Sapana P. Dubey. (Department of applied mathematics,piet, Nagpur,India) I. INTRODUCTION

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1 IOSR Joural of Mathematics (IOSR-JM) COMPETITION IN COMMUNICATION NETWORK: A GAME WITH PENALTY Sapaa P. Dubey (Departmet of applied mathematics,piet, Nagpur,Idia) ABSTRACT : We are proposig a ew etwork trasmissio model wherei we have a umber of users who wish to sed their throughput demad i the form of packets through oe or more liks with miimum cost or more efficietly tha the other users. This situatio itroduces the role of o-cooperative games i the commuicatio etwork. We itroduce three aspects which distiguish this model from other trasmissio etwork models studied elsewhere. These aspects are regardig itroductio of time (discrete) ad pealty to the users, ad cost of trasmissio to be a icreasig fuctio of time. Keywords: Covex, Nash equilibrium, No-cooperative game I. INTRODUCTION Routig i a commuicatio etwork is a game with imperfect iformatio where the complete kowledge of strategies of other users/players is impossible. I this game decisios are made simultaeously by all the users ad cost fuctios are kow to every player. There are few works usig game theoretical cocept i a commuicatio etwork. For example, i [1], A. Orda, R. Rom ad N. Shimki provide Nash equilibrium for the system of two ode multiple lik usig o-cooperative game. They have proved uiqueess of Nash Equilibrium Poit (NEP) uder reasoable covexity coditios. I this paper each user ca measure the load o the etwork liks ad chage their behavior based o the state of etwork. I. Sahi, M.A. Simaa [2] have derived a optimal flow ad routig cotrol policy for two ode parallel lik commuicatio etworks with multiple competig users. I this paper etwork cosists of several parallel liks ad each user uses differet preferece costat for differet liks. The review paper by F.N. Pavlidou ad G. Koltsidas [3] presets differet routig models that use game theoretical methodologies for covetioal ad wireless etworks as well. Time depedet behavior has a impact o the performace of telecommuicatio models ad queuig theory is also used for commuicatio perspective by Messey [4]. Bottleeck routig games ad Nash equilibrium is discussed i [5] for splittable ad usplittable flow. I [6] authors cosidered that the cost fuctio for the lik is polyomial ad they have established the uiqueess of Nash equilibrium. This work preseted here deals with routig of data packets i a commuicatio etwork. The players/users come to the "game" with the kowledge of the umber of packets they wish to sed through oe or more etwork lik over the m chaces/shots. As usual each lik at a give chace/shot has a fiite capacity to carry the packets. I all other models, the users are dissuaded from sedig umber of packets exceedig capacity of a lik by makig the cost ifiity for such a situatio ad i such a situatio there is a trasmissio failure i the sese o oe's packets are set through the lik. I our model, the cost of trasmissio remais fiite eve if the sum of packets wished to be set by the users through a give lik i a give slot/shot exceeds the specified capacity of the lik. However, we cosider a umber of scearios to deal with such a situatio. I this simple model, the game becomes competitive as the users would wat to sed their packets i earlier ad few shots possible. The game would be very competitive ad iterestig if the total umber of packets wished to be set by all the players together over all the shots exceeds the total capacity of all lik summed for all the shots. Iteratioal Coferece o Advaces i Egieerig & Techology 2014 (ICAET-2014) 1 Page

