- Email: [email protected]
Finite population cellular radio systems with directed retry
Applied Mathematical Modelling 23 (1999) 77±86
Finite population cellular radio systems with directed retry Madhu Jain
1
Department of Mathematics, Indian Institute of Technology, Delhi Hauz Khas, New Delhi 110016, India Received 15 August 1996; received in revised form 30 April 1998; accepted 17 July 1998
Abstract This paper proposes an analytically tractable model to predict the performance of cellular communication nonhomogeneous trac originated from ®nite population in cellular radio system. In many locations in cellular radio network, there is overlapping coverage, usually by nearby base stations. This coverage can be shared by mobile users of more than one base station (BS) to improve the teletrac performance characteristics. We study directed retry assignment scheme which allows a call to access the neighbouring BS in case it cannot be served by the BS in which it is located. Various system performace measures viz. blocking probabilities for dierent tracs, overall blocking probability and oered carried load are determined. Analytical results are compared with simulations to verify the validity of the proposed algorithm. It is shown that by increasing overlapping coverage area, a substantial improvement can be achieved. Ó 1999 Elsevier Science Inc. All rights reserved. Keywords: Queue; Cellular radio; Nonhomogeneous trac; Handos; Directed retry; Channel assignment; Blocking
1. Introduction Cellular technology has greatly increased the capability of coping with the increasing demand for mobile communication system. As the cellular network is called upon to take over more and more of the duties of conventional telephones, it becomes necessary for these networks to be able to handle mixed trac. This is so because a person on a mobile phone may connect his laptop to a modem and use the cellular network to transfer data. Whereas the high speed connections of the cellular networks are idle for such data communication, then arise certain issues to be solved due to nonhomogeneous nature of the trac. For the performance evaluation of cellular radio systems, the forced termination probability (the probability that a hando call is blocked) is an important criterion. The forced termination of an ongoing call is less preferable than blocking of initial access of new call. During last few years extensive study of various channel assignment strategies for handos and new attempts with a view of optimum grade of service have been done. These schemes assume that a call is served by a base station (BS) which provides best signal quality in some sense. Often a mobile unit in cellular radio system may be able to establish a communication link of acceptable quality with more than one BS. Usually nearby BSs have overlapping coverage area which can be exploited to improve teletrac performance characteristics such as blocking probability and carried trac (cf. Srivastava and Rappaport, 1991). 1
E-mail: Telex: 31 73087 IIT-IN/e-mail: [email protected]
0307-904X/99/$ ± see front matter Ó 1999 Elsevier Science Inc. All rights reserved. PII: S 0 3 0 7 - 9 0 4 X ( 9 8 ) 1 0 0 6 9 - 0
78
M. Jain / Appl. Math. Modelling 23 (1999) 77±86
Several schemes that utilize the overlapping coverage have been suggested in dierent framework. Chaudhary and Rappaport (1982) considered a general ®xed channel assignment scheme (GFCA) which allows a call to be served by any of several nearby BSs. Directed retry which allows a new call that cannot be served at one BS to attempt access via nearby BS was suggested by Eklundh (1986). Karlson and Eklundh (1989) studied a load sharing scheme which is an enhancement of directed retry by allowing calls in congested cells to be served by neighbouring cells. The exploitation of overlapping coverage areas using GFCA in a microcellular environment was taken into consideration by Goes et al. (1992), and Chu and Rappaport (1994, 1995). Recently Yum and Yeung (1995) developed models for ®nding out the call blocking probability and the probability of additional hando due to directed retry by taking single type of trac originated from in®nite population. Reuse partitioning tends to be most useful to users that are close to base stations, because they can use channels from more partitions. On the