probability and the call blocking probability. By setting limits on maximum permissible levels for the call blocking probability and the packet dropping probability, we obtain the Erlang capacity of a single cell, with and without hand-off traffic. For an illustrative RRA scheme, the Erlang capacity of a single cell is shown to be about twice that attained by a comparable fixed assigned TDMA scheme. We show that a cellular network using this RRA scheme and which applies can be no blockingof hand-offcalls, exhibits similar call capacitylevels.

1. I n t r o d u c t i o n As the demand for wireless access to digital information networks increases, efforts are made to increase the bandwidth efficiency of wireless systems by devising new packet switching based networks which employ improved multiple access protocols. Research is currently under way to design new cellular systems which use packetized demand-assigned C D M A and T D M A multiple access schemes to meet the ever increasing demand for high capacity wireless and cellular networks. To support multimedia services, packet oriented fast circuit-switched and packet-switched wireless networks are introduced. A m o n g the packet switched DAT D M A schemes under consideration for the provision of packet voice services are those which fall under the category of reservation random access (RRA). These schemes involve the use of the first few packets of a call's talkspurt to reserve a time slot (a time circuit) in a T D M A frame for the duration of the talkspurt. During voice pauses (silent periods), the user's allocated circuit is returned to the unallocated pool of the cell's circuits. Previous research studies of such schemes involve analyses which assume the total number of users in the system to be fixed (see [1,3,4]). To guarantee acceptable voice quality performance (at the packet level) it is important to employ a call level admission control procedure. Preliminary results have been presented in [6] for a system in which the total number of users in the * This work is supported by a University of California MICRO and Pacific-BellGrant No. 94-107. 9 J.C. Baltzer AG, Science Publishers

system varies over time, due to the use of a call admission control protocol. In this paper, we expand on the methods developed in [6] and present an approach for the analysis and design of R R A schemes under which calls are regulated in accordance with a call admission scheme. In section 2, we present the call admission control scheme and the R R A algorithm used in our analysis. In section 3, we model the state of single cell as a multidimensional M a r k o v Chain and derive equations which provide for an approximate calculation of the steady state distribution. Later in this section, we show how this approximation is used to design and analyze an R R A system which employs a call admission control scheme. In section 4, we apply the equations derived in section 3 to a sample case and present the results of our analysis. We compare these results with a fixed assigned T D M A system as well as with a corresponding R R A scheme which employs no call admission control. We show that the call throughput capacity of our sample system (for a single cell) is at least 1.7 times greater than that attained by a fixed-assigned T D M A (circuitswitched) system. Compared to the R R A scheme which uses no call admission control, our sample system sustains a call throughput capacity level which is about 2 Erlangs higher. Methods are also presented for the calculation of the call throughput capacity and the optim u m call admission threshold levels for such a wireless system when hand-off calls are not blocked. We compare two different call admission policies. We demonstrate the throughput effectiveness of such a network over a wide range of hand-off to in-cell call ratio levels. We show that a single cell in this system sustains an

148

/7.Rubin, S. Shambayati / Performance of a reservation random access schemefor packetized wirelesssystems

overall throughput level (supporting local and hand-off calls) which is close to that obtained by the corresponding system under which local and hand-off calls are subjected to blocking. Conclusions are drawn in section 5.

2. Call admission and reservation r a n d o m access protocols

2.1. System assumptions In our analysis we m a k e the following assumptions: 1. The channel used for transmission of voice packets from the users to the network in each cell is divided into time frames. Each frame consists of an equal number of slots. All slots are of equal duration. The duration of each frame is T seconds. The number of slots per frame is s. 2. The calls which attempt to use the above mentioned channel in a cell are divided into two categories: incell calls and hand-off calls. An in-cell call is a user who is attempting to access the network for the first time through the cell. A hand-off call is a user who has a previously established call in another cell and is attempting to transfer that call to the cell from the other cell. 3. In-cell calls arrive into a cell v according to a Poisson arrival process with arrival rate Av. 4. The holding time of a call in cell is assumed to be exponentially distributed with mean 1/#. The holding times for all users, both in-cell and hand-off, are independently and identically distributed. 5. Once a call finishes its residency in a cell u with probability p(X) (u, v) if the call is an in-cell call and with probability p(h) (u, v) if the call is a hand-off call, it is handed off to cell v. A user terminates its call in cell u with probability p(rX)(u)= 1 - ~'~vcAp(X)(u, v), if the ca 11 is an in-cell call and with probability P r( ) (u) = ~ u t p ( h ) (u, v), if the call is a hand-offcall. We set A to designate the set of all cells to which a call from cell u m a y be handed off. 6. A call which is admitted into the system switches randomly between silent and active states. The time spent in each state is exponentially distributed with p a r a m eters 0 for the silent state and "), for the active state. Given that the frame duration is 7- seconds, the probability that a silent call becomes active during a frame is given by qa = 1 - e -~ Similarly, the probability that an active call becomes silent is given by qs = 1 - e -'Y~-. 7. An active call generates one packet per frame. The length of each packet is equal to the length of a slot. 8. In each cell, signaling channels exist through which the users set up calls. These channels are also used for the transmission of the information from the base station to the users. The latter information consists of the number of calls in the cell and the number and the

identity of slots which are reserved in the current frame.