2 IOSR Joural of Mathematics (IOSR-JM) The paper comprises seve sectios. I sectio 2, we preset a mathematical modelig for commuicatio etwork. I sectio 3, Routig Scheme ad the cost fuctio is discussed showig explicitly the relatio of cost fuctio with time ad umber of packets, beig trasmitted. I sectios 4, Exteded cost fuctio is itroduced for packets more tha capacity of the lik. I sectio 5, we preset a umber of scearios to deal with the situatio of violatig the capacity of lik. I sectio 6, we prove some theorems to show existece of Nash equilibrium poit i this game ad i sectio 7 we provide some cocludig remarks. II. MATHEMATICAL MODELING I the preset commuicatio etwork model, we cosider p umber of users sharig k parallel liks coectig a commo source ode to a commo destiatio ode. We assume that the etwork/liks are available to the users over a discrete rage of time kow as time slots. We assume that there are m time slots i a sigle cycle. Let L = {1, 2, 3,. k} be the set of parallel liks, N = {1, 2, 3, p} be the set of users ad S={1,2,3,..m} be the time slots. We assume that users are ratioal ad selfish for this competitive game. Each user N has throughput demad D () which he/she wats to ship from source to destiatio. A user seds his/her throughput demad i the form of data packets through the commuicatio liks ad is able to decide at ay time how the data packets will be trasmitted ad what fractio of throughput demad should be set at that time through each lik. We assume that each lik is available for all the users after a uique iterval of time. Capacity λ l of each lik is fixed. Let P l t deotes the umber of packets trasmitted through lik l at time t by user. P l t satisfies the followig coditios : P 1 : P l t 0 (No egative costrait) P 2 : P l t λ l (Capacity costrait for each lik l) P 3 : D () = t P l t (Demad costrait for each user ) Turig our attetio to a lik l L, let P l (t) be the total trasmitted packets o that lik at time t ad s l be the speed of data packets for lik l, which remais fixed i time. We visualize this problem as a ocooperative game i which each user wats to miimize his/her cost. Let C (), deote the cost bore by th user.the followig geeral assumptios o the cost fuctio C () of each user N will be imposed throughout the paper. A 1 : C () = C(P l, t) t l It is a sum of cost of routed packets over each lik l L by user N. A 2 : Cost fuctio is o- liear ad o- egative fuctio. A 3 : Cost fuctio is strictly icreasig with o. of packets i ad time t.we shall cosider the cost of trasmittig packets (i i umber) at time t to be of the type C pi = f t. (i, i ) Additioal assumptios cocerig the time fuctio f t are T 1 : Each user get m umber of istaces called discrete time slots Iteratioal Coferece o Advaces i Egieerig & Techology 2014 (ICAET-2014) 2 Page

3 IOSR Joural of Mathematics (IOSR-JM) T 2 : Flow of packets is cotiuous which implies that there is o cogestio i the system. T 3 : Users ca trasmit more tha oe packets o the lik at the same time slot. They must obey the capacity costrait (P 2 ) ad o-egative costrait (P 1 ). III. ROUTING SCHEME AND COST FUNCTION We describe routig scheme i a sigle lik commuicatio etwork for users. Users ca route oe or more packets o this lik at time slot t. (fig. 3.1) fig. 3.1: Available space at t = t 1 The expected cost C for the user at t th time slot depeds o the umber of packets i to be routed by user ad umber of packets i by other users. C t () = f t. (i, i ) = t ( ei 1) λ l +1 i+i whe i + i λ l Where t S. Therefore total cost for user to sed i packets through lik l is C () = t f t. ϕ i, i i The time fuctio f t is the square root of time slot (or istace). IV. EXTENDED COST FUNCTION The cost fuctio is preseted as lik availability ad time based formulatio. The capacity costrait (P 2 ) for each lik l is provided that the total umber of packets by all users o the lik l should be less tha or equal to lik capacity. I Wardop s first priciple (1952) states that every user seeks miimum travel cost uder their idividual prospectio. The followig facts may arise: 1. The game is cosidered as game with imperfect iformatio ad o-cooperative, therefore the complete kowledge of strategies of other players is impossible. I this game decisios are made simultaeously by all the users ad hece without iteractio it is very difficult to maitai the capacity costrait i.e. i + i λ l. 2. Whe i + i > λ l i.e. umber of packets shipped o the lik exceeds lik capacity the effect o the cost fuctio is ot defied. 3. If umber of packets exceeds the lik capacity the there may be a possibility that some packets will be trasmitted or oe of them will be trasmitted. 4. Sice decisios are made simultaeously by all the users, it is difficult to decide resposible user for violatig capacity costrait as well as to decide who has to pay more? 5. Will game be over or ot for that resposible user? If game is over for the user the what about the packets of that user? To cosider all above poits there is eed to exted the cost fuctio for user to trasmit i packets at the time slot t. Now the exteded cost fuctio is Iteratioal Coferece o Advaces i Egieerig & Techology 2014 (ICAET-2014) 3 Page