other hand use of overlapping coverage area in more than one BS may be bene®tted to mobile users who are at distant from BS. Chu and Rappaport (1997) studied the combined use of overlapping coverage and reuse partitioning for cellular communication system. In overlapping between two adjoining cells, the calling set has the ability to contact either base station. However after crossing the centre of overlapping region, the signal from new BS may be stronger than and therefore preferable to the old one. In case of heavy trac in the second cell, the calls may be blocked due to unavailability of channels. In such case the directed retry scheme directs the caller to contact a neighbouring cell. Actually speaking the calling set is given the contact channel frequency of a few neighbouring cells. If any one of these BSs responds with a strong enough signal, a channel in that cell is allocated. The blocking probability is reduced slightly and also utilization factor is improved by using directed retry scheme. The purpose of our study is to modify Yum and Yeung's (1995) model to study nonhomogeneous trac (mixture of voice and data) originated from ®nite population. We develop an analytically tractable model using ®xed channel assignment with hando priority in the presence of directed retry to analyze communication trac performance and hando issues of cellular system. The blocking probability for each type of trac and for overall system and carried load are computed by using iterative procedure. The validity of proposed scheme is established by comparing the analytical results with simulation. 2. The trac model The queueing model for a cellular radio system with mixed trac originated from ®nite population using directed retry is described as follows: · The two types of trac namely voice and data originated from ®nite population of size M. In each cell four classes of requests arrive in poisson fashion with mean rate (i) knv for new voice calls, (ii) knp for new data packets, (iii) kvh for hando voice calls, (iv) kph for hando packets. · There are total C channels are allocated to each cell. A single call/packet may occupy only one channel, and call/packet holding times are exponentially distributed. The service rate for voice (packet) is denoted by lv lp . · The times for which mobile remain in the hando area termed as dwell times are exponentially distributed with mean 1=ld .
M. Jain / Appl. Math. Modelling 23 (1999) 77±86
79
·
In each cell the calls also arrive from its neighbouring cells due to directed retry with rate rnv rvh for new voice calls (voice hando calls). · For directed retry, it is assumed that a random subscriber has a constant probability of hearing one extra transmitter. This probability is in¯uenced by percentage of overlapped area in a cell. · Consider the cellular structure of seven cells with a fraction (fi ) of overlapping area for a cell i and the set of neighbouring cells Si . We denote the number of neighbouring cells of a cell i by n Si . · To give priority to handover attempts, we allocate Ch channels reserved for hando. · Denote 1=l 1=lp 1=lv , K Kh Kn , where Kh kph kvh rvh ; Kn knp knv rnv . Let Pn be the steady state probability that there are n customers in the system. 3. The mathematical analysis Using birth death process, the steady state probabilities are given by n 8 M K > > P0 ; 0 6 n 6 C ÿ Ch ; > > > l n > > > < M! KCÿCh KnÿCÿCh h P0 ; C ÿ Ch 6 n 6 C; Pn 1 n l M ÿ n! > > > > > > KCÿCh KCh knÿC M! > ph > Q : P0 ; C 1 6 n 6 M; Cl kld M ÿ n!C!lC nÿc k1 PM where P0 can be obtained by using n0 pn 1. We de®ne the following blocking probabilities: bnvi is the probability that a new voice call is blocked in cell i. bnp is the probability that a new packet attempt is blocked. bph is the probability that a packet hando blocked. bvhi is the probability that a voice hando call is blocked in cell i. B is the overall blocking probability that any type of attempt is blocked. By using steady state probability given in Eq. (1), we compute various blocking probabilities as follows: M X pn ; 2 bnp bnvi bvhi
M X
nCÿCh
pn ;
3
nC
bph pM :
4