2.2. Call admission protocols We consider two call admission protocols in this paper. Under the first call admission protocol, Protocol 1, a user is admitted into a cell by the cell's base station if the user's call is a hand-off call or if the user's call is an in-cell call and the total number of in-cell calls in the cell is less than a threshold, m. Under this protocol, we readily observe that the call process in the cell follows the statistics of a M / M / m / m queuing system for the in-cell users and that of a G / M / o c queuing system for the hand-off calls. Under the second protocol, Protocol 2, a user is admitted into a cell if the user's call is a hand-off call or if the user's call is an in-cell call and the total number of users (hand-off plus in-cell) is less than a threshold, m. Note that again under this protocol the cell acts as a G / M / c o queuing system for the hand-off calls. However, for the in-cell calls, the queuing model is more complex.

2.3. Medium access control protocol for transmission o f the packets A call which is admitted into a cell switches randomly between active and silent states. Once a call becomes active, it generates a single voice packet per frame. In order for the call to transmit its packets across the cell's channel, it first needs to reserve a time slot in a frame (i.e., a time circuit). Once the call is successful in obtaining this reservation, it can transmit its future packets over the same time slot position over subsequent frames until it switches back to silent state. Subsequently, the slot is released and returned to the pool of unallocated time slots. In order for a call to obtain a reservation once it becomes active, it first "flips a coin" to see whether or not it will transmit its packet over the next frame. I f the user decides to transmit its packet over the next frame, it selects a time slot at r a n d o m from the pool of unallocated time slots in that frame, and transmits its packet over that time slot. I f the call is the only call which transmits over that time slot, the transmission is successful and the call reserves that time slot for the duration of its current activity period (talkspurt). I f one or more other users transmit their packets over the same time slot, a collision is said to have occurred and the transmission is unsuccessful. In this case, the call does not manage to reserve a slot. A packet which is not transmitted or incurs a collision, is assumed lost (due to the latency requirements of voice packets no queueing of packets occurs in this system). This process is repeated over subsequent frames until either the user succeeds in reserving

I. Rubin, S. Shambayati / Performance o f a reservation random access scheme for packetized wireless systems

a time slot or until its talkspurt ends. An active call with no reservation is said to be in "contention."

3. The M a r k o v state chain a n d p e r f o r m a n c e equations 3.1. The M a r k o v state process We consider the state of a single cell. The cell's call process is represented as a discrete time Markov chain X = {X~, n _> 0} where __Xnis the vector state of the process at the start of the nth frame. The state variable consists of the component variables shown in Table 1. (Note that a variable which is not attached to a subscript n represents the steady state version of the corresponding random variable.) To describe the cell's call process and calculate its transition probability function, we divide the related activities in each frame into four consecutive phases: 1)Reservation. 2)Departure. 3)Transition. 4)Arrival. During the Reservation phase, those calls which are in contention attempt to obtain reservations. During the Departure phase, those users which have finished their residency in the cell depart from the cell. During the transition phase, active calls become silent and silent calls become active. During the Arrival phase, those calls which qualify for admission into the cell are admitted. To formalize the above process, we use the auxiliary variables shown in Table 2 to derive the state equations presented in eq. (1). These equations relate the state variables at the start of the (n + 1)st frame to the corresponding variables at the start of the nth frame. They are thus used to derive the transition probability function for __X.For n >_ 0, we have Nx,n+l : N x , n - D(nx'R) - D(nx'c) - D(nx's) ,4(x,A) , + A~x's) + .~.

Table 1 State variables. Variable

Description

Nx•n

The total n u m b e r of in-cell calls at the start of the nth frame

Nh,n

The total n u m b e r of hand-off calls at the start of the n th frame

149

Nh,n+, = Nh,n- Z)(? R> - Z ) U > - Z)U> + AU> + A(: 'AI , N(R) = N(R) ,n+l x,n

Dn

Nh(R) ,n+l

Dn(h,R) + R~h) - S(,h'R) ,

AT(R) -= ~'h,n

(x,R) + R(X) -- S~x,R) ,

Nx(C) ,n+l = N(C) x,n - R}X) - D.(~,c) - S~(x,C)

+ r(, x) + A ,XA), Nh(C)

~r

,n+l ~-~'h,n

_ R~h) _ D(h,c)

_ sU1

+ T!ht +

(1)

3.2. The system "s performance indices There are two performance index functions of interest in assessing the performance for a single cell. The first performance index measures the performance of the call admission control process. This index is the incell call blocking probability. The second index measures the quality of the supported voice connections and, hence, the effectiveness of the R R A medium access control (MAC) mechanism. This index is the packet dropping probability. The in-cell call blocking probability is defined as the fraction of time during which the cell cannot admit a new in-cell call. Under Protocol 1, this is equal to PB = P r { N x = m}.

(2)

Under Protocol 2, the in-cell call blocking probability is equal to PB = Pr{Ux + Uh >_ m } .

