Page 1

Queueing model for soft-blocking CDMA

systems

Lachlan L. H. Andrew, Donald J. B. Payne and Stephen V. Hanly

Department of Electrical and Electronic Engineering

University of Melbourne, Parkville, Vic 3052, AUSTRALIA

Ph +61 3 9344 9208Fax +61 3 9344 9188

fL.Andrew,djbp,S.Hanlyg@ee.mu.oz.au

Abstract—A new model is proposed for the analy-

sis of CDMA systems. This model is a birth and death

process whose birth process considers the new call ar-

rival rate, the blocking rate, the effect of soft handoff

and the effect of the band allocation strategy in multi-

band (multicarrier) CDMA systems. This model accu-

rately predicts the distribution of the number of calls

connected to a base station.

I. INTRODUCTION

Code division multiple access (CDMA) systems

are characterised by “soft blocking”. New calls are

blocked when the total same cell plus other cell in-

terference is too high. Thus they can neither be con-

sidered to be finite queues, which accept arrivals un-

til a fixed threshold is reached, or infinite queues,

which never block arrivals. The traditional M/M/

queue model [1] does not reflect this soft blocking

behaviour of a CDMA system. This paper develops

an alternative model, in which each cell blocks newly

arriving calls with a state-dependent probability. In

its simplest form this model is similar to that used

(but not directly evaluated) in [2].

This model treats each cell of a multi-cell system

independently. The state of the cell is the number

of users currently in that cell. Other cells simply

contribute random interference. This random inter-

ference causes blocking with some probability,

which is assumed to depend only on the state

current cell.

The state of the cell can be modelled as a birth

and death process [3]. Deaths (call departures) have

negative exponential inter-event times with rate pro-

portional to the current number of calls. Births also

have a negative exponential inter-event time, with

rate equal to the arrival rate thinned by the blocking

probability,

In this paper,

?

B

i,

i of the

????B

i

?.

B

i are determined by simulation.

This work was funded in part by the Australian Research

Council (ARC)

The question arises “if

model the system?” Modelling each cell as indepen-

dent, and treating all other cells as an independent

random process is an approximation. If the simula-

tion results agree with the Markov model, this vali-

dates that approximation. In future, closed form ex-

pressions for the

analytic solution will be available.

B

iare from simulation, why

B

is will be determined, and a fully

II. HARD HANDOFF

When hard handoff is used, new calls are allo-

cated to the nearest base station. This gives rise to

the very simple birth and death process with the ar-

rival rate to state

ture rate

chain “Markov chain A”. Figure 2 shows the proba-

bility distribution for the state of a cell according to

an event driven simulation of the system and the state

distribution of Markov chain A. For comparison, it

i being

????B

i

?, and the depar-

i?, shown in Figure 1. Call this Markov

also shows the distribution for an M/M/ ? model.

Clearly the proposed model gives a much better fit

than the traditional model. This simulation was for a

?

and calls accepted if the signal to interference ratio

(SIR) exceeds 6dB. Log-normal shadowing was as-

sumed, with standard deviation

tipath fading was ignored. The load was 14 Erlangs.

The overall blocking probability was 12.7%, both

by simulation and the Markov A model. (Note that

very high blocking probabilities are used through-

out this paper to highlight the difference in the mod-

els.) When the blocking probability is low, it may

be approximated by the probability that the number

of calls exceeds a maximum permissible value in an

M/M/

is

cessing gain,

all blocking probability,

to other-cell interference in a uniformly loaded sys-

?? hexagonal grid with a spreading factor of 128,

???dB, but mul-

? model. The permissible value typically used

W ????? ???B?f???, where

W is the pro-

? is the required SIR,

B is the over-

f is the ratio of same-cell

Page 2

21n0

32n(1-) (1-) (1-)(1-) (1-)

(n+1)Fig. 1. Markov chain for hard handoff CDMA.

0

0.02

0.04

0.06

0.08

0.1

0.12

0510

Number of Calls

15 2025 30

Probability

simulation

Markov A

M/M/inf

Fig. 2.State distribution for hard handoff: simulated,

proposed Markov model and M/M/? model.

tem, and

threshold depends on the blocking probability, an it-

erative solution is required (although in many cases

theapproximation

??? is the load on the system. Since the

B??suffices). Forhard handoff,

f

approximately 50%, the probability of exceeding ap-

proximately 14 calls. Clearly the approximation that

????, and in this case the blocking probability is

the distribution is approximately that of an M/M/ ?

queue is not valid in this case, and thus the result is

substantially different from the true value and is of

no use in network planning.