The overall blocking probability in the cell can be obtained by taking into consideration where the call is originated. We consider the cluster of seven hexagonally shaped cells to compute the overlap covering area of the central cell i with the adjecent j j 1; 2; . . . ; 6 cells. The analysis is done for individual cell i wherein calls are assumed to arrive from ®nite population of size M. Let fi denote the fraction of total area of cell i which is overlapping with nearby cells. The overall blocking probability for voice hando to be blocked is X Bvh 1 ÿ fi bvhi fi bvhi bvhj : 5 j
80
M. Jain / Appl. Math. Modelling 23 (1999) 77±86
The overall blocking probability for new voice call to be blocked is X Bnv 1 ÿ fi bnvi fi bnvi bnvj :
6
j
Thus knv Bnv knp bnp kvh Bvh kph bph : K The carried load is given by B
CL
knv 1 ÿ Bnv knp 1 ÿ bnp kvh 1 ÿ Bvh kph 1 ÿ bph : K
7
8
4. Calculation of directed retry trac We provide a technique to compute arrival rates due to directed retry. It should be noted that a packet is sent for retry only when it cannot have a free channel in the cell from which it has the strongest signal and there is a nearby cell which has strong enough signal to set up the link. Since fi is not too large so that there are relatively little directed retry, therefore we can assume that blocking probabilities in cell i and cell j are independent to each other. Let Si be the set of adjecent cells of cell i and jSi j denote the number of elements in set Si . Following Yum and Yeung (1995), we obtainPtrac due to directed retry as follows: rnvi j knvj Prob. (cell j is blocked and cell i is not blocked for new voice trac). Prob. (a new voice call is directed to cell i/cell j is blocked and cell i is not blocked for new voice calls) X fi ; j 2 Sj ; knvj bnvj 1 ÿ bnvi 9 jSj j j Similarly rvhi
X fi ; kvhi bvhj 1 ÿ bvhi jSj j j
j 2 Sj ;
10
We ®nd that results for blocking probabilities include arrival rates due to directed retry. Therefore there is need to adopt some iterative approach to compute blocking probabilities. We outline the iterative algorithm to compute performance characteristics as follows: Step 0: rnvi 0:; rvhi 0; d 1. Step 1: if jdj < (say. 00001), go to Setp 4. Otherwise go to Step 2. Step 2: Compute Bvh and Bnv using Eqs. (5) and (6). Step 3: Compute new rnvi and rvhi using Eqs. (9) and (10), respectively. Let d be the dierence between the old rvhi and the new rvhi . Go to Step 1. Step 4: Compute bnp ; bph using Eqs. (2) and (4) respectively. Also compute Bvh ; Bnv , B and CL using Eqs. (5)±(8). 5. Numerical experiments The analytical method described in earlier section is used to compute numerical performace results. We have also used a discrete time simulation to verify the validity of the proposed algorithm. The programming of the simulation is done in C- language in the UNIX environment.
M. Jain / Appl. Math. Modelling 23 (1999) 77±86
81
For simulation we generate exponentially distributed random numbers for each type of event. A call is simulated as a packet with the following information in it: (i) cell id. to identify the calling mobile station. (ii) type ®eld to identify the type of call. (iii) location tag to identify whether a call is in the region where it can be sent for directed retry. (iv) arrival time and service time of the packet. (v) Retry ¯ag to indicate whether a packet is already a retry. The ¯ow chart for call handling algorithm is shown in Fig. 1. We ®x the parameters M 17, C 11, Ch 4 and ld 10 l. Figs. 2±6 depict the comparision of the overall blocking probability (B) obtained by analytic method and simulation by varying dierent parameters. In Fig. 2, by varying service times, we note that there is increase in blocking probability with mean service time. There is notable similarity in the results obtained by simulation and analytic method. Figs. 3(a)±(d) reveal the decreasing trend of blocking probability with increasing mean arrival times of new data, new voice, data hando and voice hando attempts, respectively. Note that the decrease in blocking probability is not signi®cant for larger value of mean arrival times (i.e. for light trac). Intitutively one expect blocking to increase with increasing mean arrival rate. Figs. 4(a)±(d) display the eect of variation of mean arrival times of new data, new voice, data hando and voice hando attempts, respectively on the blocking probability when service rate is taken dierent for data and voice calls. We set lp 0.01 and lv 0.02. The composit service rate
Fig. 1. Flow chart for call handling algorithm in simulation.
