Queueing Theory (Part 3)
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1 Queueig Theory art 3 M/M/ Queueig Sytem with Variatio M/M/, M/M///K, M/M//// Queueig Theory-
2 M/M/ Queueig Sytem We defie λ mea arrival rate mea ervice rate umber of erver > ρ λ / utilizatio ratio We require λ <, that i ρ < i order to have a teady tate Rate Diagram λ λ λ λ λ λ λ λ λ Queueig Theory-2
3 M/M/ Queueig Sytem Steady-State robabilitie / / + / ad C Ue Birth Death rocee Rate I Rate Out Coefficiet are eay to remember if you thik of rate diagram Example: 3 λ λ λ 2 λ 3 4 λ where C +, +,, $ * + ' % & * + ' % &,, 2,..., +, + 2, C C C 2 2 & % $ ' C 3 & % $ ' C 4 & % $ ' C 5 & % $ ' C & % $ ' Queueig Theory-3 3 % & $ 3' % & $ ' 2 3
4 Queueig Theory-4 M/M/ Queueig Sytem L, L q, W, W q 2 2 / + L q $ % & ' ' * * +, > /.. / / / e e t t t How to fid L? W? W q? Ue L q to fid W q L q λw q : W q L q /λ Ue W q to fid W: W W q + / Ue LλW to fid L i term of L q : L λw λw q + / λl q /λ + / L q + λ/ t q e t $ % & ' > * If λ / the thi term i t
5 M/M/ Example: A Better ER A before, we have Average arrival rate patiet every ½ hour λ 2 patiet per hour Average ervice time 2 miute to treat each patiet 3 patiet per hour ow we have 2 doctor 2 Utilizatio ρ λ/2 2/6 /3 Before, ρ2/3 Queueig Theory-5
6 I teady tate, what i the M/M/ Example: ER Quetio. probability that both doctor are idle? probability that exactly oe doctor i idle? 2. probability that there are patiet? 3. expected umber of patiet i the ER? Queueig Theory-6
7 I teady tate, what i the M/M/ Example: ER Quetio. probability that both doctor are idle? & % $ ' & & % % $ ' $ ' probability that exactly oe doctor i idle? probability that there are patiet?, &. %. $ ' % 2&. $ 3' -. &.% $ '. 2 * % & / $ 3' 2 3. expected umber of patiet i the ER? 2 % & 2 $ 2' * *2 2 % & $ 3' if < 2 if L q L W L q + Queueig Theory-7
8 M/M/ Example: ER Quetio I teady tate, what i the 4. expected umber of patiet waitig for a doctor? 5. expected time i the ER? 6. expected waitig time? 7. probability that there are at leat two patiet waitig i queue? probability that a patiet wait more tha 3 miute? Queueig Theory-8
9 M/M/ Example: ER Quetio I teady tate, what i the 4. expected umber of patiet waitig for a doctor? L q 2 2/3 2 3 $ 2 2 / expected time i the ER? W L/λ 3/4/2 3/8 hour 22.5 miute 6. expected waitig time? W q L q /λ /2/2 /24 hour 2.5 miute 7. probability that there are at leat two patiet waitig i queue? 4 patiet i ytem 2 3 ½ - /3 /9 / probability that a patiet wait more tha 3 miute? 2 $ q > t e ' q > 2 hour q > 3mi % & % t 2 ' * e t & 3 6 e4 t * +.22 Queueig Theory-9
10 erformace Meauremet 2 ρ 2/3 /3 L 2 3/4 Lq 4/3 /2 W hr 3/8 hr Wq 2/3 hr /24 hr at leat two patiet waitig i queue a patiet wait more tha 3 miute Queueig Theory-
11 Travel Agecy Example Suppoe cutomer arrive at a travel agecy accordig to a oio iput proce ad ervice time have a expoetial ditributio We are give λ./miute, that i, cutomer every miute.8/miute, that i, 8 cutomer every miute If there wa oly oe erver, what would happe? λ/ > Cutomer would balk at log lie ever reach teady tate - loe cutomer - go out of buie? How may erver would you recommed? Calculate, L q ad W q for 2, 3, ad 4 Queueig Theory-
12 ω>t ω q >t Queueig Theory-2
13 ω>t ω q >t Queueig Theory-3
14 ω>t ω q >t Queueig Theory-4
15 Sigle Queue v. Multiple Queue Would you ever wat to keep eparate queue for eparate erver? Sigle queue v. Multiple queue Queueig Theory-5
16 Bak Example Suppoe we have two teller at a bak Compare the igle erver ad multiple erver model Aume λ 2, 3, L L q W W q ρ λ/2 / λ`/ λ//3 /3 Queueig Theory-6
17 Suppoe we ow have 3 teller Agai, compare the two model Bak Example Cotiued M/M/3 Three M/M/ queue λ2, 3 λ λ/3 2/3, 3 ρλ/ 2/9 M/M/: ρλ /3 2/9 ρ i the ame L.676 L.286 3L.858 L q.9 L q.63 3L q.89 W.338 W.429 W q.5 W q Queueig Theory-7
18 M/M///K Queueig Model Fiite Queue Variatio of M/M/ ow uppoe the ytem ha a maximum capacity, K We will till coider erver Aumig K, the maximum queue capacity i K Some applicatio for thi model: Truk lie for phoe call ceter Warehoue with limited torage arkig garage Draw the rate diagram for thi problem: λ λ λ λ λ λ λ - + K 2 - Queueig Theory-8
19 λ M/M///K Queueig Model Fiite Queue Variatio of M/M/ λ λ λ λ λ K Balace equatio: Rate I Rate Out State : λ State : λ λ+ State 2: λ λ+2 2 State 3: λ λ+3 3 State K-: λ K K λ+3 K- State K: λ K- 3 K C C C C $ ' $ C 4 & ' & % 33 % $ ' $ C & % 33 * ' & % C K K C C for + + K Queueig Theory-9
20 M/M///K Queueig Model Fiite Queue Variatio of M/M/ Solvig the balace equatio, we get the followig teady tate probabilitie: for, 2,..., + / + / K $ + for, +,...,K > K Verify that thee equatio match thoe give i the text for the igle erver cae M/M///K Queueig Theory-2
21 M/M///K Queueig Model Fiite Queue Variatio of M/M/ L q / [' 2 ' K ' ' K ' K ' ' ], where / L ' * + L q & + $ ' % ' * To fid W ad W q : Although L λw ad L q λw q becaue λ i ot equal for all, L W ad L q Wq $ where K Alo, becaue there i a fiite umber of tate, the teady tate equatio do hold, eve if ρ> Queueig Theory-2
22 M/M//// Queueig Model Fiite Callig opulatio Variatio of M/M/ ow uppoe the callig populatio i fiite, We will till coider erver Aumig, the maximum umber i the queue capacity i, o K doe ot affect aythig If i the etire populatio, the the maximum umber i ytem i. Aume K ad Applicatio for thi model: Machie replacemet Draw the rate diagram for thi problem: λ -λ --λ -λ -+λ 2 + λ Queueig Theory-22
23 M/M//// Queueig Model Fiite Callig opulatio Variatio of M/M/ λ -λ -2λ -3λ -4λ λ Balace equatio: Rate I Rate Out 3 3 State : λ è λ/ State : λ λ+ è 2 /2λ/ -λ/ C C & % $ ' C 2 2 C 3 3 & % $ ' 2 2 & % $ ' 3 Queueig Theory-23
24 Queueig Theory-24 M/M//// Reult $ % & ' + $ % & ' $ > % % & ' * +, - & ' * +, - - for for,,..., for q L $ % & ' + + ' ' q L L
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