4 IOSR Joural of Mathematics (IOSR-JM) C t () = f t. (i, i ) = λ l + 1 i + i whe i + i λ l whe i + i > λ l 0 whe i = 0 (1) Also we cosider that game will ot be over util all the packets are trasmitted or fixed umber of time slot is over. Ad hece the efficiecy of user is measured by the term performace which depeds o total cost, umber of packets ad time slots. i.e. Performace = Total Cost = C( ) = t i f t.ϕ i,i No.of trasmitted packets ( No.of Time Slots ) ( t i ) m P t l t m Efficiet user = mi Performace user1, user2 V. PENALTIES To avoid the violatio of capacity costrait we itroduce the term pealty for the users. Now we will discuss differet types of pealties. 5.1 Type A: Trasmissio failure for all user (whe i + i > λ l ) 1. No packet will be trasmitted at that time slot. 2. Cost for user ca be expressed as C t () = t. (e i 1) Pealty costat x will give the umber of packets which should be routed i the ext time slot by the user with maximum packet i the trasmissio failure situatio. i.e. x = max i, i m 1 EXAMPLE: Let the capacity of lik l is 3 (i.e. λ l = 3) ad user u 1 ad u 2 wats to trasmit 4 packets i 4 time slots. Trasmissio may be fail at ay time slot. I the Trasmissio failure situatio pealty x = max i, i m 1 will be activated for the ext time slot. The user is more efficiet if he/she will trasmit maximum packets (approximately 4 packets) i less umber of time slots. The efficiecy of user is measured by the term performace i the followig games. Where Performace = Total Cost = t i f t.ϕ i,i No.of trasmitted packets ( No.of Time Slots ) P t l t m Efficiet user = mi (Performace user1, user2 ) GAME -1 User 1 User 2 Situatio Time Slot U1(No. of U2 (No. of Cost(U1) packets) packets) Cost(U2) No Trasmissio No Trasmissio Trasmitted Trasmitted Total Iteratioal Coferece o Advaces i Egieerig & Techology 2014 (ICAET-2014) 4 Page

5 IOSR Joural of Mathematics (IOSR-JM) performace Efficiet user = User Type B: Trasmissio failure oly for user with maximum packets (whe i + i > λ l ) I type B, we cosider that data packets will ot be trasmitted for the user with maximum umber of packets, but other user s data packets will be trasmitted. 1. No packet will be trasmitted for user with maximum packets at that time slot. P l t = max(p l 1 t, P l 2 t ) or i = max (i 1, i 2, i 3,.. ) 2. Cost for user ca be expressed as C t () = t. (e i 1) C t ( 1 ) = t (ei 1 1) λ l +1 i 1 +i For ay other user where i o. of packets except i which is ot beig trasmitted. Therefore total cost for user to sed i packets through lik l is C = t i f t. ϕ i, i Example : Cosider the same example i which coditio is applied accordig to type B Game 1( Pealty Type B) User 1 User 2 Time Slot U1 Cost(U1) Situatio U2 Cost(U2) Situatio No Tras Trasmitted Trasmitted Trasmitted Trasmitted Trasmitted Total performace Efficiet user = mi Performace user1, user2 = user Type C: Adjustmet of packets by retrasmissio o the free lik (whe i + i > λ l ) We ca cosider a smart system which ca retrasmit data packets i aother free lik if umber of packets exceeds the lik capacity. This smart system maitais a record of status (st) of all liks available i the etwork. If a lik l has o free space the its status is set as st =1 otherwise st = 0. Whe i + i > λ l system search aother lik with status st = 0, ad fid i = max (i 1, i 2, i 3,.. ). It divides i ito two parts i ad i " such that i + i = λ l ad i " will be retrasmitted i the lik with st = 0. Iteratioal Coferece o Advaces i Egieerig & Techology 2014 (ICAET-2014) 5 Page

