Channel assignment for GSM half-rate and full-rate traffic

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Channel assignment for GSM half-rate and full-rate traffic

P. Lin, Y.-B. Lin*

Department of Computer Science and Information Engineering, National Chaio Tung University Hsinchu, Taiwan, ROC

Abstract

Global System for Mobile Communications (GSM) supports full-rate and half-rate calls. In this paper, we propose analytical and simulation models to study the performance of four channel assignment schemes for GSM half-rate and full-rate traffic. Our study indicates that among the four schemes, the repacking scheme has the best performance for mixing half-rate and full-rate traffic. We also observe that good performance is expected if the standard derivation of the cell residence time for a mobile station is large.q 2000 Elsevier Science B.V. All rights reserved.

Keywords: Global system for mobile communications; Full-rate; Half-rate

1. Introduction

Global System for Mobile Communications (GSM) [1] is a standard adopted by cellular systems widely developed in Europe and Asia. In a GSM network, a mobile station (MS) initiates a communication session by making an access request to a base station (BS), if the MS is in the cell (the radio coverage area) of the BS. If no channel is available at that cell, the call is blocked. If the MS moves to another cell during the conversation, then the radio link to the old BS is disconnected and a radio link to the new BS is required to continue the conversation. This process is called handoff [2,3]. If the new BS does not have any idle channel, the handoff call is dropped or forced to terminate. Handoff requests and new call requests compete for radio channels in a cell. Several channel assignment schemes [4] have been proposed to reduce call blocking and call dropping.

GSM combines time division multiple access (TDMA) and frequency division multiple access (FDMA) for radio channel allocation. In this approach, a frequency carrier is divided into eight time slots per frame, which are used to support speech and data transmission. GSM supports full-rate calls and half-full-rate calls. A full-full-rate call uses one time slot in every frame, while a half-rate call uses one time slot in every two frames. Once an MS initiates a full-rate (half-rate) call request, the MS will operate in full-rate (half-(half-rate) mode until the call is terminated. A call may alternate between full-rate and half-rate channels [5]. Such an approach is not considered here. In this paper, mixing

full- and half-rate calls in a frequency carrier result in eight full-rate calls, 16 half-rate calls, or any feasible combi-nations. To simplify the description, we view a GSM time slot as a “full” time slot that can be divided into two half time slots. Fig. 1 shows a feasible combination. In this figure, time slots 4 and 7 are idle. Time slots 1, 2 and 5 are occupied by full-rate calls F1, F2 and F3, respectively. Time slot 3 is occupied by two half-rate calls H2 and H3. Time slots 0 and 6 are occupied by half-rate calls H1 and H4, respectively. These two time slots are referred to as “partially occupied” time slots. The channel allocation stra-tegies for incoming calls may significantly affect the perfor-mance. For example, if eight half-rate calls occupy eight different full time slots in a frequency carrier; that is, the eight time slots are partially occupied, then the next incom-ing full-rate call will be blocked. On the other hand, if these half-rate calls are packed into four full time slots, then the frequency carrier can accommodate four extra full-rate calls. In this paper, we evaluate four GSM channel assign-ment schemes described in Ref. [6]: random, repacking, fair-repacking and best-fit. These schemes are elaborated as follows.

Random: all full-rate and half-rate calls are assigned to

any free time slots without any control.

Best-Fit: each incoming full-rate call is allocated an

empty full time slot. A half-rate call is always assigned a partially occupied time slot that has already contained a half-rate call. If no such time slot exists, then an empty full time slot is assigned to the half-rate call. Note that when a half-rate call departs, it is possible that more than one partially occupied time slots exist.

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Repacking: this scheme is similar to the best-fit scheme

except that when a full-rate call arrives to a cell, the scheme repacks the half-rate calls if two partially occu-pied time slots exist. Repacking is achieved by intracell handoff technology.

