Performance Analysis of Cellular Radio System Using Artificial Neural Networks

Transcription

1 Amercan Journal of Neural Networks and Applcatons 27; 3(): do:.648/j.ajnna ISSN: (rnt); ISSN: (Onlne) erformance Analyss of Cellular Rado System Usng Artfcal Neural Networks Krt rya Gupta, *, Madhu Jan 2 Symboss Centre for Management Studes, NOIDA Faculty of Management, Symboss Internatonal Unversty, une, Inda 2 Department of Mathematcs, Indan Insttute of Technology (IIT), Roorkee, Inda Emal address: krt.gupta@scmsnoda.ac.n (K.. Gupta), madhujan@sancharnet.n (M. Jan) * Correspondng author To cte ths artcle: Krt rya Gupta, Madhu Jan. erformance Analyss of Cellular Rado System Usng Artfcal Neural Networks. Amercan Journal of Neural Networks and Applcatons. Vol. 3, No., 27, pp do:.648/j.ajnna Receved: December 26, 26; Accepted: January 6, 27; ublshed: March 7, 27 Abstract: In ths paper, we explot one of the fastest growng technques of Soft Computng,.e. Artfcal Neural Networks (ANNs) for obtanng varous performance measures of a cellular rado system. A prortzed channel scheme wth subratng s consdered n whch a fxed number of channels are reserved for handoff calls and n case of heavy traffc, these reserved channels are subrated nto two channels of equal frequency to deal wth more handoff calls. Two models dealng wth nfnte and fnte number of subscrbers are consdered and the blockng probabltes of new and handoff calls are computed analytcally as well as by usng ANNs. A feedforward two-layer ANN s consdered for obtanng the blockng probabltes. The backpropagaton algorthm s used for tranng the ANN. The analytcal and ANN results are compared by takng the numercal llustratons. Keywords: Artfcal Neural Networks, Cellular Rado System, Handoff, Reserved Channels, Subratng, Backpropagaton. Introducton Among the varous paradgmatc changes n scence and technology that have taken place n ths century, one such change concerns the concept of Soft Computng (SC). Soft computng provdes flexble nformaton processng capabltes for handlng real lfe ambguous stuatons. Hard computng has the characterstcs of precson and categorcty whle the soft computng has the propertes of approxmaton and dspostonalty. Soft computng explots the tolerance for mprecson and uncertanty to acheve tractablty, lower cost, hgh Machne Intellgence Quotent (MIQ) and economy of communcaton. One of the most powerful technques of soft computng s Artfcal Neural Networks, whch ams to perceve and comprehend the sgnfcance of the data wth whch they are traned. ANN approach s frequently employed to analyze a varety of problems and s best dstngushed from other SC technques n that t s non-rule-based and can addtonally be made stochastc so that the same acton does not necessarly take place each tme for the same nput. A stochastc behavor allows a neural network to explore ts envronment more fully and potentally to arrve at a better soluton than the conventonal methods. ANN s a powerful data-modelng tool that s able to capture and represent complex nput/output relatonshps. The propertes of ANN lke learnng and adaptaton, classfcaton, functon approxmaton etc. have made them of extreme use n solvng varous mathematcal problems. Neural networks have been successfully appled to broad spectrum of data-ntensve applcatons, such as Sgnal rocessng [], Chp Desgnng [2], optmzaton problems [3] and n many engneerng problems [4]. ANNs are also used for solvng problems that are too complex for conventonal technologes e.g., problems that do not have an algorthmc soluton or for whch an algorthmc soluton s too complex to be found. There are multtudes of dfferent types of ANNs. Some of the more popular nclude the Multlayer erceptrons (MLs), whch are generally traned wth the backpropagaton algorthm [5]. Ths type of neural network conssts of multple layers and s known as a supervsed network because t requres a desred output n order to learn. The goal of ths type of network s to create a model that correctly