6 IOSR Joural of Mathematics (IOSR-JM) I this case cost for th user with i data packets C t () = t(e i 1) + 2. t (e i" 1) λ l 1 +1 i +i λ l 2 +1 i" +i VI. EXISTENCE OF NASH EQUILIBRIUM The followig theorems establish the existece of Nash Equilibrium Poit for the commuicatio etwork. Theorem 1: I a commuicatio etwork cost fuctio C t () : I I R + for each time slot, defied as (1) is covex. Proof : To prove that cost fuctio C t () is covex we will use followig theorem (by [8]) A fuctio f(x) is covex if the Hessia matrix H X = 2 f X x i x j defiite, the fuctio f(x) will be strictly covex. By equatio (1) exteded cost fuctio ca be expressed as is positive semidefiite. If H(X) is positive C t () = f t. (i, i ) = λ l + 1 (i + i whe i + i λ ) l whe i + i > λ l 0 whe i = 0 For a fixed time slot, f t is costat, therefore C t () (for simplicity cosider C t () = C) will be a fuctio of two variables i ad i, ad Hessia Matrix for C is H C = i 2 i i i i i 2 Case I: Whe i + i λ l t(e i 1) C = λ l + 1 (i + i ) Let t = K ad λ l + 1 i + i = M == K i 2 ei , M M 2 M 3 M 3 = 2K ei 1, i 2 M 3 i i = K ei 1 M M 3 2 M 3 Now Determiat of Hessia Matrix H(C) = K 2 e i M 4 [ei 2] > 0 ( e 1 = ) Case II: Whe i + i > λ l C = = K (e i 1) = i 2 K ei ad 2 C = 2 C = 0 i 2 i i Case III: Whe i = 0 C = 0 Therefore H(C) = 0 which is o-egative. Iteratioal Coferece o Advaces i Egieerig & Techology 2014 (ICAET-2014) 6 Page

7 IOSR Joural of Mathematics (IOSR-JM) = 2 C = 2 C = 0 i 2 i 2 i i Therefore H(C) = 0 which is o-egative. Sice H(C) is positive defiite i all cases therefore the fuctio C t () will be strictly covex for each time slot. Theorem 2: Cost fuctio C t () is cotiuous i each time slot. Proof : We will use ε δ defiitio to prove that C t () : I I R + is cotiuous i each time slot. Let (i o, i o ) is ay poit i the domai of C () t = C(let) such that i i o < δ ad i i o < δ where δ > 0, δ > 0 Now C(i, i ) C(i o, i K o ) = (e i 1) K (e io 1) λ l +1 (i+i ) λ l +1 (i o +i o ) K (e i 1) K (e i o 1) = K e i e i o Where ε = K eλl δ > 0. Hece C is cotiuous i each iterval. K eλ l i i o < K eλl δ < ε sice K is positive iteger. VII. CONCLUSION I this work, we attempted to preset mathematical modelig of trasmissio i a commuicatio etwork, usig game theoretical cocept with multiple chaces available to users. The cost fuctio ivolves the umber of packets to be routed ad time variables, i a o-liear fashio. Pealty to the users are also itroduced for smoothig the game makig the cost fiite. Each user is give the flexibility to route their data at differet time slots. The examples also demostrate the cost fuctio whe packets sed by the users through a lik at a give shot exceeds the capacity of the lik. Despite the results accomplished so far, there is space for more detailed ivestigatio for multiuser; complex etwork with o-symmetrical liks (i.e. liks with differet speed). Furthermore, differet demads ad differet source ad destiatio seem to play a critical role i this packet trasmissio that has ot bee ivestigated i detail yet. REFERENCES [1] A.Orda, R. Rom, N. Shimki, Competitive Routig i Multiuser Commuicatio Networks,IEEE/ACM tras. Networkig 1(1993) [2] I. Sahi, M. A. Simaa, A flow ad routig cotrol policy for commuicatio etwork with multiple competitive users, Joural of Frakli Istitute 343(2006) [3] F.N. Pavlidou, G. Koltsidas, Game theory for routig modelig i commuicatio etworks A survey, Joural of commuicatio ad etworks Vol. 10 N0. 3 Sept [4] W. A. Massey, The aalysis of Queues with time varyig rates for telecommuicatio models, Telecommuicatio System 21 : 2-4, ,2002. [5] R.Baer, A. Orda Bottleeck Routig Games i Commuicatio Networks [6] E. Altma, T. Basar, T. Jimeez, N. Shimki, Competitive Routig i Networks with polyomial Cost,IEEE Trasactios o Automatic Cotrol, Vol.47, No.! Jauary 2002, pg 92 [7] E. Altma, L.Wyter, Equilibrium, Games ad Pricig i Trasportatio ad Telecommuicatio Networks, Networks ad Spatial Ecoomics,4 (2004) 7-21 [8] Sigiresu S. Rao Egieerig Optimizatio: Theory ad Practice, Fourth Editio Iteratioal Coferece o Advaces i Egieerig & Techology 2014 (ICAET-2014) 7 Page