Fair-Repacking: this scheme is a variation of the

repack-ing scheme. The only difference between repackrepack-ing and fair-repacking is that in fair-repacking, if only one half time slot is left in a cell, the next incoming half-rate calls will be blocked. In Ref. [6], the authors claimed that with fair-repacking, the blocking/force-termination probabil-ities of full- and half-rate calls are likely to be equal for mix traffic. Our study will indicate that when the number of channel in a cell is small, fair-repacking significantly degrades the performance of the full-rate calls without improving the half-rate call performance. On the other hand, the performance of fair-repacking is similar to repacking for a GSM cell with a large channel number. Since the implementation complexity for fair-repacking is higher than that for repacking, fair-repacking may not be appropriate for a practical GSM network.

The above four algorithms have been evaluated in Ref. [6] without considering the MS mobility. By accommodat-ing the MS mobility, this paper proposes an analytical model for repacking and simulation models for the four schemes.

2. Input parameters and output measures

This section lists the input parameters and output measures used in this paper. The input parameters include

• lf(lh): the new full-rate (half-rate) call arrival rate to a cell

• 1/mf(1/mh): the expected full-rate (half-rate) call holding time

• hf(hh): the full-rate (half-rate) MS mobility rate • c: total number of time slots in a cell

The output measures include

• lh,f(lh,h): the handoff full-rate (half-rate) call arrival rate to a cell

• pf,f(pf,h): the force-termination probability for the full-rate (half-full-rate) call

• pb,f(pb,h): the new call blocking probability for a full-rate (half-rate) call

• pnc,f(pnc,h): the probability that a full-rate (half-rate) call is not completed (either blocked or forced to terminate)

• pnc lfpnc_;f1lhpnc_;h=lf1lh : the probability that a

full-rate or half-rate call is not completed

3. An analytical model for repacking

This section proposes an analytical model for the repack-ing scheme, which accommodates MS mobility. We assume that the full-rate (half-rate) call arrivals to a GSM cell form a Poisson process. Consider the timing diagram in Fig. 2. Let tci be the call holding time for type i call where i f (full-rate) or h (half-rate), which is assumed to be exponen-tially distributed with the density function

fcitci mie

2mitci _{for i} f or h 1

and the mean call holding time is Etci 1=mi: The cell residence time of an MS (for type i call service) at a cell j

Fig. 1. An example for full- and half-rate traffic.

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is tmi;j: In this figure, tmi;0is the time that the MS resides at cell 0, and tm_i_;j(where j$ 1 is the residence time at cell j.

We assume that tm_i_;0; tm_i_;1; tm_i_;2; tm_i_;3; tm_i_;4; …; tm_i_;k are

inde-pendent and identically distributed random variables with a general density function fmitmi;j with mean 1=hi: Let f

p mis be the Laplace transform of the cell residence time distribu-tion. Then fmpis Z∞ t_mi;j0 fmitmi;j e 2stmi;j_dt mi;j for i f or h: 2

Consider type i calls. Letlh,ibe the handoff call arrival rate to a cell, pb,ibe the new call blocking probability, pf,ibe the force-termination probability, and pnc,i be the probability that a type i call is not completed. From Ref. [7],lh,ican be expressed as:

lh;i

1 2 pb;ihi1 2 fmpimili

mi1 2 1 2 pf;ifmpimi

for i f or h: 3 For the moment, we assume that pb,iand pf,iare known. Both probabilities are derived by using an iterative algorithm to be described later. From Ref. [7], type i call trafficrito a cell is: ri li mi 12 hipb;i1 2 f p mimi mi1 2 1 2 pf;ifmpimi ( ) for i f or h 4 and pnc,iis pnc;i pb;i1 1 2 pb;ihi1 2 fmpimi mi1 2 1 2 pf_;ifmpimi ( ) pf;i for i f or h 5

Finally, the average probability pncthat a call (either full-rate or half-full-rate) is not complete can be computed as follows:

pnc l

fpnc;f1lhpnc;h

lf1lh

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We will use pnc,iand pncas the major output measures in our performance study.