82
M. Jain / Appl. Math. Modelling 23 (1999) 77±86
Fig. 2. Blocking probability (B) vs. mean service time.
l which is used to generate the blocking probability is computed in two ways (i) by simple average of two service rates lp and lv (ii) weighted average of service rates lp and lv , taking the arrival rates as weights. Curves are drawn for blocking probability using two approximate analytical results having composit service rate l using the manner described above and simulation results. The graph pattern is similar to Figs. 3(a)±(d). The analytical results by using second approximation is much closure to simulation results. The results are satisfactory for the variation of new data and new voice attempts however there is discrepency for high trac of data handos. This may be due to use of approximation rather than exact result for service rate. Directed retry is much useful, in general, for the system in which adjoining cells are unevenly loaded, i.e. one cell has heavy trac while adjoining one is relatively free. The limitation of our analysis and simulation is the assumption of same load in adjoining cells. Thus the eect of directed retry in Figs. 5(a)±(b) are not notably dierent for low trac when overlap area is varied. However for high trac, we observe the trend of decreasing blocking probability with the increase in overlap area. The reservation of a certain number of channels for hando calls (guard channels) is a strategy applied to decrease the blocking of hando calls but at the cost of new calls. That is the reason, while the hando blocking is decreasing, there may be actually an increase in the overall blocking probability as a result of the priority scheme. We observe from Fig. 6(a) that for high trac (lower value of mean arrival times) there is no signi®cant drop in blocking probability whereas for low trac, there is an unwanted rise in blocking probability as fewer channels are available for the new calls. Thus the number of overall blocked calls is higher here as there would be a case when half the channels are free and yet an incoming call is blocked. Again the trend of the simulation results in Fig. 6(b) matches with the result predicted analytically. 6. Discussion We have developed directed retry scheme for a cellular radio system having mixed trac. The various performance characteristics have been computed by using iterative algorithm and sensitivity analysis is carried out by varying dierent parameters. The comparision with simulation results con®rm the validity of the algorithm. The eect of directed retry can be seen in terms of reduction in blocking probability and improvement in carried load. However the improvement is
Fig. 3. Blocking probability (B) by varying mean arrival time of (a) new data calls (b) new voice calls (c) data hando calls (d) voice hando calls.
M. Jain / Appl. Math. Modelling 23 (1999) 77±86 83
hando calls.
Fig. 4. Blocking probability (B) with distinct service rate by varying mean arrival time of (a) new data calls (b) new voice calls (c) data hando calls (d) voice
84 M. Jain / Appl. Math. Modelling 23 (1999) 77±86
M. Jain / Appl. Math. Modelling 23 (1999) 77±86
85
Fig. 5. Eect of overlapping area on blocking probability (B). (a) Analytical, (b) simulation.
accomplished at the expense of those subscribers who can not make use of the directed retry facility due to their location in the cell. It should be noted that only mobile units in the overlap covering area can take advantage of the directed retry facility. The reason for the improvement in carried load is due to sharing of high trac in one cell with the adjoining lesser stressed cells. The cost of directed retry is an increase in the number of handos. A single cell may need several attempts before it is allocated a channel or may be even blocked. Also there is increased complexity of the call handling algorithm in a BS. As a fallout of this, some of the users will ®nd that the performance of the system has actually gone down while others, in the fringes of a call will ®nd it as improved one due to lesser blocked attempts. Acknowledgements This work is supported by University Grant Commission, New Delhi (India) under UGC career award project No F. 10 - 29/93 (SA-III). The author is thankful to unknown referee for their helpful comments for the improvement of the present paper.
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M. Jain / Appl. Math. Modelling 23 (1999) 77±86
Fig. 6. Eect of reserved channels for handos on blocking probability (B). (a) Analytical, (b) simulation.