Figure 3 shows the blocking probability in each

state for the above simulation.

blocking probability is greater for heavily loaded

cells than for cells with a moderate load. A more

unexpected result is that blocking is also higher for

very lightly loaded cells. This is because of the cor-

relation in blocking events. If the other-cell interfer-

ence is high when a call arrives, it will probably also

be high when the next call arrives. Thus when the

other-cell interference is high for a prolonged period,

the occupancy of the cell will drop as users leave

but no new users are admitted. While it is surpris-

ing that blocking is high when the number of calls in

a cell is low, the converse (that the number of calls

in a cell is low when the blocking is high) is intu-

itively obvious. To verify this explanation, the figure

also shows the blocking per state for soft handoff.

(The load was increased to obtain per-state blocking

of the same order of magnitude as for hard handoff,

although the overall blocking was not matched.) In

As expected, the

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0510 15 20 25 30

Blocking probability

currents calls

hard handoff, 14 Erlangs

soft handoff, 20 Erlangs

Fig. 3. Blocking probability in each state for the simula-

tion used in Figure 2 and, for comparison, the block-

ing probability per state for a soft handoff system.

soft handoff, the other cell interference is greatly re-

duced since users are decoded by the base station at

which they have the highest SIR. Hence the total in-

terference will drop more markedly as the cell emp-

ties than is the case for hard handoff, and prolonged

periods of high interference are less common. This

phenomenon is not captured by the statistical models

used in [2] and elsewhere.

III. SOFT HANDOFF

In soft handoff, a mobile connects simultaneously

to several base stations, and is decoded by the base

station with the highest SIR. This greatly improves

system capacity. The state of a cell is then the num-

ber of users which are currently being decoded by

that cell. Applying the blocking probabilities mea-

sured forsoft handoff to Markov chain A gives a very

poor fit to the soft handoff simulation. This can be

seen in Figure 4, which shows the results for aload of

20 Erlangs. The overall blocking rate is 7.3%, which

is the same order of magnitude as the simulated value

of 5.6%, but that is largely due to the cancellation

of the overestimate of blocking due to very heavily

loaded cells by the underestimate of blocking due to

moderately loaded cells.

Soft handoff produces a distribution with more

cells having approximately the average number of

calls than hard handoff. That is because calls will

typically have a higher SIR in more lightly loaded

cells, and thus will be more likely to connect to the

corresponding base stations, leading to a higher ef-

fective arrival rate at base stations with fewer calls.

This can be modelled by making the arrival rate de-

pend on the current state of the cell. A suitable and

effective heuristic for this arrival rate will now be

Page 3

outlined.

For this purpose it is reasonable to make the ap-

proximation that all of the neighbouring cells have

an equal number of users

sumed that

that the effective arrival rate will be proportional to

the true arrival rate, and so the effective arrival rate

in state

not change when the entire load on the system scales

uniformly, so it will depend on

the ratio

loaded, and the arrival rate at all cells will simply be

the true arrival rate, yielding

. Initially it will be as-

n????. It is also reasonable to assume

i will be

k

i?n

????B

i

?. Similarly,

k

i?nshould

i and

n only through

i?n. When

i?n??, the system is uniformly

k

i?n

??. For

i?n??,

k

nect to neighbouring base stations, rather than the

base station of interest. Finally, for

will be around 2, since the base station will get all of

the calls in its own cell, and those calls in the neigh-

bouring cells for which the base station of interest is

the second nearest. One functional form that meets

these requirements is

pirically it has been found that this provides a good

fit with

against

tive to the mean load. Thus the overall arrival rate

is

i?nwill be almost zero, since most users will con-

i?n??,

k

i?n

??????i?n?

a

? for

a??. Em-

a??. Figure 5 shows this scaling factor

i?n, the occupancy of the current cell rela-

??

???i????

?

???B

i

?

in state

chain B”. Once an estimate of the overall block-

ing probability,

be improved by approximating the occupancy of

neighbouring cells as

“Markov chain B with two iterations”. This may

be repeated until the blocking probability converges.