2 6 Krt rya Gupta and Madhu Jan: erformance Analyss of Cellular Rado System Usng Artfcal Neural Networks maps the nput to the output usng hstorcal data so that the model can then be used to produce the output when the desred output s unknown. A three-layer feedforward ANN wth sgmodal actvaton functons n the hdden layer and traned usng the backpropogaton algorthm, s able to approxmate an arbtrary nonlnear functon [6]. ANNs have been appled to the problem of traffc predcton, adaptve control of nonlnear traffc etc. [7, 8, 9,, ]. Researchers have used ANNs for bandwdth allocaton [2], admsson control [3, 4] and for computng the optmal number of channels to be allocated to varous users n GRS [5]. Several researchers have used ANNs and other soft computng technques for studyng channel assgnment problems n cellular networks [6, 7, 8, 9]. Some researchers have also used ANN for locaton detecton and predcton n cellular networks [2, 22]. In ths paper, we consder a cellular rado system wth a prortzed scheme n whch some channels are fxed exclusvely for the handoff calls. Also the reserved channels are subrated nto two channels of equal bandwdth for servng more handoff calls n case of heavy traffc. A feedforward ANN wth three layers s employed to compute the blockng probabltes of new and handoff calls. The backpropogaton algorthm s used for tranng the network. The rest of the paper s organzed as follows: In secton 2, the basc archtecture of an ANN s descrbed along wth the backpropogaton algorthm. The analytcal model for the cellular rado system s dscussed n secton 3. In secton 4, the ANN approach for computng the blockng probabltes of the cellular system, s dscussed. The results obtaned from the analytcal method and ANN are compared n secton 5 by takng the numercal llustratons. Fnally, the concluson s drawn n secton Archtecture of ANN ANNs are closely modeled on bologcal processes for nformaton processng, ncludng specfcally the nervous system, and the neuron. A mathematcal model of the neuron s depcted n Fgure. It shows n nputs wth assocated weghts v j (j=,2,,n) and the bas v. The output y can be expressed as n vjxj v () j= y = σ( + ) Fgure. Mathematcal Model of a Neuron. where σ(.) s a dfferentable functon known as the actvaton functon whch s selected dfferently n dfferent applcatons. Fgure 2 exemplfes a graphcal representaton of a threelayer ANN. The frst layer s known as the nput layer wth n number of nputs and the second layer s known as the hdden layer, wth L number of hdden-layer neurons. The thrd layer s known as the output layer wth m number of neurons. ANN wth multple layers are known as MLs. The computng power of MLs s sgnfcantly enhanced over the two-layer ANN whch conssts of only nput and output layers. The output of the three-layer ANN s gven by L n y = σ ( w σ ( v x + v ) + w ), =,2,..., m (2) 2 l lj j l l= j= Fgure 2. Three-Layer Artfcal Neural Network. where v lj s the weght for the j th nput to the l th neuron of the hdden layer and w l s the weght from the l th neuron of the hdden layer to the th neuron of the output layer. σ (.) s the actvaton functon for the hdden layer and σ 2 (.) s for the output layer. The ML and many other ANNs learn usng an algorthm called backpropagaton. Wth backpropagaton, the nput data s repeatedly presented to the neural network. Wth each presentaton, the output of the ANN s compared to the desred output and an error s computed. Ths error s then fed back (backpropagated) to the neural network and used to adjust the weghts such that the error decreases wth each teraton and the ANN gets closer and closer to producng the desred output. Ths process s known as tranng. The backpropagaton algorthm for a two layer ANN s descrbed below: Backpropagaton Algorthm Inputs: Number of nputs, n; Input pattern, X; Number of neurons n the hdden layer, L; Number of neurons n the output layer, m; Desred output pattern, Y; Actvaton functons σ and σ 2 ; Learnng rate, η; Number of epochs, NE; Error goal to be reached, ε. rocess:

3 Amercan Journal of Neural Networks and Applcatons 27; 3(): Step : Intalze E = and e = ; Step 2: resent the nput pattern X to the ANN; Step 3: Repeat Steps 4 to 8 untl E < ε or e > NE Step 4: Intalze weghts v lj and w l randomly; Step 5: Compute the outputs of the two layers as n z = σ ( v X ), l =,2,..., L and X = ; l lj j j= L y = σ ( w z ), =,2,..., m and z = ; 2 l l l= Step 6: Compute the sum-squared error as m E = ( Y y) 2 = 2 Step 7: Update the weghts n layers 2 and respectvely accordng to E wl = wl η ; =,2,..., m; l =,2,..., L; w l E vlj = vlj η ; l =,2,.., L; j =,2,... n; v lj Step 8: e = e + ; Output: Updated weghts for the two layers. 3. Analytcal Model for rortzed Scheme n Cellular Rado System We consder a cellular system wth a prortzed channel scheme n whch, a fxed number of channels are reserved exclusvely for the hand-off calls. In order to deal wth heavy traffc condtons, these reserved channels are also subrated.e. a reserved channel s dvded nto two channels of equal frequency. Jan and Rakhee [2] studed cellular system wth subratng. Two models are consdered wth fnte and nfnte subscrbers respectvely. Both models are dscussed later n ths secton. The arrval rates of all the calls are assumed to be osson and the servce tmes are dstrbuted exponentally. The mean call holdng tmes and call resdence tmes also follow exponental dstrbuton. Followng notatons are used for mathematcal formulaton of the analytcal model: M Number of subscrbers C Total number of channels n the cellular system r Number of channels reserved for handoff calls /µ Mean call-holdng tme /η Mean cell resdence tme of each portable λ ν Arrval rates of new calls λ η Arrval rates of handoff calls λ Arrval rate of calls; λ = λ ν + λ η Steady state probablty that there s no call n the system Steady state probablty that there are calls n the B n B h system Blockng probablty of new calls Blockng probablty of handoff calls 3.. Model wth Infnte Number of Subscrbers (ISM) In ths model, the number of subscrbers n the system s assumed to be fnte. The steady state probabltes are obtaned as follows: λ!( µ + η) = c r ( c r) λ λh!( µ + η) c r c r + c + r where s computed by usng normalzaton condton as = + c r c+ r c r ( c r) λ λ λh =!( µ + η ) = c r+!( µ + η) (3) (4) 3.2. Model wth Fnte Number of Subscrbers (FSM) In ths model, the number of subscrbers s taken as fnte,.e. M. The steady state probabltes are gven by the followng equatons: M λ ( µ + η) = M c r ( c r) λ λh ( µ + η) Where c r c r + c + r M M λ λ λ c r c+ r = + = ( µ + η ) = c r + ( µ + η) c r ( c r) h (5) (6) erformance Measures The blockng probabltes of new and handoff calls for both the models are calculated as and B n c+ r = (7) = c r Bh c + r = (8) 4. The ANN Approach for Computng Blockng robabltes Now, we descrbe the ANN model for computng the performance measures of the cellular system dscussed n the prevous secton. We consder a two-layer feedforward ANN wth L neurons n the hdden layer and one neuron n the output layer. The actvaton functons at the hdden layer and output layer are assumed to be sgmod and lnear