References Chaudhary, G.L., Rappaport, S.S., 1982. Cellular communication schemes using generalised ®xed channel assignment and collision type request channels, IEEE Trans. Veh. Tech. VT-31, 53±65. Chu, T.P., Rappaport, S.S., 1994. Generalised ®xed channel assignment in microcellular communication systems. IEEE Trans. Veh. Tech. 43 (3), 713±721. Chu, T.P., Rappaport, S.S., 1995. Overlapping coverage and channel assignment in microcellular communication systems. IEEE Proc. Comm. 142 (5), 323±332. Chu, T.P., Rappaport, S.S., 1997. Overlapping coverage with reuse paritioning in cellular communication systems. IEEE Trans. Veh. Tech. 46 (1), 41±54. Eklundh, B., 1986. Channel utilization and blocking probability in a cellular mobile telephone system with directed retry, IEEE Trans. Comm. COM-34, 329±337. Goes, P.B., Kawashima, H., Sumita, U., 1992. Analysis of a new two way communication system with discrete minimal zone, IEEE Trans. Comm. COM-40, 754±764. Karlson, J. and Eklundh, B., 1989. A cellular mobile telephone system with sharing an enhancement of directed retry, IEEE Trans. Comm. COM-37, 530±535. Srivastava, N., Rappaport, S.S., 1991. Models for overlapping coverage area in cellular and micro-cellular communication systems, IEEE GLOBECOM'91, Phoenix, AZ Dec 2±5, 26.3.1-26.3.5. Yum, T.S.P., Yeung, K.L., 1995. Blocking and hand-o performance analysis of directed retry in cellular mobile systems. IEEE Trans. Veh. Tech. 44 (3), 645±650.
Finite population cellular radio systems with directed retry Madhu Jain
1
Department of Mathematics, Indian Institute of Technology, Delhi Hauz Khas, New Delhi 110016, India Received 15 August 1996; received in revised form 30 April 1998; accepted 17 July 1998
Abstract This paper proposes an analytically tractable model to predict the performance of cellular communication nonhomogeneous trac originated from ®nite population in cellular radio system. In many locations in cellular radio network, there is overlapping coverage, usually by nearby base stations. This coverage can be shared by mobile users of more than one base station (BS) to improve the teletrac performance characteristics. We study directed retry assignment scheme which allows a call to access the neighbouring BS in case it cannot be served by the BS in which it is located. Various system performace measures viz. blocking probabilities for dierent tracs, overall blocking probability and oered carried load are determined. Analytical results are compared with simulations to verify the validity of the proposed algorithm. It is shown that by increasing overlapping coverage area, a substantial improvement can be achieved. Ó 1999 Elsevier Science Inc. All rights reserved. Keywords: Queue; Cellular radio; Nonhomogeneous trac; Handos; Directed retry; Channel assignment; Blocking
1. Introduction Cellular technology has greatly increased the capability of coping with the increasing demand for mobile communication system. As the cellular network is called upon to take over more and more of the duties of conventional telephones, it becomes necessary for these networks to be able to handle mixed trac. This is so because a person on a mobile phone may connect his laptop to a modem and use the cellular network to transfer data. Whereas the high speed connections of the cellular networks are idle for such data communication, then arise certain issues to be solved due to nonhomogeneous nature of the trac. For the performance evaluation of cellular radio systems, the forced termination probability (the probability that a hando call is blocked) is an important criterion. The forced termination of an ongoing call is less preferable than blocking of initial access of new call. During last few years extensive study of various channel assignment strategies for handos and new attempts with a view of optimum grade of service have been done. These schemes assume that a call is served by a base station (BS) which provides best signal quality in some sense. Often a mobile unit in cellular radio system may be able to establish a communication link of acceptable quality with more than one BS. Usually nearby BSs have overlapping coverage area which can be exploited to improve teletrac performance characteristics such as blocking probability and carried trac (cf. Srivastava and Rappaport, 1991). 1
E-mail: Telex: 31 73087 IIT-IN/e-mail: [email protected]
0307-904X/99/$ ± see front matter Ó 1999 Elsevier Science Inc. All rights reserved. PII: S 0 3 0 7 - 9 0 4 X ( 9 8 ) 1 0 0 6 9 - 0
78
M. Jain / Appl. Math. Modelling 23 (1999) 77±86