The results for soft handoff with a load of 20 Er-

langs are shown in Figure 4. After one iteration,

Markov chain B produced a much better approxima-

tion to the true state distribution than Markov chain

A. However, it slightly overestimated the number of

calls in the cell (that is, the mean of the distribution

was shifted to the right), and it predicted an overall

blocking rate of 7.4%, which is very similar to that

of Markov chain A, but substantially different from

the true value of 5.6% obtained by simulation. After

a second iteration, the model produced a very good

approximation to the simulated state distribution, and

predicted a blocking probability of 5.4%. For com-

parison, the results with no blocking are included.

The fact that results from Markov chain B match

the simulated curve well but those from Markov

chain A do not shows that the good match is not “in-

i. Call this modified Markov chain “Markov

B, is known, the Markov chain can

n?????????B?, yielding

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

05 10 15 20 2530

Probability

# Calls in cell

simulation

Markov A

Markov B

2 iterations

no blocking

Fig. 4.State distribution for soft handoff: Simulated,

original Markov chain, modified Markov chain.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.51 1.5

relative load, i/n

2 2.53 3.54

scaling factor, k

Fig. 5. Scaling factor applied to arrival rate, as a function

of

i?n.

evitable” given the per-state blocking probabilities,

but rather that the choice of Markov chain is respon-

sible for the good match.

Note that it was not necessary to alter the arrival

rates in this way for soft handoff in [2]. That is be-

cause base station selection there was based on the

measured path gain, rather than the SIR. Thus the

base station to which a new call connected depended

only on the propagation conditions and was indepen-

dent of the load on the cell, causing the arrival rate

(before blocking) to be indpendent of the state.

IV. MULTI-BAND OPERATION

The capacity of a CDMA system may be easily in-

creased using multi-band (or multi-channel) CDMA,

in which a second slice of spectrum is used, and calls

are allocated to one or other band [4–6]. The perfor-

mance of such systems is influenced by the way new

calls are allocated to bands, and this difference in

performance can be predicted by the proposed tech-

nique. The Markov chain will now be two dimen-

Page 4

2µ

µ

µ

µ

2µ

2µ

2µ

3µ

3µ

3µ

1,0 1,11,21,3

2,02,12,2 2,3

3,03,1 3,2 3,3

0,00,10,2 0,3

...

...

::

λ00λ01λ02

λ12λ11λ10

λ20 λ21λ22

λ30 λ31 λ32

λ30 λ20λ10 λ00µµµ

2µ 2µ 2µ

3µ 3µ3µ

λ01λ11 λ21λ31

λ32 λ22 λ12 λ02 3µ

2µ

µ

3µµ

::

...

...

Fig. 6. Markov chain for two band system.

sional (or

ity, this paper will only treat the case of hard handoff.

One obvious strategy is to assign a call to a band

randomly. If the call is blocked in the chosen band, it

istried inthe next band. Thisgives rise tothe Markov

chain of Figure 6, where

band in state

rate of arrivals to band

overall arrival rate (

tially selecting band

of not being blocked on band

state

the probability of initially selecting the other band

N dimensional for

N bands). For simplic-

?

ijis the arrival rate to the

j from either state

?i?j? or

?j?i?. The

k from a given state is the

??) times the probability of ini-

k ( ????), times the probability

k given that band

k is in

j ( ???B

j), plus the overall arrival rate times

( ?

not being blocked on band

???), but being blocked on that band ( ?B

i) and

k ( ???B

j). Thus

?

ij

?

????????

By symmetry, this collapses to the chain of Figure 7,

but now with

cause here the probability of being in state

the probability of having

bands, while the probability of being in state

?B

j

??B

i

???B

j

???????B

j

????B

i

???.

?

iidoubled to

????B

?

i

?. That is be-

?i?i? is

i calls in each of the two

?i?j?,

j

the first band and

the first band and

not require knowledge of the blocking probability for

each state, but only the marginal blocking probability

given there are

Another strategy which has proven successful is

to assign new calls to the band which currently has

the fewest calls [5], with blocked calls again trying

the other band. This can again be represented by the

collapsed Markov chain of Figure 7, but now with

??i, is the probability of either having

i calls in

j in the second, or having

j calls in

i in the second. Note that this does

i calls in a particular band.

?

ij

?????B

j

? if

i?j,

?

ij

? ?B

i

???B

j

?

2µ

2µ3µ

3µ

1,11,21,3

2,22,3

3,3

0,00,10,2 0,3

...

...