To derive the new call blocking probability pb,i, we consider a stochastic process with state n nh; nf where

nhand nfrepresent the numbers of the outstanding half-rate and full-rate calls in a cell, respectively. Suppose that there are c full time slots in a cell where a full time slot can be used by one full-rate call or two half-rate calls, then in the repacking scheme, the following constraints must be satis-fied:

nh1 2nf # 2c; 0 # nh# 2c; and 0 # nf # c

The state space S of the stochastic process is

S {nh; nfunh1 2nf # 2c; 0 # nh# 2c; and 0 # nf

# c} 7

According to Zachary [8] and Kelly [9], the stationary probability of the state n nh; nf can be computed as

pn G21 r nf f nf! rnh h nh! 8 where GX n[S rnf f nf! rnh h nh! 9

The second and third terms of the right hand side of Eq. (8) are the weights contributed by the full-rate call traffic and the half-rate call traffic, respectively. G in Eq. (9) is a normalized factor to ensure thatPn[Spn 1:

With the above stochastic process model, pb,fis computed as follows. When a full-rate call arrives at a cell, it is blocked if no more than one half time slot is left in that cell. That is, nh1 nf  2c or nh1 nf  2c 2 1 when the

full-rate call arrives. Define E1as

E1 {nh; nfunh1 2nf  2c or nh1 2nf  2c 2 1; 0# nh# 2c; 0 # nf # c} 10 Then we have pb;f X n[E1 pn 11

Similarly, when a half-rate call arrives at a cell, it is blocked if all time slots are busy. That is, nh1 2nf  2c when the

half-rate call arrives. Let

E2 {nh; nfunh1 2nf  2c; 0 # nh# 2c; 0 # nf # c} 12 then pb;h X n[E2 pn 13

With Eqs. (3)–(6), (11) and (13), we use an iterative algo-rithm [7] to computelh;f;lh;h; pnc;f and pnc;h:

The Iterative Algorithm.

Step 1: Select initial values forlh,handlh,f.

Step 2:Compute pb,fand pb,hby using Eqs. (4), (11) and (13).

Step 3:lh;f ;old←lh;f andlh;h;old←lh;h:

Step 4: Computelh,fandlh,hby using Eq. (3).

Step 5: If ulh;f2lh;f;oldu .dlh;f and ulh;h2lh;h;oldu .

dlh;h then go to Step 2. Otherwise, go to Step 6. Note

thatd is a pre-defined threshold. In our study,d is set to 0.00001.

Step 6: The values forlh_;f;lh_;h; pb_;f; and pb_;h; converge.

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4. Discrete event simulation models

This section describes a discrete event simulation model for repacking, best-fit, fair-repacking and random. In our simulation experiments, the GSM network is configured with k2 BSs connected as a k× k wrapped mesh [10], where k 6 is found adequate to simulate a large-scale GSM network. We assume that an MS resides at a cell for a period, and then moves to one of the four neighboring cells with the same routing probability (i.e. 0.25). The full-rate (half-rate) call arrivals to each cell form a Possion process with arrival ratelf(lh).

We develop a discrete event simulation model for these

schemes. Three types of events are defined to represent a new call arrival, a call completion, or a handoff call moving from the old cell to the new cell. An event is either for a full-rate call or a half-full-rate call. The events are inserted into an event list, and are deleted/processed from the event list in non-decreasing timestamp order. A simulation clock ts is maintained to indicate the progress of the simulation. In each experiment, more than 100,000 incoming calls are simulated to ensure that the results are stable. Several coun-ters are maintained in the simulation including the number

Nf(Nh) of full-rate (half-rate) call arrivals, the number Nb,f (Nb,h) of blocked new full-rate (half-rate) calls and the number Nf,f (Nf,h) of force-terminated handoff full-rate (half-rate) calls. These counters are used to compute pnc,f and pnc,h: pnc;f Nb;f1 Nf;f Nf and pnc;h Nb;h1 Nf;h Nh