Several schemes that utilize the overlapping coverage have been suggested in dierent framework. Chaudhary and Rappaport (1982) considered a general ®xed channel assignment scheme (GFCA) which allows a call to be served by any of several nearby BSs. Directed retry which allows a new call that cannot be served at one BS to attempt access via nearby BS was suggested by Eklundh (1986). Karlson and Eklundh (1989) studied a load sharing scheme which is an enhancement of directed retry by allowing calls in congested cells to be served by neighbouring cells. The exploitation of overlapping coverage areas using GFCA in a microcellular environment was taken into consideration by Goes et al. (1992), and Chu and Rappaport (1994, 1995). Recently Yum and Yeung (1995) developed models for ®nding out the call blocking probability and the probability of additional hando due to directed retry by taking single type of trac originated from in®nite population. Reuse partitioning tends to be most useful to users that are close to base stations, because they can use channels from more partitions. On the other hand use of overlapping coverage area in more than one BS may be bene®tted to mobile users who are at distant from BS. Chu and Rappaport (1997) studied the combined use of overlapping coverage and reuse partitioning for cellular communication system. In overlapping between two adjoining cells, the calling set has the ability to contact either base station. However after crossing the centre of overlapping region, the signal from new BS may be stronger than and therefore preferable to the old one. In case of heavy trac in the second cell, the calls may be blocked due to unavailability of channels. In such case the directed retry scheme directs the caller to contact a neighbouring cell. Actually speaking the calling set is given the contact channel frequency of a few neighbouring cells. If any one of these BSs responds with a strong enough signal, a channel in that cell is allocated. The blocking probability is reduced slightly and also utilization factor is improved by using directed retry scheme. The purpose of our study is to modify Yum and Yeung's (1995) model to study nonhomogeneous trac (mixture of voice and data) originated from ®nite population. We develop an analytically tractable model using ®xed channel assignment with hando priority in the presence of directed retry to analyze communication trac performance and hando issues of cellular system. The blocking probability for each type of trac and for overall system and carried load are computed by using iterative procedure. The validity of proposed scheme is established by comparing the analytical results with simulation. 2. The trac model The queueing model for a cellular radio system with mixed trac originated from ®nite population using directed retry is described as follows: · The two types of trac namely voice and data originated from ®nite population of size M. In each cell four classes of requests arrive in poisson fashion with mean rate (i) knv for new voice calls, (ii) knp for new data packets, (iii) kvh for hando voice calls, (iv) kph for hando packets. · There are total C channels are allocated to each cell. A single call/packet may occupy only one channel, and call/packet holding times are exponentially distributed. The service rate for voice (packet) is denoted by lv lp . · The times for which mobile remain in the hando area termed as dwell times are exponentially distributed with mean 1=ld .
M. Jain / Appl. Math. Modelling 23 (1999) 77±86
79
·
In each cell the calls also arrive from its neighbouring cells due to directed retry with rate rnv rvh for new voice calls (voice hando calls). · For directed retry, it is assumed that a random subscriber has a constant probability of hearing one extra transmitter. This probability is in¯uenced by percentage of overlapped area in a cell. · Consider the cellular structure of seven cells with a fraction (fi ) of overlapping area for a cell i and the set of neighbouring cells Si . We denote the number of neighbouring cells of a cell i by n Si . · To give priority to handover attempts, we allocate Ch channels reserved for hando. · Denote 1=l 1=lp 1=lv , K Kh Kn , where Kh kph kvh rvh ; Kn knp knv rnv . Let Pn be the steady state probability that there are n customers in the system. 3. The mathematical analysis Using birth death process, the steady state probabilities are given by n 8 M K > > P0 ; 0 6 n 6 C ÿ Ch ; > > > l n > > > < M! KCÿCh KnÿCÿCh h P0 ; C ÿ Ch 6 n 6 C; Pn 1 n l M ÿ n! > > > > > > KCÿCh KCh knÿC M! > ph > Q : P0 ; C 1 6 n 6 M; Cl kld M ÿ n!C!lC nÿc k1 PM where P0 can be obtained by using n0 pn 1. We de®ne the following blocking probabilities: bnvi is the probability that a new voice call is blocked in cell i. bnp is the probability that a new packet attempt is blocked. bph is the probability that a packet hando blocked. bvhi is the probability that a voice hando call is blocked in cell i. B is the overall blocking probability that any type of attempt is blocked. By using steady state probability given in Eq. (1), we compute various blocking probabilities as follows: M X pn ; 2 bnp bnvi bvhi