4 8 Krt rya Gupta and Madhu Jan: erformance Analyss of Cellular Rado System Usng Artfcal Neural Networks respectvely. The backpropagaton algorthm s employed for tranng the network. For studyng the effect of dfferent parameters on the performance measures of the analytcal models, ANNs wth dfferent combnatons of nput and output neurons are used, whch are descrbed n fgures 3-6. The ANNs descrbed n fgures 3a and 3b are used to study the effect of λ ν on B n and B h respectvely for both models where, λ ν s the nput neuron and B n and B h respectvely are the output neurons. In the ANNs shown n fgures 4a and 4b, C s the nput neuron and B n and B h are the outputs respectvely. The ANNs n fgures 5a and 5b have two nput neurons.e C and r, and the output neurons are B n or B h. For studyng the effect of M for FSM, the ANNs used have M as the nput neuron and B n and B h as the output neuron as demonstrated n fgures 6a and 6b. Fgure 4b. ANN Model for calculatng B h takng C as nput. Fgure 3a. ANN Model for calculatng B n takng λ ν as nput. Fgure 5a. ANN Model for calculatng B n takng C and r as nputs. Fgure 3b. ANN Model for calculatng Bh takng λν as nput. Fgure 5b. ANN Model for calculatng B h takng C and r as nputs. Fgure 4a. ANN Model for calculatng B n takng C as nput. Fgure 6a. ANN Model for calculatng B n for FSM takng M as nput.

5 Amercan Journal of Neural Networks and Applcatons 27; 3(): Fgure 6b. ANN Model for calculatng B h for FSM takng M as nput. 5. Numercal Experment In ths secton, we compare the analytcal results obtaned Table. ANN arameters. n secton 3 wth the ANN results by takng some numercal llustratons. Frstly, we determne the performance measures for the models ISM and FSM by usng the analytcal results. Then these results are valdated by usng the ANN models dscussed n secton 4. For llustraton, we assume C=3, r=2 and the arrval rate of handoff calls to be 2% of that of the new calls,.e. λ ɳ =2%λ υ. For ISM, µ s taken as.5 and ɳ s assumed to be.6. For FSM, we take µ=.5, ɳ =.6 and M=46. For all ANN models, the learnng rate (lr) s taken as.. Other ANN parameters for varous results are summarzed n table. For all ANN models, the backpropogaton algorthms are run on entum IV usng MATLAB 5.2. Fg. No. 7(a) 8(a) 9(a) (a) (a)-(d) 2(a) & 2(b) No. of Epochs No. of Epochs after whch SSE s calculated 2 5 No. of neurons n hdden layer (L) Error goal Fgures 7a and 8a exhbt the analytcal as well as ANN results for B n and B h of ISM respectvely by varyng λ υ Smlarly, B n and B h for FSM by varyng λ υ are shown n fgures 9a and a. Obvously, both B n and B h ncrease wth λ υ for both the models. The varaton of the sum-squared error wth the number of epochs for each computaton s demonstrated n fgures 7b-b correspondng to the fgures 7a-a. We notce that SSE decreases wth the ncrease n the number of epochs and fnally SSE reaches the requred error goal. The respectve error surface graphs are also shown n the fgures 7c-c. These graphs represent those values of the weghts and bases for the ANNs, whch gve the lowest error. In each of these graphs, we note that the error surface has a global mnmum at the center of the plot and the sde valleys lead to local mnma. Fgures 2a and 2b depct the effect of M on B n and B h respectvely for FSM by takng analytcal and ANN results as well. We notce the obvous result that both B n and B h ncrease wth M as expected. Fgure 7b. SSE vs. Epochs for Fg. 7a. Fgure 7a. Bn by varyng λn for ISM. Fgures a d dsplay the ANN results for B n and B h for ISM and FSM by varyng C and r both. We note that for both models ISM and FSM, B n decreases wth C and ncreases wth r. Also, B h decreases wth r and s almost constant wth C. These results are qute comparable wth the analytcal results. Fgure 7c. Error Surface Graph for Fg. 7a.