In the simulation model, a cell j is modeled as an object

Cell(j). Every cell object contains c time_slot sub-objects

and a member function Channel_allocation used to allocate channels based on the four different schemes described in Section 1. The data structure of time_slot consists of a state variable and an array of channel allocation times. The state indicates whether the time slot is empty, occupied by one half-rate call, two half-rate calls, or a full-rate call. The channel allocation times indicate when the time slots are occupied by the corresponding calls. The simulation flow is similar to that in Ref. [11] with the following exceptions: Before channel allocation, the repacking and the fair-repacking simulations repack the time slots of the time_slot object. When Channel_allocation function is invoked, the simulation manipulates the time_slot objects according to the channel assignment schemes described in the previous section.

Fig. 3 plots the pnc curves obtained from the analytical model (the dashed curves) and the simulation model (the solid curves) of the repacking scheme. The figure indicates that the analytical and simulation results are consistent.

5. Performance evaluation

This section investigates the performance of the four GSM channel assignment schemes based on the perfor-mance models developed in Sections 3 and 4. In our study, the considered input parameters lf,lh, hf, hhand

mhare normalized bymf. For example, if the expected full-rate call holding time is 1=mf 2 min; then lf 2mf

means that the expected full-rate inter call arrival time is 1 min.

5.1. Effect of the proportion oflf

Fig. 4(a) plots pncas a function of the ratiolf=lf1lh;

where hf hh 2mf; mhmf and c 7: In this figure,

the net call arrival rate is a fixed value lf1lh 4mf: Fig. 3. Comparison of the analytical and the simulation resultslf1lh

4mf;mhmf; c 7:

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The figure indicates that the order, from the best to the worst, of the pnc performance for the four schemes is: repacking, best-fit, random and fair-repacking. Note that the performance differences are most significant when

lf=lf1lh 0:5; where the repacking scheme results

in 20% improvement over fair-repacking, 14.2% improve-ment over random and 9.68% improveimprove-ment over best-fit. It is clear that if the GSM network only has single type traffic, i.e. when l_f 0 or l_h 0; the performance of the four schemes is the same.

5.2. Effect of incoming call traffic

Fig. 5(a) plots pnc as a function of lf, where lf

lh; mhmf; hf hh 2mf and c 7: The performance

superiority among these four schemes is the same as what we observe in Fig. 4. The performance differences among these schemes become more significant aslf(lh) increases. Fig. 6(a) and (b) plots pnc,fand pnc,has functions oflf, where

lf lh; mhmf; hf hh 2mf and c 7: Fig. 6(a)

indicates that the pnc,f performance from the best to the worst is in the following order: fair-repacking, repacking, best-fit and random. The performance differences among the four schemes become more significant aslf(lh) increases. Fig. 6(b) indicates that the pnc,hperformance from the best to the worst is in the following order: random, best-fit, repacking and fair-repacking. The performance differences among the four schemes also become more significant aslf (lh) increases. Fig. 6(a) and (b) indicates that, when pnc,ffor a scheme is large, the scheme will have a small pnc,hvalue compared with the other schemes. The reason is that when more time slots are occupied by full-rate (half-rate) calls, the half-rate (full-rate) calls are more likely to be blocked. Note that pnc,hfor repacking is much lower than that for fair-repacking. On the other hand, pnc,ffor both repacking and fair-repacking are about the same. In other word, “fairness” of fair-repacking is achieved by significantly degrading the half-rate call performance without improving the full-rate call performance. Thus, it is clear that repacking is better than fair-repacking.

5.3. Effect of MS mobilitieshfandhh

Fig. 4(a) shows the pncperformance for high MS mobility hf hh 2mf; and Fig. 4(b) shows the pncperformance for low MS mobility hf hh 0:1mf: For each of the

four schemes, pncdecreases as the MS mobility decreases. When the MS mobility is high, the performance differences among the four algorithms are not consistent with that for low MS mobility. For example, consider the case when

lf=lf 1lh 0:5: For hfhh 2mf; the repacking

scheme results in 9.68% improvement over best-fit, 14.2% improvement over random and 20% improvement over fair-repacking. On the other hand, forhf hh 0:1mf;

repack-ing results in 15.8% improvement over best-fit, 18.34% improvement over random and 19.43% improvement over fair-repacking. Fig. 5(a) and (b) indicates similar results.

