Written while studying the course Advanced Computer Networks: Queuing theory Queueing theory is the mathematical study of waiting lines, or queues.[1] A queueing model is constructed so that queue lengths and waiting time can be predicted.[1] Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service.

1. ASSMAteDate
Page L
INTRODUCTLON:AUELOINGTH EORY
S
7oY Smins 0 7x5fo.3 X 20=
hoomvh ya ekpect to Wait on
475
3oY. 20mins
avesaq
2
Ctoettng tonmeLo
2-S
total tme 7xs + 20* 3
5 60 S mins
avg.watingtme 3sx 60 20/
7.23
0.8 5 min 2x30+_8xs
.22 mins 60
avg tattHngHme 05 4 030 ,
AK erLang Caoa)
SHudieeand ormalized queing theorybor tale phone-
networKS robbic_enggenagCirpor take-oh londing

2. alassMAte
Data
13 %20t
Paga
PROBABLITN THEORy
events probabilt Sample Space
F p
F C 2 P Fg2
i s discrete F 2**
ih s centinuoLK F 2
AXIOMS OF PROBABILITY
pCa):L
E0E E,EeF
hen pCE) + p (E p CEvE)
CoROLLARIES
iACB
p(A) p(B)
Au (en A)
Since pC)o A0CE

3. elAsSmAte
Date 13 82r2D
Page
2 CEUE p ) p(E) ENE)
pCEUE) E
pE v (EaE))
pCE)p (E,E EnE2
pE)p(CE,OE) CEVE.))
EOE (EE
NE)p)-pE, 0E
PCE DE)pCE) (SE)
p (E p(E)-plEOE)
CoNDITLDNALPROBABILTY
CABD pr bability of evet A given B as ocured
A BeF
Q tS dkcrete
Sat
auu outcomes qually KalyaB
p (AIB) NCANB
N)AnS

4. classmate
Data138.2020
Paga
PROBABILITY MASS FONCILON PME
C)=
HWrOll_a_dice. wtcaj
X (Mi,n)>t na-
(XESu, 123 pi(5,) L6,5, C,O
36
ContinUOUS_random vaCiahle
noX
S
l i m
Ux
probabt lity alens'ty fpnethian-
CPDE)
cmulative distribution_Aunction (CDEE
E() Cxs) E Co
FxC-s)-o

5. edasSmate
Date 3,3-22
( Page
ECa E (X)
ACBCA
eR ()E O1 asSupp.ort opX-
auSS distrilbutoy
potssion distyibutto)
C-
27
O Standavd deulato).
O T tS Catlednormaldietrdbvtion

6. elassmate
Data'17:8:202p
Paga -
ExPECTATION
HI0. Aindout. binomial distribuhon-g0cmetrit distvebution
prove that E Cantb): af b
VE ax+b
y Ccw) a XCw +o
RECAP
probabilty axioms
derivedeD ¢mportant resolks
Condiionok probabilitp
random v artableS Cdiscret& ContinuoUs)
pm Pdg, cdt-
Co):1
EL,E e'E
pE AUE pCE tp(E,)
CALB)p (ADB)ipB)2o
PCB)

7. elAsSMAte
Date 72.2026,
Page
qau sslan diStributon
2
2 t 2
V2a
M0anmean T Standard oleviet'on0here2,
4uo t s ts called Dormal ouktrtbu
ExPECTATION OF RVV
X
hime3. b manuheadl_we ekpect)toss a_Coin
N
HTH..H T 2"Outorn
R x o,L,2,.nt
( ) =_# heads_dn wto
Semple
Kexperiments lxp erimevt S sSingcan n Hme
14
Knz hmes

8. JasSMe
vnte (7.22020
ECx) 0.K t_K2 4 n.Kn+
Samp Ov9
im 2 n.m kne6 im_K
Ko K
im
um K
K
pX-o))
m p (X=(-) C - n
K
erpecteion
Ex pX-)
neX(en) hor DRV
Xw)
Ex. da pectation -
dCRV
HO ind out binomial distribution
gcomctri distbtHm
Provethat E CoX+b) aEx+b
= aX+b
Nw X(w)+6
(l YC)
C a X()+b=y)

9. elassMAte
Date 18 202
Page
bernoull RV
Cec toss pCH)p
X
HX1
1-p
Prob
ELpA O. Ct-p):p. y S.143 -2e
V 5x+3 TY: S0 +33.
Ey 2p +3(1-)
Eis imear opArator,
Spt3
binomtal RV Chp.)
Soces ces n thdepeolent bernoili tralk
Y *hbernoulle trtal s Succea
O_eihertote
EY2EX 2P.p

10. alnssmate
Data 17-32020
Paga 5
poiss.on RV
X o2
px)
K!
Ex2 Ke k k-1)
e
VARIAN CE
much Spread he distribunoo valves Qvenow Prove
ECX-EX) E-[Ex1
M St momenk
Mean Cu alled nsret
mome RV
moment generahng
nctionsSataAE tS Called asthe Se ond
momen
E s taued he ih maman_