M X
nCÿCh
pn ;
3
nC
bph pM :
4
The overall blocking probability in the cell can be obtained by taking into consideration where the call is originated. We consider the cluster of seven hexagonally shaped cells to compute the overlap covering area of the central cell i with the adjecent j j 1; 2; . . . ; 6 cells. The analysis is done for individual cell i wherein calls are assumed to arrive from ®nite population of size M. Let fi denote the fraction of total area of cell i which is overlapping with nearby cells. The overall blocking probability for voice hando to be blocked is X Bvh 1 ÿ fi bvhi fi bvhi bvhj : 5 j
80
M. Jain / Appl. Math. Modelling 23 (1999) 77±86
The overall blocking probability for new voice call to be blocked is X Bnv 1 ÿ fi bnvi fi bnvi bnvj :
6
j
Thus knv Bnv knp bnp kvh Bvh kph bph : K The carried load is given by B
CL
knv 1 ÿ Bnv knp 1 ÿ bnp kvh 1 ÿ Bvh kph 1 ÿ bph : K
7
8
4. Calculation of directed retry trac We provide a technique to compute arrival rates due to directed retry. It should be noted that a packet is sent for retry only when it cannot have a free channel in the cell from which it has the strongest signal and there is a nearby cell which has strong enough signal to set up the link. Since fi is not too large so that there are relatively little directed retry, therefore we can assume that blocking probabilities in cell i and cell j are independent to each other. Let Si be the set of adjecent cells of cell i and jSi j denote the number of elements in set Si . Following Yum and Yeung (1995), we obtainPtrac due to directed retry as follows: rnvi j knvj Prob. (cell j is blocked and cell i is not blocked for new voice trac). Prob. (a new voice call is directed to cell i/cell j is blocked and cell i is not blocked for new voice calls) X fi ; j 2 Sj ; knvj bnvj 1 ÿ bnvi 9 jSj j j Similarly rvhi
X fi ; kvhi bvhj 1 ÿ bvhi jSj j j
j 2 Sj ;
10
We ®nd that results for blocking probabilities include arrival rates due to directed retry. Therefore there is need to adopt some iterative approach to compute blocking probabilities. We outline the iterative algorithm to compute performance characteristics as follows: Step 0: rnvi 0:; rvhi 0; d 1. Step 1: if jdj < (say. 00001), go to Setp 4. Otherwise go to Step 2. Step 2: Compute Bvh and Bnv using Eqs. (5) and (6). Step 3: Compute new rnvi and rvhi using Eqs. (9) and (10), respectively. Let d be the dierence between the old rvhi and the new rvhi . Go to Step 1. Step 4: Compute bnp ; bph using Eqs. (2) and (4) respectively. Also compute Bvh ; Bnv , B and CL using Eqs. (5)±(8). 5. Numerical experiments The analytical method described in earlier section is used to compute numerical performace results. We have also used a discrete time simulation to verify the validity of the proposed algorithm. The programming of the simulation is done in C- language in the UNIX environment.
M. Jain / Appl. Math. Modelling 23 (1999) 77±86
81
For simulation we generate exponentially distributed random numbers for each type of event. A call is simulated as a packet with the following information in it: (i) cell id. to identify the calling mobile station. (ii) type ®eld to identify the type of call. (iii) location tag to identify whether a call is in the region where it can be sent for directed retry. (iv) arrival time and service time of the packet. (v) Retry ¯ag to indicate whether a packet is already a retry. The ¯ow chart for call handling algorithm is shown in Fig. 1. We ®x the parameters M 17, C 11, Ch 4 and ld 10 l. Figs. 2±6 depict the comparision of the overall blocking probability (B) obtained by analytic method and simulation by varying dierent parameters. In Fig. 2, by varying service times, we note that there is increase in blocking probability with mean service time. There is notable similarity in the results obtained by simulation and analytic method. Figs. 3(a)±(d) reveal the decreasing trend of blocking probability with increasing mean arrival times of new data, new voice, data hando and voice hando attempts, respectively. Note that the decrease in blocking probability is not signi®cant for larger value of mean arrival times (i.e. for light trac). Intitutively one expect blocking to increase with increasing mean arrival rate. Figs. 4(a)±(d) display the eect of variation of mean arrival times of new data, new voice, data hando and voice hando attempts, respectively on the blocking probability when service rate is taken dierent for data and voice calls. We set lp 0.01 and lv 0.02. The composit service rate
Fig. 1. Flow chart for call handling algorithm in simulation.