Fig. 7 plots pnc,fand pnc,hof repacking for the case when

hf ±hh: In this figure, lf lh; mhmf and c 7: Fig.

7(a) and (b) plots pnc,fand pnc,has functions ofhhandlfby fixinghf 2mf: This figure indicates that changinghhdoes not affect pnc,f significantly. On the other hand, pnc,h increases ashhincreases. In Fig. 7(c) and (d), we fixhh

2m_f: The figure indicates that changing hf has significant effect on pnc,f, and only has insignificant effect on pnc,h. We conclude that the full-rate (half-rate) MS mobility has

Fig. 5. Effect of incoming call trafficc 7; mhmf;lhlf:

Fig. 6. Effect of incoming call traffic (Cont.; c 7;hfhh 2mf;

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Fig. 7. Effect of MS mobilitieshfandhhc 7;mhmf;lhlf:

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significant effects on pnc,f(pnc,h), but only has insignificant effects on pnc,h(pnc,f) for the half-rate (full-rate) calls. 5.4. Effect of c

Fig. 8 plots pncas a function oflf, wherelf lh; mh

mf; hf hh 0:1mf and c 21: In both Figs. 5(b) and 8,

the offered loads to a GSM cell are selected such that pncis bounded by 2.25%. From Fig. 8, we observe that with large

c, pncperformance from the best to the worst is in the follow-ing order: repackfollow-ing, fair-repackfollow-ing, best-fit and random. This order is different from that with small c as shown in Fig. 5(b).

When c 21; the repacking scheme results in larger improvement over the best-fit and random schemes. For example, when lf lh 9:6mf; the repacking scheme

results in 25.8% improvement over the best-fit scheme and 32.7% improvement over the random scheme. As c increases, fair-repacking behaves more like the repacking scheme. That is, it becomes “less fair” and has much better

pncperformance.

5.5. Effect of standard deviation of the cell residence time

In Fig. 9, we use the same Gamma cell residence time distributions [12,13] for both full-rate and half-rate MSs. We assume that the Gamma distribution has the mean value 1=h; the standard deviations 1=hpa; wherea is the shape parameter for MS cell residence times. Fig. 9 shows the effect ofs on the pncperformance for repacking, wherelf lh; mhmf; and c 7: Fig. 9(a) plots pncas a function of with high MS mobility h 2m_f: This figure indicates that if it is sufficiently small (i.e.s# 1=h; then pnc is not sensitive to the change ofs. On the other hand, for

s. 1=h; pnc decreases as s increases. Fig. 9(b) plots pnc with low MS mobility h 0:1mf: This figure indicates

results similar to that in Fig. 9(a) except that the effect is not as significant. We conclude that high MS mobility with large variation (standard deviation) has significant effect on

pnccompared with low MS mobility with large variation.

6. Conclusion

We proposed analytical and simulation models to inves-tigate GSM channel assignment performance for half-rate and full-rate traffic. The channel assignment schemes under evaluation are random, best-fit, repacking and

fair-repack-ing. Our study indicated that the repacking scheme can significantly improve the pnc performance over the other three schemes (about 20% improvements are observed). The probability pnc increases when the proportion of full-rate call traffic increases. We also observed that changinghf (hh) has significant effect on pnc,f(pnc,h), and only has insig-nificant effect on pnc,h (pnc,f) in the repacking scheme. Furthermore, good pnc performance is expected when the variation (standard deviation) of the MS residence time is large.

Acknowledgements

P.L.’s work was supported in part by National Science Council, Contract No. NSC88-2213-E009-079.

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