(3)

The packet dropping probability is defined as the probability that a packet generated during a frame is dropped. It is equal to the average total number of packets dropped per frame over the average total number of packets generated per frame. Thus E[N(x c)] + E[N~ c)] - E[R(X)] - E[R(h)] Pdrop = E[N(xC)] + E[N~C) ] + E[Nx(R)] + E[N~R)] -

(4)

3.3. Calculation o f the system's performance indices

N:(c) ,n

The total n u m b e r in-cell calls in contention at the start of the nth frame

Nh~C~ ,n

The total number hand-off calls in contention at the start of the nth frame

N (xRn

The total n u m b e r in-cell calls with reservation at the start of the nth frame

Nh(e) ,n

The total n u m b e r hand-off calls with reservation at the start of the nth frame

In order to calculate the system's performance indices, we need to calculate the steady state probability distribution of the system state for a single cell. From the state transition equations in subsection 3.1, we can derive the transition probability functions for the state process of a single cell. The computation of these functions is, however, analytically complex due to the large size of the underlying state space. Therefore, we have devised a method for the approximate calculation of this steady state distribution. We then apply this approximation to the calculation of the performance indices.

150

L Rubin, S. Shambayati / Petformance of a reservation random access schemefor packetized wireless systems

Table 2 Auxiliary variables. Variable

Description The number of in-cellcalls which were in contention at the beginning of the nth frame and obtain reservation during the reservation phase of the nth frame The number of hand-off calls in contention at the beginning of the nth frame which obtain reservation during the reservation phase of the nth frame

D(JI

Number of in-cellcalls which have a reserved circuit after the reservation phase and which depart the system during the departure phase of the nth frame

D(h,R)

Number of hand-off calls with reservation since the reservation phase which depart the system during the departure phase of the nth frame

D(X,c)

Number of in-cellcalls in contention after the reservation phase which depart the system at the end of the nth frame

D(h,c)

Number of hand-off calls in contention after the reservation phase which depart the system during the departure phase of the nth frame

D ,sl D(f, s)

Number of in-cellcalls that are silent after the reservation phase and depart the system during the departure phase of the nth frame Number of hand-off calls that are silent and depart the system at the end of the nth frame Number of silent in-cellcalls in the cell which become active after the departure phase of the nth frame

T(hl

Number of silent hand-off calls in the cell which become active after the departure phase of the nth frame

S~X,C)

Number of in-cellcalls in contention which become silent during the transition phase of the nth frame

S (nh,c)

Number of hand-off calls in contention which become silent during the transition phase of the nth frame

s(x,R)

Number of in-cellcalls w~threservation which become silent during the transition phase of the nth frame

s(nh,R)

Number of hand-off calls with reservation which become silent during the transition phase of the nth frame

A(X,S)

Number of in-cell calls which arrive into the cell during the arrival phase of the nth frame and which are silent

A~h,s)

Number of hand-off calls which arrive into the cell during the arrival phase of nth frame and which are silent

A (nX,A)

Number of in-cellcalls which arrive into the cell during the arrival phase of the nth phase and which are active

A(f, A)

Number of hand-off calls which arrive into the cell during the arrival phase and which are active

O u r a p p r o a c h is as follows. W e first note that, for this system, the call a d m i s s i o n p r o t o c o l is i n d e p e n d e n t of the M A C protocol. Hence, u n d e r certain a s s u m p tions the j o i n t p r o b a b i l i t y d i s t r i b u t i o n of the n u m b e r of h a n d - o f f calls a n d the n u m b e r o f in-cell calls are readily calculated. Secondly, we note t h a t the packet d r o p p i n g p r o b a b i l i t y depends only o n the total n u m b e r of a d m i t t e d users in the cell, since all a d m i t t e d users have the same packet access privileges. Hence, once we find the expectations o f the total n u m b e r o f calls with reserv a t i o n a n d calls in c o n t e n t i o n , given the total n u m b e r of a d m i t t e d calls (regardless of type), we readily calculate the packet d r o p p i n g p r o b a b i l i t y . Since the d e r i v a t i o n s o f these expectations are highly complex i n v o l v i n g calc u l a t i o n o f very large M a r k o v t r a n s i t i o n matrices, we derive a n a p p r o x i m a t e expression for the steady state d i s t r i b u t i o n o f the total n u m b e r calls in c o n t e n t i o n a n d the total n u m b e r o f calls with r e s e r v a t i o n given the total

n u m b e r of calls in a single cell. We a s s u m e the latter p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n to be a p p r o x i m a t e d by the j o i n t p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n of the n u m b e r of calls in c o n t e n t i o n a n d the n u m b e r of calls with reserv a t i o n s for a single cell in which the total n u m b e r of a d m i t t e d calls in the cell does n o t vary. Thus, we set

P r { N (R) = j , N (c)

=kiN=i}~ P R c , i ( j , k ) ,

(5)

where N (R) represents the total n u m b e r of calls with reservation, N (c) is the total n u m b e r of calls in c o n t e n tion, N is the total n u m b e r o f calls in the cell a n d PRc,i(j, k) is the steady state p r o b a b i l i t y that a single celt which employs the prescribed R R A scheme a n d has a fixed total n u m b e r o f a d m i t t e d calls equal to i, h a s j calls with reservations a n d k calls in c o n t e n t i o n . I n the next s u b s e c t i o n we derive the p r o b a b i l i t y dist r i b u t i o n for the total n u m b e r of calls in a single cell. I n

I. Rubin, S. Shambayati / Performance of a reservation random access schemefor packetized wireless systems

the subsequent subsection, we derive the Markov transition matrix required for the calculation of PRc,i(j, k).