82
M. Jain / Appl. Math. Modelling 23 (1999) 77±86
Fig. 2. Blocking probability (B) vs. mean service time.
l which is used to generate the blocking probability is computed in two ways (i) by simple average of two service rates lp and lv (ii) weighted average of service rates lp and lv , taking the arrival rates as weights. Curves are drawn for blocking probability using two approximate analytical results having composit service rate l using the manner described above and simulation results. The graph pattern is similar to Figs. 3(a)±(d). The analytical results by using second approximation is much closure to simulation results. The results are satisfactory for the variation of new data and new voice attempts however there is discrepency for high trac of data handos. This may be due to use of approximation rather than exact result for service rate. Directed retry is much useful, in general, for the system in which adjoining cells are unevenly loaded, i.e. one cell has heavy trac while adjoining one is relatively free. The limitation of our analysis and simulation is the assumption of same load in adjoining cells. Thus the eect of directed retry in Figs. 5(a)±(b) are not notably dierent for low trac when overlap area is varied. However for high trac, we observe the trend of decreasing blocking probability with the increase in overlap area. The reservation of a certain number of channels for hando calls (guard channels) is a strategy applied to decrease the blocking of hando calls but at the cost of new calls. That is the reason, while the hando blocking is decreasing, there may be actually an increase in the overall blocking probability as a result of the priority scheme. We observe from Fig. 6(a) that for high trac (lower value of mean arrival times) there is no signi®cant drop in blocking probability whereas for low trac, there is an unwanted rise in blocking probability as fewer channels are available for the new calls. Thus the number of overall blocked calls is higher here as there would be a case when half the channels are free and yet an incoming call is blocked. Again the trend of the simulation results in Fig. 6(b) matches with the result predicted analytically. 6. Discussion We have developed directed retry scheme for a cellular radio system having mixed trac. The various performance characteristics have been computed by using iterative algorithm and sensitivity analysis is carried out by varying dierent parameters. The comparision with simulation results con®rm the validity of the algorithm. The eect of directed retry can be seen in terms of reduction in blocking probability and improvement in carried load. However the improvement is
Fig. 3. Blocking probability (B) by varying mean arrival time of (a) new data calls (b) new voice calls (c) data hando calls (d) voice hando calls.
M. Jain / Appl. Math. Modelling 23 (1999) 77±86 83
hando calls.
Fig. 4. Blocking probability (B) with distinct service rate by varying mean arrival time of (a) new data calls (b) new voice calls (c) data hando calls (d) voice
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Fig. 5. Eect of overlapping area on blocking probability (B). (a) Analytical, (b) simulation.
accomplished at the expense of those subscribers who can not make use of the directed retry facility due to their location in the cell. It should be noted that only mobile units in the overlap covering area can take advantage of the directed retry facility. The reason for the improvement in carried load is due to sharing of high trac in one cell with the adjoining lesser stressed cells. The cost of directed retry is an increase in the number of handos. A single cell may need several attempts before it is allocated a channel or may be even blocked. Also there is increased complexity of the call handling algorithm in a BS. As a fallout of this, some of the users will ®nd that the performance of the system has actually gone down while others, in the fringes of a call will ®nd it as improved one due to lesser blocked attempts. Acknowledgements This work is supported by University Grant Commission, New Delhi (India) under UGC career award project No F. 10 - 29/93 (SA-III). The author is thankful to unknown referee for their helpful comments for the improvement of the present paper.
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Fig. 6. Eect of reserved channels for handos on blocking probability (B). (a) Analytical, (b) simulation.
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