λ00λ01 λ02

λ12 λ11

λ22

λ30λ20 λ102µµµ

2µ 4µ

6µ

λ21λ31

λ32

3µµ

...

...

Fig. 7. Collapsed Markov chain for two band system.

if

(“LL”) strategy has been analysed thoroughly under

the assumption of no blocking [6].

Figures 8 and 9 show the actual (i.e., simulated)

and Markov state distributions for each of the two

i?j, and

?

ii

?????B

?

i

?. This “least load”

strategies, along withthe M/M/ ?results forcompar-

ison. The spreading factor was halved to 64, so that

the same total bandwidth was used as for the single

band case, and the offered load was 17 Erlangs. For

simplicity, the graph shows the marginal probability

of having

distribution. Clearly there is very good agreement

between the model and simulation. Least load allo-

cation shows a more peaked distribution, while ran-

dom allocation shows a heavier tail, with more cells

having a large number of calls. This behaviour is

captured well by the proposed Markov model.

The blocking probabilities for the simulated (re-

spectively Markov) cases were 16.1% (15.6%) for

least load, and 16.6% (13.4%) for random. The fig-

ures for the Markov model are an approximation,

given by the square of the marginal per-band block-

ing probability. This is a fair approximation because

a call must be rejected on both bands to be blocked.

To show once again that the good agreement with

simulation is not solely due to the use of accurate

per-state blocking probabilities, Figures 10 and 11

show the Markov distribution for each strategy us-

ing the per-state blocking probabilities fromthe other

strategy. Each graph shows the two sets of simula-

tion results and the results using one of the Markov

chains with both sets of measured blocking probabil-

ities. In each case, the predicted results match the

i calls in the first band, rather than the full

Page 5

0

0.02

0.04

0.06

0.08

0.1

Simulated vs. Markov and M/M/ ? results for

random band allocation strategy.

0.12

0.14

0.16

0246810 12 141618

Probability

Number of Calls

Sim, Rnd

Markov, Rnd

M/M/inf, Rnd

Fig. 8.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

02468 10 12141618

Probability

Number of Calls

Sim, LL

Markov, LL

M/M/inf, LL

Fig. 9.Simulated vs. Markov and M/M/ ? results for

least load band allocation strategy.

simulation to which the Markov model corresponds

more closely than the one from which the per-state

blocking probabilities were taken. This confirms that

the model is not overly sensitive to the exact per-state

blocking probabilities. Thus this model can be used

to predict the overall blocking probability for differ-

ent band allocation schemes based on the different

Markov chains and a single set of measured per-state

blocking probabilities.

V. CONCLUSION

It is possible to model soft blocking in multi-cell

CDMA systems as an independent birth and death

process at each cell. The birth process is thinned by

the state-dependent blocking probability. The birth

process can also be modified to model soft handoff,

and the process can be extended to analyse multi-

band systems.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

02468 101214 16 18

Probability

Number of Calls

Markov

Sim, Rnd

Sim, LL

Fig. 10. Predicted state distributions for random Markov

model using least load blocking probabilities.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0246810 1214 1618

Probability

Number of Calls

Markov

Sim, Rnd

Sim, LL

Fig. 11.Predicted state distributions for least load

Markov model using random blocking probabilities.

ACKNOWLEDGEMENT

The authors thank Chaitanya Rao forcomments on

this manuscript, and for pointing out an error in the

original arrival rates for the multiband Markov chain

with random assignment.

REFERENCES

[1] A. J. Viterbi, Principles of Spread Spectrum Communica-

tion. Reading, MA: Addison-Wesley, 1995.

[2] Y. Ishikawa and N. Umeda, “Capacity design and perfor-

manceof calladmission control incellularCDMAsystems,”

IEEE J. Select. Areas Commun., vol. 15, pp. 1627–1635,

Oct. 1997.

[3] M. Kijima, Markov processes for stochastic modeling. Lon-

don: Chapman and Hall, 1997.

[4] L. L. H. Andrew, “Measurement-based band allocation in

multiband CDMA,” in Proc. Infocom ’99, (New York),

1999, pp. 1536–1543.

[5] T. Dean, P. Fleming, and A. Stolyar, “Estimates of multi-

carrier CDMA system capacity,” in Proc. Winter Sim. Conf.,

(Washington, DC), 1998, pp. 1615–1622.

[6] P. J. Fleming and B. Simon, “Heavy traffic approximations

for a system of infinite servers with load balancing,” to ap-

pear.