6 Krt rya Gupta and Madhu Jan: erformance Analyss of Cellular Rado System Usng Artfcal Neural Networks Fgure 8a. Bh by varyng λn for ISM. Fgure 9a. Bn by varyng λn for FSM. Fgure 8b. SSE vs. Epochs for Fg. 8(a). Fgure 9b. SSE vs. Epochs for Fg. 9(a). Fgure 8c. Error Surface Graph for Fg. 8(a). Fgure 9c. Error Surface Graph for Fg. 9(a).

7 Amercan Journal of Neural Networks and Applcatons 27; 3(): 5-3 Fgure a. Bh by varyng λn for FSM. Fgure a. Bn by varyng C and r for ISM. Fgure b. SSE vs. Epochs for Fg. (a). Fgure b. Bh by varyng C and r for ISM. Fgure c. Error Surface Graph for Fg. (a). Fg. c. Bn by varyng C and r for FSM.

8 2 Krt rya Gupta and Madhu Jan: erformance Analyss of Cellular Rado System Usng Artfcal Neural Networks cellular rado system. A prortzed channel scheme wth subratng has been consdered for the cellular system. The blockng probabltes of handoff and new calls have been determned by usng a three-layer feedforward neural network. The backpropagaton algorthm has been used for tranng the network. The numercal smulatons show that the results obtaned by ANNs are comparable wth the analytcal results. We also conclude that once the ANN s traned aganst a data set, t takes less computatonal tme than the conventonal methods for calculatng the requred results whch ndcate that ANNs provde an easy and fast soluton technque and are better than the conventonal methods. We have used ANNs for obtanng the performance measures of a cellular system. ANNs can be further used for takng handoff decsons for practcal moble cellular networks. Also, other soft computng technques vz. Genetc Algorthms and Neuro Fuzzy Systems can be explored for modelng the performance of cellular networks. Fgure d. Bh by varyng C and r for FSM. References [] Cchock, A. and Unbehauen, R. (993). Neural networks for optmzaton and sgnal processng, Wley, NY, USA. [2] Clarkson, T. G., Ng, C. K. and Guan, Y. (993). The pram: an adaptve VLSI chp, IEEE Trans. Neural Networks, Specal Issue on Neural Network Hardware, 4 (3), [3] Hopfeld, J. J. and Tank, D. W. (995). Neural computaton of decsons n optmzaton problems, Bol. Cybern., 52, Fgure 2a. Bn by varyng M for FSM. [4] Onyagha, C. G., Krasnq, X. and Clarkson, T. G. (June 996). robablstc RAM neural networks n an ATM multplexer n solvng engneerng problems wth neural networks, roc. Internatonal Conference Engneerng Applcatons of Neural Networks, [5] Hecht-Nelsem, R. (Jan. 989). Theory of back-propagaton neural networks, roc. IEEE Internatonal. Conf. Neural Networks, Washngton, USA,, [6] Hornk, K. (989). Multlayer feedforward networks are unversal approxmators, Neural Networks, 2, [7] Tarraf, A. A., Habb, I. W. and Saadaw, T. N. (993). Neural networks for ATM multmeda traffc predcton, roc. Internatonal. Workshop on Applcatons of Neural Networks to Telecommuncatons,, Fgure 2b. Bh by varyng M for FSM. We conclude from the above results that the results obtaned from the ANNs are qute accurate and are at par wth the analytcal results. 6. Concluson and Scope of Further Research In ths paper, we have nvestgated the potental of artfcal neural networks for analyzng the performance of a [8] Moh, W. M., Chen, M. J., Chu, N. M. and Lao, C. D. (995). Traffc predcton and dynamc bandwdth allocaton over ATM: a neural network approach, Comput. Commun., 8 (8), [9] Drossu, R., Lakshman, T. V., Obradovc, Z. and Raghavendra, C. (995). Sngle and multple frame vdeo traffc predcton usng neural network models, Computer Networks Archtectures and Applcatons, [] Edwards, T., Tansley, D. S. W., Frank, R. J. and Davey, N. (997): Traffc trends analyss usng neural networks, roc. Internatonal. Workshop on Applcatons of Neural Networks to Telecommuncatons,

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