11. lAssmate
Date 17:20
Page 6-
MEM ORYLESSRV
expenantial RV )2e
O-0
Cc Ey)1
-
eA
p IB AOB)PCB)
P(x>t
P(x)to)
p(AC) - pCA)_ eato
- ( - E
- C1-s)p(t t.)
Cxt|X4 pX t-t
markov prepa

12. goma pay eorning elassate
phisics engine Dete 17.3.2t
Poge
X.ime atcwhith_ (S packet arYivel
0 0 t los
t 1oSec
ooesn_matfe
momory e.
mar kov
serversS2ved
savorage cr xpected packet costcmersrevesh d
2rvers
Aima Spas
the uHe
L -l(-
avG or ixpeCkEd ) theSustern
LtLs

13. clsSMAte
Date 17.4-2ta
Page
averoge rate o arrival
moAE)
Aron t t+At
pdhe amval At o (at)
o_ptmore thon ene armival
V4
IN avrivals in non-overlapping hterval ara
cndepent oeach other
Stata n U fime t npackek _Qrmved
H=prob. that thare om k arnvak ti time. b_
4 (-2At)
A t )
EAE)- )_ AQ--A
r K-0
+At -2Ae)

14. lassmate
Date17:8.2020
Page
9CtHAL t ) a)
n da,) -2a)im
dt
, )
Co) C
C ) e
im
Ao
da)
dt
daCt 29,ce)29,4)_
) (a)
k
(mepvyle

15. elSSMAt
Date 2-8.»a
Page
UE UIN THEORy
RECAP
memorless andem vaYiahloS
expenental (2) poiess.n (At) geometri
t h
twoSucces.Sive airivals indepenolent
At
pLexaty one arrival i At AAt + oAt
mare than one armival in At soCat)
hen the proCassis a PotsSon process oith
param2ter 2A._
n_arrarrivals in t tima units) e ( t
nl
a
vindhya
KCIS

16. claMAt
Dote 202202o
Paga
urd , 12,)t).nKarnivals cn t time units) e
+ O cn Second queu
E K mhtrst qveve
E k- in Lirst quevet n Second qveve
E th trstqueve K n Second GUve
pK packets in queweI& K-K packets en quwe2)
K 0
tdepenclt
(A (ate
Ko K1! Ck-K)I
e
K
poisson C2+2)
poisson RV X(A) Ex 2L
xpenentinl RvX(p) Ex V

17. AsSMAteDate 20 822
Page 3
tTt ho packas ànt.
ext pa.ckat arrival time
-Tst) PT 1-
exponeryHiat-
A=5 packels Second
O. 03
5
5 packok h se.conda) e 1o
0:175 packais in Lsa.cond e 5
51
lo pLCKak in econd ey O183
A/B/c/D/E_ r A/B/
AOrmivalprocess mean 2
servite rate ()
Servey
max no o CUxfaners on packek he Sstem,
pepulahien max. nc o Cusfomer toill ouer need fb
Serv

18. elasSMAte
Dota20:8.2020/
Poge 4-
M/M
M-memcrylessCpotsson)2
M-Senuice process memoryleg Cu exponential
-oneServer
cobur copopulattonsize
BIRTH-DEATH PRoCESS
load bactor P A
P(E probabikity onceluctomers ch th system at
Ame t
n
t4At)At) : honew arrial9noneu) olepartre
yo were in State n at time t._
OR one ne arrival, yoULwer in Mate.
at ime t
R M neu olepartora
&OUwerr in_State nH at ime

19. SSMAte
Date 26.82020
Page
AL)
AAt )+uAt PTt
kaking lim_At>o
dp, (Atup,t) +Agt) 4uE)
or Steady &iate dnct)o
olt
Calculatinq loP
et

20. assnate
Date20.2.2220
Page 6
or D=
M/M
P A P
CtR
P A P -
P

21. elassMAteDate 20 2:0
Page
M/Mtlizathon -P P-
1
.9999 0,93 75
b.9
3
5
p (o 3)
-28 35p C2 Col2
tPtP+P+ P+R
-6.999
12timeA
(6
2
. 5
2 5 S O = S .
. 2 2
b12
2
O-0 2

22. clasSMAate
Date 21: 8.2020
Page
UEUIN GTHEORY_
RECAP
poLsSon process
on pois&on PrOcessS
2 potssooProceseA
birthcleafh proces
Stare n ocostornars/ packeis in Hhe sstem
AlBlc/D/E
A- arrval pracess (A) mean
B Service processs Cmean uD
C senverS
D- t maxek Cusfonarspacketk (bohtersize
E*populafion &ize thct uorl need service
AlBl A/BleD
Eo
MM on Serven
afivalproes
smemes
he service proASA-
s mema le
paiscan arivo orth_ Cexpoantiad Oth Seviea
Yote

23. elassnateData 24:826
Page
dpC
d
tCo we
willassome dh o
AP tALPE (ltu_p
loadacto
.
C-pP 2x Ct-p) +

24. aAssMte
Data 2482020
Paga 3-
LpL (1p)p(tP+¢2
L .9 L
0 - 5 4
P:5
S07
L
L
20Mbpt 99 Mbpss
SDMbps
L Lo t Ls tnivy 2 mn
A Lmin
09ost min

