O ZLLWWPd********************* QUEUE1 *********************;A, - (}*( ǠŠ-(r)(!WAITING LINES OCCUR IN ALMOST ANYR($SYSTEM FROM THE SUPERMARKET CHECKOUTr(COUNTER TO THE RUNWAY AT ANp,($AIRPORT. IN ANALYZING A WAITING LINEP(SITUATION, ON THE ONE HAND LONGp(WAITING LINES CAUSE LOSS OFo-(%BUSINESS EVEN IF ONLY DUE TO CUSTOMERQ(DISSATIFACTION WHILE INCREASINGo(SERVICE INCREASES COSTS.  IN THIS  B! (}(s*("WE ASSUME THERE IS A SINGLE SERVERT(%(E.G. CHECKOUT PERSON),WHO CAN SERVE,s(ON AVERAGE, A GIVEN NUMBER2t,($OF CUSTOMERS PER TIME. THE CUSTOMERSV(%ARRIVE RANDOMLY AT SOME AVERAGE RATE.t(THE WAITING LINE IS FIRST<q)(!COME FIRST SERVED. WE EXAMINE THER($PROBABILISTIC CHARACTERISTICS OF THEq(WAITING LINE OR QUEUE. THEFu(( AVERAGE SERVICE RATE IS THE MEANK(SERVICE RATE WHILE THE AVERAGEu(%ARRIVAL RATE IS THE MEAN ARRIVAL RATEP(Z B_t (}.( ONE REQUIREMENT IS THAT THE MEANX(%SERVICE RATE BE GREATER THAN THE MEANt(ARRIVAL RATE. OTHERWISEd++(#THE QUEUE WOULD GET INFINITELY LONGn(sr'(WHAT AVERAGE MEANS IS SOMETIMESK(LONGER SOMETIMES SHORTER BUT IFr("SOMEONE LOOKED AT THE QUEUE EVERY xn(( DAY FOR A MONTH THEN THE AVERAGEN(!(MEAN) QUEUE LENGTH FOR THE MONTHn(WOULD BE CLOSE TO WHAT WAS( COMPUTED (HERE("(PRESS RETURN TO BEGIN" (}J,($ŠǠŠΠҠҠG( J(B A 4(! ENTER MEAN ARRIVAL RATE OF UNITS8B BB A04(! ENTER MEAN SERVICE RATE OF UNITS8B B?)2(NO NEGATIVES ALLOWED5(? A L  .( SERVICE RATE MUST AT LEAST EQUAL?( ARRIVAL RATEB(L A K6-'+&,'6-#@'+$+&,,96-@'+&,C6-$'K6-',(6DD(AVERAGE LENGTH OF THE QUEUE = P:$B%?P,'B@DD(AVERAGE LENGTH OF THE SYSTEM = P:$B%?P,'BJEE( AVERAGE WAITING TIME ON QUEUE = P:$B%?P,'BTFF(!AVERAGE TIME WAITING IN SYSTEM = P:$B%?P,'B^DD(PROBABILITY SYSTEM IS BUSY = P:$B%?P,'Bh?( (.( SYSTEM TIME IS WAITING TIME PLUS?( SERVICE TIMEr%(!(PRESS RETURN FOR MORE%| (} At'+!(PRESS RETURN TO CONTINUE%((+$D:QUEUE1