3.3.1. Steady state probability of the total number of calls in a single cell In order to calculate the probability distribution of the total number of users in a cell we assume that the blocking probability for in-cell calls (regardless of the admission protocol used) is low enough so that the network of cells is approximated by a Jackson network without call blocking [2]. Using the properties of such a network, the hand-off call arrivals into cell u is a Poisson arrival process with arrival rate A(uh), where (for a network covering a space S of cells):

A~h) = Z[p(X)(v,u) 9 ( 1 - P(Bv))A.o+p(h)(V,U) . A~h)]. (6) vG.A

The set of simultaneous linear equations described in (6) needstobe solvedto obtain {A(,h), u E S}. Subsequently, under Protocol 1, the stefidy state distribution of the total number of admitted calls in cell u, P~) (i), is given by the convolution of the following two probability distributions:

P(~) (i) = P(xu) (i) * p~U)(i),

(7)

where p(U)(i), the distribution of the total number of incell calls in cell u, is given by:

p~'//i! P(")(i) =

2_.j=0p~ /j. O,

Pt~ h

_

(~)

p .t

,

i!

i__>0,

(9)

and

p(x") =

u/ ,p2u

=

(10)

Under Protocol 2, P~) (i) is given by

p(T.)(i) =

0 < i < m,

i]

~,P) •

) Ph

i!

"rr Iu)

i=0

PRc,i(Y, z,j, k) = Pr{N~ R)+I= Y' ~'n+l~V(C)= zlN~R) = j, N(nC) = k } =

Z

ib(k,s-j,l),

i]

p(u), mp(u)(i_m),~ -1

min(z,i-j-k)

k - l

l (1-qs)Yqj+t-Y.

) ( l _ qs)Z_rqsk_l z+r.

r=max(Oz-k+l) k - l - z + r (13)

(11)

for min(k, s - j ) > max(O, y - j ) , y <_rain(s, k § and y + z <_ iandj + k <_ i. In the above equation ~ ( k , s - j , l ) represents the probability that out of k contending calls 1 obtain reservations, given that s - j slots (per frame) are free. Hence, this probability represents the dynamics of the reservation phase. This probability is related to tg(a, b, c), the probability that out of a calls in contention which actually transmit their packets, c are successful in obtaining reservations, given that b slots are (yet) unreserved, by the following equation:

,~(k,s - j , 1) = ~

i!

(1 -pa)k-~par'~(r,s-j,l), (14)

where Pa is the probability that a contending call will transmit its packet over a tagged frame. This probability can be adjusted according to the total number of users in the cell and the number of free slots in a frame so that the packet dropping probability is minimized. The function tg(a, b, c) is given by

',9(a,b,c) i=m+l

j

l=max(O,y-j)

m
where P~)(0) is given by

(

Following the same arguments presented in section 3.1, we can conclude that the frame by frame state of a single cell in which the total number of calls in the system is fixed,{N(C),N(R),n >_ 1} forms a two dimensional Markov Chain. The latter's states are: N (c), the total number of calls in contention, and N (R), the total number of calls with reservations. The transition probability function of this Markov chain is used to calculate the steady state distribution PRc,i(]', k). In order to simplify the derivation of this Markov chain we consider two phases: 1. Reservation phase, in which calls in contention attempt to obtain reservations. 2. Transition phase, in which active calls may become silent and silent calls may become active. Using this the probability transition matrix for this system is given by

(8) otherwise.

P~") (i), the probability distribution of the hand-off calls in cell u, is given by ()e_phi

3.3.2. The Markov transition matrix and the calculation of PRc,i (j, k)

(i-Jr-k)(1--qa)i-j-k-rqa r

i
151

]

--

f(a,b,c)

(15)

ab

wheref(a, b, c) is the number ways that a distinct objects

152

L Rubin, S. Shambayati / Performance of a reservation random access scheme for packetized wireless systems

I - q~

co at least r percent of the time; where co is the smallest such level. To put it formally,

1 q~

q,

p(r) drop = Pr{Packet DroppedlNx + Nh = co} qo

Pr{Nx + Nh <_ c~ - 1} < r/lO0

Fig. 1. Voice activity model of an admitted call.

<_ Pr{Nx + Nh <_ co}. are put into b distinct boxes such that c of the boxes have exactly one object in each. A method for the recursive calculation off(a, b, c) is provided in the appendix. The terms other than O ( k , s - j , l ) represent the dynamics of the transition phase.

