Question

Consider a birth/death process with . - 2 customers per time unit, n = 1, 2, ..., 10 = 3 arrivals per time unit, 24 = 2.12 =1, and 2n=0, n = 3, 4, ... (note that, when there are 3 customers in the systems, the arrival rate is zero). (a) (5 points) Display the transition rate diagram, including the transition rates. (b) ) Calculate the steady state probabilities. (©) Calculate L, LG, W, W, (d) (5 points) Use Kendall notation to define this queueing model and its parameter values. (Hint: note that the arrival rate decreases with the number of customers in the queueing system; the number of customers in the input source depends on the number of customers inside the queueing systems; note also that, if the arrival rate eventually becomes zero, it is because there are no more customers in the input source)

Answer 1

Solutions I heory of Birth death process: (2n terms óf Queueing process) The state of single queue can be described by a birth death process which describe the arrivals and I departures from the queue, along with the number of jobs 'l ullstomers currently in the system. An arrival increases the number of jobs by 1 and a departure decreases k By 12 BUYEMDIE DATOEST Teacher's Signature –

with s general_transition transition rate is .. Doa , rate diagram below TK-2 741 - My My pre - The value in the circles represent the state of the birth death process, K. The system transitions between Values of k by birth and deaths which occus at rates Veiven by various value of ai Carrival rates) and efi (departures rates)... The Steady state equations for birth and death process are as follows (Pon denotes steady state probability to be in State n. ♡ : 12.00 nolai_ = 1 I Pn = Pot: Po & ll llit nro : 4 .1-0. where pozi - 1 + 8 F ni n=1 i-o llit and Pn = An-12.9-2.... no e Un lnt... lls. NOW: We described the 804mula :-) 0 L AW ® La = a hq - Siulangide 1 Teacher's Signature

W W = Wq + 1 9 Lq = . p² 2 i-p 2 p = 2 cis number of server in all . Parallel. where, ai mean crate of arrival. Mi meam service rate ... pi portion of time server is busyt L i mean number of customers in Systent Lqe mean: number of customers in queue, Wi mean waiting time in system Wq it mean waiting time in queue. Now we deal with given question.

given that in birth death process lapas Customenis polime (4t ,1)=1,2, 20=3 avuvals per time unit, 2,- 2,12 1, 112=0, 93,4%..... We doow transilion guate diagram ... including the transilim) gates' Pi (1=0,1,...) and li (**), 2, ...). as below? -20-3 2,221,3 2302=()__amar -0 4,= 2 M = 2 2 lls: 2 Un=2. Dg2 Teacher's Signature

PAGE NO.: y Now We Calculate ._Steady state There are total n state. To we calculate n a probabilities to Š Pn . since 2n-1 An-2 nano no P. ln un-i.... ll . An-t = an- 2 = c... = 72 = 0 . o. Pn = Pozi But NE) 1 0 Ö (: :2; = 0 , 16 13,4... } Here also + 11 = ... Posiſ les probability & probability of of oth steady State - 0. oth steady state = 1 I Now we calculate. n, l. l, L, Lq W, Wq ii 20+ a, tone tanins || - 3+2+1 + Ontorn.+0 - 6+0 . . but here (So we 23=24 = h an Fo. take n=3). n - Fluisna 279rosa Teacher's Signature -

ON SIDATI M = Motel tonnotlin = 2+2+..+262 times) - 20 - p = : a Cu C = 3 /3 customers are in System ) = 2 3x2 = 0.33 = 0 Lo= p lp II . = 10.33) 1 - (0.33 = 0.1089 Vi 0.67 = 0.1625 ni Lq = 0,1625 Wq = La == 0.1625 = 0.08125 - - Suisnpie 2791589 Teacher's Signature -

I W = Wq + ! . - 30.08125 + 1 = 0.08125 + 0,5 => W = 0.58125 - L · _ => aw. = 2 x 0.58125 1.1625 L = 1.1625 Now we define the's queueing model using T kendall notation. - Single queueing nodes are usually described I using Kendall's notation in the form. Al 5c where A describes the distribution of durations between each arrival to the quilleb, s the distribution of service times. for jobs and a the number of servers at the node. → It has three parameters AS and c. for this question . A is described as To = 3, 9g = 2, 22 = 1, 71320, - an-ro s is C=3 mnie -- XP- described as een=2. (N=1,2,....) (3 customeres are in service) - X Teacher's Sianature" 00152: