Data Networks Solutions


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SOLUTIONS MANUAL Second Edition Data Networks DIMITRI BERTSEKAS Massachusetts Institute a/Technology ROBERT GALLAGER Massachusetts Institute a/Technology II PRENTICE HALL. Englewood Cliffs. New Jersey 07632 ã © 1993 by PRENTICE-HALL, INC. A Paramount Communications Company Englewood Cliffs. New Jersey 07632 All rights reserved 10 9 8 7 6 5 4 3 ISBN 0-13-200924-2 Printed in the United States of America CHAPTER 3 SOLUTIONS 3.1 A customer that carries out the order (eats in the
  SOLUTIONSMANUAL SecondEdition Data Networks DIMITRI BERTSEKAS MassachusettsInstitutea/Technology ROBERT GALLAGER MassachusettsInstitutea/Technology II PRENTICEHALL. EnglewoodCliffs.NewJersey07632  ã 1993 by PRENTICE-HALL,INC.AParamountCommunicationsCompanyEnglewoodCliffs.NewJersey07632Allrightsreserved 109876543 ISBN 0-13-200924-2 Printed in theUnitedStates of America  CHAPTER3SOLUTIONS 3.1 Acustomerthatcarriesouttheorder(eatsin the restaurant)staysfor5mins (25mins). Thereforetheaveragecustomertime in thesystemisT=0.5*5 + 0.5*25= 15. By Little'sTheoremtheaveragenumber in thesystemisN= A*T = 5*15=75. 3.2 Werepresentthesystemasshown in the figure. Thenumber of fl1es in theentiresystem is exactly one at all times.Theaveragenumber in nodeiis AiRi and theaveragenumber in node3is'IolPl + A2P2. Thereforethethroughputpairs (AhA2) mustsatisfy (in addition to nonnegativity)theconstraint If the systemwereslightlydifferent and queueingwereallowed at node 3, while nodes1and2couldtransmitatwill,adifferentanalysiswouldapply.Thetransmissionbottleneckforthefiles of node 1impliesthat 1 A S- } R} Similarlyfornode2 we get that Node3 can work on onlyonefileatatime. If we lookatthefllereceivingserviceatnode3asasystemandletNbetheaveragenumberreceivingserviceatnode3, we conclude from Little'stheoremthat  and NS1 This impliesthat AIPI+A 2 P 2 S1 3.3 Working Machines ... -. R Machines . Waiting Repair Q Repairmen We representthesystem as showninthefigure. In particular,onceamachine breaks down, it goesintorepair if arepairperson is available at thetime,andotherwisewaitsinaqueue for arepaiIpersontobecomefree.Notethat if m= 1 thissystemisidentical to theone of Example3.7.Let A be thethroughput of thesystemandlet Q be theaveragetimeabrokendownmachinewaitsforarepairperson to become free. ApplyingLittle'stheorem to theentiresystem,weobtain A(R+Q+P) =N fromwhich A(R+P) SN (1) (2) Sincethenumber of machineswaitingrepair canbe atmost (N-m), theaveragewaitingtime AQ is at mosttheaveragetimetorepair(N-m)machines,whichis(N-m)P.Thus,fromEq. (1) weobtain A(R+ (N - m)P +P) ~ N ApplyingLittle'stheoremtotherepairpersons,weobtain APSm (3) (4)
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