(16)

4. Numerical results In order to verify the validity of the evaluation method developed here and to provide an example of how this method can be used for the design and analysis of a cellular system utilizing the proposed call admission control protocols we have performed simulations and calculations for a sample system. For this system, each frame consists of 20 slots and has a duration of 40 ms. The voice activity parameters, qs and qa, based on the parameters in [3], are calculated to be 0.04 and 0.03, respectively. We have evaluated the call throughput capacity of a single cell using the method described in this paper (without hand-off loading), for both call admission protocols for three performance objective alternatives: 1. Call blocking probability less than 0.02, and average packet dropping probability less than 0.01. 2. Call blocking probability less than 0.02, and 85percentile packet dropping probability less than 0.01.

3.3.3. Tail blocking probabilities as measures of performance We have introduced two measures of performance for the system, the call blocking probability and the packet dropping probability. These measures are based on long term averages. However, while the system may display an acceptable behavior on the average, (e.g., the average packet dropping probability may be within the prescribed limits), the length of the periods during which the packet dropping probability is unacceptable may be quite long. To describe such performance fluctuations, we examine the 85-percentile and 99-percentile packet dropping probabilities. Consider a single cell. The r-percentile packet dropping probability for is defined as the packet dropping probability observed by the an admitted call in the system when the total number of calls in the system is equal to co. The total number of calls in the system is equal to

1.00E-1 -

..j

$~u~io~ 20 ~cond5 M 9

HoI~ TM

~ a ~ o n ~ I 0 Mince M ~ Holdm~T~ne

t~

1.00E-2

-r

_a;a~. N)

~f ..~

1.00E-3

1.00E-4

t

10.00

I

20.00 30.00 Total Number of Admitted Calls

F

40.00

Fig. 2. Analytical packet dropping probability and packet dropping probability obtained through simulation vs. the total number of admitted calls.

I. Rubin, S. Shambayati / Performance o f a reservation random access schemefor packetized wireless systems

3. Call blocking probability less than 0.02, and 99percentile packet dropping probability less than 0.01. The packet dropping limit of 0.01 is selected in accordance to the results of qualitative tests which are documented in [1]. We have also calculated the call Erlang throughput of a single cell for packet dropping probabilities varying between 0.001 and 0.01, for the three performance objectives mentioned above. In addition we have calculated the call throughput capacity of a single cell under objectives 2 and 3 for various levels of call hand-off loading. Performance results for these cases are presented in the following. We have compared the values of packet dropping probability, Pd~op, given the number of admitted calls, Nx, as calculated by our method with those obtained through simulation. In the simulations, we vary the mean call holding time (1/#) between 20 seconds and 10 minutes while keeping the Erlang loading of the cell constant. In doing so, the mean time between changes in the number of admitted calls varies as well. The results of these simulations are presented in Fig. 2 which includes five curves. The first curve represents Pd~op (given Nx) vs. Nx as calculated by our method (identified as analytical). The other curves represent the corresponding results as obtained by simulations for various mean call holding times. As observed from Fig. 2, our method is highly accurate over a wide range of mean call holding times. Furthermore, as expected, as the mean call holding times increase, the results calculated by our method become more accurate. Fig. 3 exhibits the behavior ofpd~op given Nx vs. Nx as calculated by our method. We use the results exhibited by this curve to calculate the Erlang capacity of a single cell (with no hand-off loading) for the performance objectives mentioned above. This is done by varying the call threshold m. For a given m, we increase

30.00

"~

25.00

~,

20.00

j/

j/

L~

.9 15.00

153

10.00 20.00

30.00

40.00

50.00

Call Threshold

Fig. 4. Maximum call throughout (Erlangs) vs. call threshold, no handoffs, call blocking probability less than 0.02, packet dropping probabifity < 0.01.

the Erlang call loading of the cell until either the packet dropping probability limit or the call blocking limit is reached. The ensuing Erlang call loading is recorded. We then select the value of m (denoted by mopt) which gives the highest Erlang throughput. The latter is defined as the call throughput capacity (also identified as the Erlang capacity) for the cell. The results obtained through this process are displayed in Figs. 4 through 6. From Fig. 4 we see that under criterion 1, the call throughput capacity for a single cell is equal to 28.59 Erlangs. This is obtained when the threshold level, m = m o p t , is set equal to 38. Fig. 5 shows that under criterion 2 the call throughput capacity is 26.35 Erlangs. This is achieved under a call threshold level, m = mopt, of 36. 30.00 -

1.00E+0

1.00E-I

= ,z~ o

//

25.00

/

/

1.00E-2

,

-

/

/

/

20.00

1.00E-3

--

t r

-

i

/

/

15.00

1.00E-4

/

!

I i

1.00E-5 0.00

10.00

20.00 30.00 Number of Admitted Calls

10.00 -

L

I 40.00

20.00 50.00

Fig. 3. Packet dropping probability vs. the number of admitted calls.

i

i 30.00

40.00

50.00

Call Threshold

Fig. 5. Maximum call throughout vs. call threshold, no handoff, call blocking probability < 0.02, 85 oYopacket dropping probability < 0.01.

