Penguk ur an L ong-R un K iner j a Sist em A n- t r ian
6.6 Penguk ur an L ong-R un K iner j a Sist em A n- t r ian
² Main long-run measur es: – L: long-run t ime-aver age number of cust omers in t he syst em
–L Q : in t he queue – w: long-run t ime aver age t ime spent in t he syst em –w Q : in t he queue – ½: server ut ilizat ion
² Ot her measures:
– long-run proport ion of cust omers who are delayed in queue – longer t han t 0 t ime unit s
– long-run proport ion of cust omers t urned away – long-run proport ion of t ime t he wait ing line cont ains more t han
k 0 cust omers T im e-aver age num ber is syst em L
in‡uenced by init ial condit ions at t ime 0 and t he run lengt h T. – L(t ) : number of cust omers in syst em at t ime t .
–T i : t ot al t ime during [0; T] in which t he syst em cont ained exact ly
– As T !
1 ; L hat ! L (long-run t ime-average number is syst em) –L Q (t): number of cust omers wait ing in line –L Q : long-run t ime-average number wait ing in line. –L Q hat : observed t ime-average number of cust omers in line from
t ime 0 t o t ime T. –T Q
i : t ot al t ime during [0; T] in which exact ly icust omer s are wait - ing in line.
A ver age t im e spent in syst em p er cust om er , w similar t o L Conser vat ion equat ion ( L it t le’s L aw) L=¸!
– ¸ : long-run aver age arrival rat e; – ¸ hat : observed average arrival rat e; – ¸ hat ! ¸ as T !
1 and N ! 1
– The average number of cust omers in t he syst em at an arbit rary point in t ime = t he average number of arrivals per t ime unit * average t ime spent in t he syst em.
Ser ver ut ilizat ion proport ion of t ime t hat t he server is busy. Ser ver ut ilizat ion in G=G=1=1 =1
– arrival rat e: ¸ (cust omers per t ime unit ) – service rat e: ¹ (cust omers per t ime unit ) – server can be considered as a queuing syst em in it self, so L = ¸ !
can be applied. – What is ! for t he server subsyst em, i.e., average server t ime?
!=¹ ¡1 – L hat : observed average number is server subsyst em –L s : average number in server subsyst em or busy servers – In general, for a single-ser ver queue, L s = ½= ¸ ! = ¸ =¹
¤ also called t he o¤ered load; a measure of workload
– c ident ical ser vers in parallel: t he choice of server might be made at random.
– Maximum service r at e for is c ¹ : all servers are busy. –L s = ¸ E (S) = ¸ =¹ (average # of busy servers = c) – ½= L s =c = ¸ =c¹ (long-run average server ut ilizat ion = 1: pro-
por t ion of t ime an arbit rary server is busy in t he long run.) – For t he syst em t o be st able, c > ¸ =¹ – A st able queue can st ill have long lines. – Trade-o¤ analysis: server ut ilizat ion vs. cust omer sat isfact ion
Cost s in Queuing Pr oblem s – Cost can be associat ed wit h variousaspect s of t hequeueor servers.
– $x: cost per hour per cust omer; $y per hour while busy – Average cost per cust omer: $x¤ b w Q
– Average cost per hour: $x ¤ b L Q / hour – Cost for a set of c parallel servers
¤ Server is busy: $y ¤ c½ ¤ Server is idle: $y ¤ c(1 ¡ ½)
² Object ive: minimize t ot al cost s by varying above paramet ers (# of
servers, service rat e, syst em capacit y)