L Rubin, S. Shambayati / Performance of a reservation random access schemefor packetized wireless systems

154

As shown in Figs. 4~6, each throughput curve exhibits two distinct parts: an increasing linear part and a decreasing non-linear part. The linear part spans those values of m for which the maximum Erlang throughput is obtained when the call blocking limit of the criterion 20.00 under consideration is reached before the packet dropping limit is realized. The non-linear part of the curve spans those values ofm for which the reverse is true. Note that for large values of m, under criteria 2 and 3, the call throughput capacity of the system is 26.19 15.00 and 20.324 Erlangs, respectively. This indicates that if i we require the call blocking probability to be very low (approaching zero), the system capacity is equal to 26.19 Erlangs (0.16 Erlangs less than that attained when the maximum call blocking probability level is set equal 10.00 to 0.02) for criterion 2 and 20.324 Erlangs (2.09 Erlangs 20.00 30.00 40.00 50.00 less than that obtained under a prescribed call blocking Call Threshold probability of 0.02) for criterion 3. Fig. 6. Maximum call throughput vs. call threshold, no handIn order to demonstrate the effects of different voice off, call blocking probability < 0.02, 99% packet dropping probability quality requirements on the capacity of the R R A < 0.01. scheme represented here, we vary the packet dropping probability limits under the three criteria from 0.001 to Fig. 6 shows that under criterion 3 the call through0.01. The results are illustrated in Figs. 7-9. Note that if put capacity for the system is 22.37 Erlangs. This is obtained when the call threshold level, rn = rnopt, is set the resources of this system are used for a FA-TDMA system, a single cell call system would behave like a M/ equal to 31. For a fixed assigned TDMA (FA-TDMA) system M/20/20 queuing system. On the other hand, for the with 20 slots per frame and a maximum call blocking proposed R R A system where the slots are shared statisprobability equal to 0.02, the cell's Erlang capacity is tically among the offered voice packets, a single cell obtained from M / M / m / m queuing system computa- call service system behaves like a M/M/mopt/mopt queutions and is calculated to be equal to 12.92. Compared ing system. Therefore, if for a given packet dropping to this TDMA system, the Erlang call capacity attained probability limit, the mopt level selected for the R R A under the regulated R R A system is higher by a factor system is greater than 20, the R R A system supports a of 221% times under criterion 1,204% times under cri- higher call throughput capacity than that accommoterion 2, and 173% times under criterion 3. dated by a comparable FA-TDMA system. Otherwise, 25.00 -

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7

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Fig. 7. C a l l t h r o u g h o u t c a p a c i t y a n d o p t i m u m call t h r e s h o l d vs. m a x i m u m

20.00 1.20E-2

a l l o w a b l e p a c k e t d r o p p i n g p r o b a b i l i t y , n o h a n d o f f s , call b l o c k i n g

probability< 0.02.

L Rubin, S. Shambayati / Performance of a reservation random access schemeforpacketized wireless systems

ets in an effective fashion. As a result, in these ranges, an FA-TDMA operation is advantageous. We also consider the call Erlang capacity of a single cell when the cell is subjected to hand-off loading. As mentioned in section 3.3.1, we assume that hand-off calls arrive into a cell at random in accordance with a Poisson process. Under this assumption we have obtained (using the same methodology outlined above for obtaining the single ce.ll call throughput capacity without hand-of f s) the call throughput capacity of a single cell for various hand-off call loading levels using Protocol 1, under performance criteria 2 and 3. Performance results are shown in Figs. 10 and 11. As we can see both curves are "sawtooth" shaped. The explanation for this phenomenon is as follows. For a given hand-off loading Ph, and a given in-cell call threshold m, the Erlang capacity of a single cell is achieved and calculated based on the following observation. We gradually increase the in-cell call traffic until either the packet dropping limit or the call blocking probability limit is reached. As we can see from Figs. 4-6 these two events do not occur simultaneously. Consider the case under which the in-cell traffic causes the call blocking probability limit to be reached before the packet dropping probability limit is met. In this case, the cell can be loaded with additional hand-off traffic while the in-cell call traffic remains constant until the packet dropping probability limit is reached. Therefore, in this case, the total call throughput vs. hand-off loading curve has a positive constant slope. On the other hand, when the incell call traffic loading causes the packet dropping limit to be reached before the in-cell call blocking probability

the former system provides a lower cell throughput capacity. Under criterion 1, when the voice quality index is determined by the average packet dropping probability, we observe in Fig. 7 that the cell's call throughput capacity is always larger than that attainable by a comparable FA-TDMA scheme over the whole range of packet dropping probabilities. This is observed from the fact that the optimum call threshold, mopt, is always greater than 20. We note that by varying the packet dropping probability requirement from 0.001 to 0.01 we more than double the Erlang call capacity of the cell. Under criterion 2, when the 85-percentile packet dropping probability limit is varied, we observe in Fig. 8 that the regulated R R A scheme has a higher capacity than that offered by the FA-TDMA system only when the 85-percentile packet dropping probability limit is above 0.0012. In this case, the call throughput capacity of the cell also more than doubles as we vary the 85-percentile requirement from 0.001 to 0.01. Under criterion 3, when the 99-percentile packet dropping probability is used as the objective function, we observe in Fig. 9 that the proposed R R A scheme performs better than the FA-TDMA system only when the 99-percentile packet dropping probability limit is higher than 0.002. Again, we note that varying the packet dropping probability limit from 0.001 to 0.01 almost doubles the Erlang call capacity of a cell using our R R A scheme. Clearly, for higher specified (85% and 99%) packet dropping probability levels, the R R A scheme cannot statistically share the bandwidth among the user pack-

30.00

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156

L Rubin, S. Shambayati / Performance o f a reservation random access scheme for packetized wireless systems

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Fig. 9. Call troughout capacity and optimum call threshold vs. 99% packet dropping probability, no handoffs, call blocking probability < 0.02.

edge of each "sawtooth" corresponds to those levels of the hand-off loading for which the reverse is true. Note that, asides from the "sawtooth" variation described above, as the hand-off loading is increased the system's capacity is generally reduced. The level to which this throughput is ultimately reduced corresponds to the limiting throughput level shown in Figs. 5 and 6 as m --+ ec. It must be noted that for relatively low levels of

limit is met, for any additional hand-off traffic, the incell call loading must be reduced as to meet the packet dropping probability target level. The rising edge of each "sawtooth" thus corresponds to those levels of hand-off loading for which mopt and its resulting maximum in-cell call loading are such that the in-cell call blocking probability limit is reached before the packet dropping probability limit is realized. In turn, the falling 40.00 - -

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; 16.00

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Fig. 10. Optimum call threshold and total call throughput capacity vs. hand-off loading, in-cell blocking probability < 0.01, call admission protocol 1.

L Rubin, S. Shambayati / Performance of a reservation random access schemeforpacketized wireless systems 40.00 -~,

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Hand-off Loading (Erlang) Fig. 11. Optimum call threshold and total call throughout capacity vs. hand-off loading, in-cell call blocking probability < 0.02, 99% packet dropping probability < 0.01, call admission protocol 1.

hand-off loading, the overall throughput of a single cell does not vary greatly. The curves in Figs. 12 and 13 display the call throughput capacity characteristics of a single cell under varying hand-off loading conditions for criteria 2 and 3, respectively, under Protocol 2. As we can see from these figures, the optimum call threshold level does not vary under criterion 3 while it increases by 1 under criterion 2. From system's design point of view, this behavior is desirable since it allows a selection of a fixed call threshold value regardless of the hand-off loading level. Figs. 12 and 13 illustrate an interesting phenomenon. As we can see from these figures, initially the total call throughput capacity of the system increases as the hand-off loading is increased. However, there is a sudden drop in the total call throughput capacity. This drop occurs when the hand-off loading becomes large enough as to affect the tail of the distribution of the total number of calls significantly. Since the 99-percentile packet dropping probability is more sensitive to the variations at the tail of the distribution of the total number calls, this drop for criterion 3 occurs at a level of handoff loading which is lower than the corresponding level attained under criterion 2. However, if the hand-off loading is viewed as a percentage of the total call throughput capacity of the cell, the throughput drops under both criteria occur when the hand-off loading constitutes about 40% of the total call throughput capacity of the cell. Figs. 14 and 15 provide a comparison between the two proposed call admission protocols. First note that under both criteria, the call throughput performance

levels exhibited under both protocols are similar. Under criterion 2, Protocol 2 shows a somewhat better performance at lower levels of hand-off loading (by at most 0.2 Erlangs) while at higher hand-off loading levels the two protocols exhibit similar throughput performance. Under criterion 3, for very low levels of hand-off loading, Protocol 1 performs better than Protocol 2. However, for any noticeable amount of hand-off loading Protocol 2 yields a call throughput capacity which is 0.5 Erlang larger than that attained under Protocol 1. The explanation for this phenomenon is as follows. The 99percentile packet dropping probability is mostly dependent on the overall loading level of the cell whereas the 85-percentile packet dropping probability is more dependent on the loading of that type of call which provides most of the loading to the cell. Therefore, under criterion 2, Protocol 2 performs better than Protocol 1 as long as the major part of the cell's loading is provided by the in-cell calls. Under criterion 3, Protocol 2 almost always performs better than Protocol 1, since the incell call regulation is done by considering the total number of admitted calls in the cell, and thus, the total loading of the cell.

5. Conclusions In this paper we have presented an approach for the design and analysis of wireless systems supporting packetized voice users and employing call admission control protocols. A reservation random access scheme is used to regulate the access and transmission of voice packets across the channel. We develop an approximate analyti-

L Rubin, S. Shambayati / Performance o f a reservation random access schemefor packetized wireless systems

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cal procedure to compute the system's performance indices. This approximation is based on the use of a quasi-stationary model derived from the observation that the packet activity of the voice users reaches steady state status between any two consecutive changes in the number of calls. We have modeled the state process of a single cell in such a system as a Markov chain. The

steady state probability distribution of this Markov chain was then approximately calculated. Using these results, we have calculated the average packet dropping probability, the 85-percentile and the 99-percentile packet dropping probabilities, and the call blocking probability, under various call control protocols and call loading levels. We have applied this technique to a - - 23.00

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Fig. 13. Optimum call threshold and total call throughout capacity vs. hand-off loading, in-cell call blocking probability < 0.02, 99% packet dropping probability < 0.01, call admission protocol 1.

L Rubin, S. Shambayati / Performance of a reservation

random access

schemeforpacketizedwireless systems

159

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)

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Fig. 14. Total call throughout capacity vs. percentage hand-off throughput, 85% packet dropping probability < 0.01, in-cell blocking probability <0.02, call admission protocols 1 & 2.

sample case and have observed that such a system, when applying call admission control sustains a call throughput capacity which is about 10 percent higher than the capacity achieved by a corresponding system which applies no call admission control. In addition we have shown that such a system has at least 1.7 times the Erlang capacity of a similar fixed assigned (circuit

switching) T D M A system. A number of call admission policies were studied and compared. In applying distinct call admission controls for local and hand-off calls we have demonstrated that this system can be employed to support high levels of hand-off loading such that no hand-off calls are blocked.

23.00 TotalCallThroughputCapacity,Protocol1 TotalCallThroughputCapacity,Protocol2

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L Rubin, S. Shambayati/ Performanceofa reservationrandomaccessschemefor packetizedwirelesssystems

160

References

Appendix: Calculation off(n, r, l) In this appendix, we derive equations for the recursive calculation o f f (n, r, l), the number of ways in which n distinct objects may be put into r distinct boxes such that I of the boxes have exactly a single object in them. First, we define f~(n, r, I) as the number of ways in which n distinct objects are put into r distinct boxes such that none of the boxes is empty and l of the boxes have exactly one object in them. Obviously, f2(n,O,O) = 1, f~(n,r,O) = O,

~2(n, 1 , 0 ) = { 0 ' 1,

(1.1)

r > n/2,

(A.2)

n=l, otherwise.

(A.3)

For r <
Z

f~(n - i, r - 1,0).

(A.4)

i=2

Furthermore, we note that f~(n, r, 1)

=

f~(n-l,r-l,O).l!,

r>l>O,n>_r.

(A.5) Finally we note that for n > 4, f~(n, 2, 0) = 2~ - 2n - 2.

(A.6)

As we can see from these equations, as n, r and t increase, the new values of f~(n, r, l) can be iteratively calculated from its past values. We calculatef(n, r, l) from f2(n, r, l) according to the following equations: f(n,r,l) O,

l>norl>r, i

f2(n, i, l) ,

Izhak R u b i n received the B.Sc. and M.Sc. from the Technion, Israel, and the Ph.D. degree from Princeton University in 1970, all in electrical engineering. Since 1970, he has been on the faculty of the UCLA School of Engineering & Applied Science, where he is currently a professor in the Electrical Engineering Department. Dr. Rubin has had extensive research, publications, consulting and industrial experience in the design and analysis of telecommunications networks; local, metropolitan, and wide-area computer communications and C3 networks. During the 1979-i 980, he served as acting chief scientist of the Xerox Telecommunications Network (XTEN). At UCLA, he is leading a large research group. Dr. Rubin has also been serving as chiefengineer of IRI Computer Communications Corporation, a leading team of telecommunications, computer communications and C3 experts that provides consulting, analysis, design and software development services. Dr. Rubin is an IEEE Fellow. He has served as co-chairman of the I981 IEEE International Symposium on Information Theory; as program chairman of the 1984 NSF-UCLA workshop on Personal Communications; program chairman for the I987 IEEE INFOCOM conference; and as program co-chair of the 1993 IEEE Workshop on Local and Metropolitan Area Networks. He has also been serving as an editor of the IEEE Transactions on Communications and of the journal on Wireless Networks. E-mail: [email protected]

n > r,

= / Ln/Zj/ r \

[ i~_nt(i/f~(n,i,O),

[(q)

[1] D.J. Goodman, R.A. Valenzuela, K.T. Gayliard and B. Ramamurthi, Packet reservation multiple access for local wireless communications, IEEE Trans. Commun. 37 (1989) 885890. [2] D. Gross and C.M. Harris, Fundamentalsof Queuing Theory, 2nd Ed. (Wiley, New York, 1985). [3] S. Jangi and L. Merakos, Performance analysis of reservation random access protocols for cellular packet communications, Proc. GIobecom'91,pp. 26.4.1-26.4.6. [4] S. Nanda, AnMysis of packet reservation multiple access: Voicedata integration for wireless networks, Proc. Globecom "90, pp. 1984-1988. [5] A.M.Viterbi and A.J. Viterbi, Erlang capacity of power controlled CDMA system, IEEE J. Select. Areas Commun. 11 (1993) 892-900. [6] I. Rubin and S. Shamabayati, Performance evaluation of a reservation random access scheme for packetized wireless systems with call control, Proc. Globecom '92, pp 16-20, Orlando, Florida, Dec. 1992.

n_l,

,,,

otherwise. (1.7)

Shervin Shambayati was Born in 1967 in Tehran, Iran. He received his BS in applied mathematics and engineering from California State University, Northridge in 1989. Subsequently, he received his MSEE and Engineer's Degree from University of California, Los Angeles in 1991 and 1993, respectively. In 1993, Mr. Shambayati joined the Deep Space Communications Systems Group at the Jet Propulsion Laboratory in Pasadena, California, where he continues to work today. Currently, Mr. Shambayati is pursuing his Ph.D. at UCLA, studying issues of multiple access and networking in wireless networks.

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