25. elasSmate
Date 2182020
Page .
QS LITTLE'SLALD
L_A avg. ma spent b each pacdet
Qvg.t pockek/Aime
Cvg. H packek|austamers
K=Vz
E EY. Ez X
not aluwauE Costemers Qrived
roein lo,t1):At
avg Rach costomRn sperds w unit of ma
total tme spent o ol the
E D0Stomors thosa uoha arcivedin to,t].
L
Fotal time Spnt by thae
Customers : lt
t Lt Atw
tmtn 2mA g0ee
temy mt 5m
L Au
t lau

26. alassmate
Data 24 8 2020/
Paga S-
0 . A: 9 customens/min
mins
A 9 Ccostomasmn MloCostornars min.
m i n : Wa twsoe
LL Lsar
L:0.9 9
Wser W
L-
3A=49 castonersmin ioCustorerimin
houo much time coil yo0 penchhe Quetx o averac22
C I min u - Wser min-0:1 0. mn 54Se.
- 1 CUstomermin 2o Customan mn_
Conct She xpectedfime to be Sp@nt in the Queue
P: 0.9
1-P
6.5min18
05-O.05 _o15min 21se2D

27. elasSmate
Date28-2020
Page
GAl9costormes/min 20 Costdmer/min
cohat is
L:19 tO -L
20 20
2custmers min = loCUstomare/min
O.2
min SO
20 Cust/min
to ceminHOO
wating ime u attng HmR
2e tstmy
A/MLN
AA-
-4 R
P u 'N

28. lAsSmate
Date 24. 2020
Paga
dt
O N
-P P
1-P
-pN
GUIZ
bloc kine prababilif
mon / hu.
M// MM/m 0n mooodle,_
sla buS: Dehob:theor
mml

29. classmate
Date 27:202t
Page
QUEOINTHEORY-
RECAP
N (N+1)
M/M/1/N
A/elcin/e
N)
max t CUStoMAri hthe
APE Pra
P
NP
-p Po
t1-P
oNblotkingprDpabilhy P
(T1-pNH
Pe.Chen quee i holl
or M/M[I_ P 2 N 2
2

30. elassMAte
Data27.8.2C20/
Paga
M/M//N
qprobable.N+J N+1
o9 P x 1o
- 0-S o.9999
0
0 5 P 508 P O.0l6
P 67
LP=2 = :ol6 PNE OSO8
L5 5 P P::0002
o . 9
N 4 7
2 PN 3
N 20
N (00b P 27x1o
L E n) . +LP +2 P t NRNR
P + 2p2 + 3p3 NN
NH)
+ 2 N)e+N

31. elassAte
Date 27.2 207
Page 3
LPL p+ p N
N
N+
LC-P P -pN) NpNN_pN
LC- N oNA (1-
Ne Nt
pN+
N 2
LpN+I
or LaA+2NNN+1
M/M/oo UEING SVSTEM Seruer
Ce StomeNS
L Lo+ Ls L ACo 0populatio
L -(N)
Cu tCH

32. elasSMte
Data27.2c20/
Paga
2
2
OgOrVêr
K PrO Server2 nALP
OServer n
p
n
L
n=
n'!
L

33. elASSMAt
Date
Page.Menoryles seruiCA
MM/ QUELNG SYSTEM
m servers
omemery4leA arnival
2
m
mA P m
P n>mmiBem-m
n-
m LJ
m-mCA)
in MM/m
m
Cn-m) Pm
Cn-m).1n= m1
m m-
:PenBLmI C1-)

34. elassmate
Data 27.8.220/
Paga 6
aHer qveue siartskillingth he SySMem etartsbebeavin
Simarto M/M/1twith ServireYate ML
M/Mm/m UEUING SYSTEM
Memery less memaru leK- m server mM t Customes
Qrriva Service pacok (n Sidem
MG1 M/D/1
enera Seruice olo terminicHo Service
procesSrecess

35. dassmate
Date 312 2
Page
L min P
386 3
78
2C-068)
L: w minO
2C1-P
o,99x + O.olx30 0.43min3
0.olx3o 3 0
063 3 O63
0. 1715 14-115

36. classmate
Date 3t.82020
Page.
SO wHAT DID WESEE_LAST_CLASS
RECAP
M/MLlN 4ueve
M/M/0
L
P blo.cking probabit
N
NtI
o
L N/2
M/M/m
s P h
L4 1-P m-p)
Pcob:oh ueLing9
Pa Pm Pmz
L Cn-m)p 2 Cn-m)m"p" p
M/Mm m max m packets n4S1em
mser vers L
memory iesS_arVa memoA(ess Servic

37. sSMAte
Date 31.8.2a
Page 2
(o)pz uDP AAlc-p
A L
I+ An C ult
2 m
ua)=2u )-3uu
1
+

38. elasSMate
Date 21-8. 2026
3Page
M/G M- mmorles arava
C- qeneroseuice PCAL.
One Serer
tsepi time faken fo provicle ervia
to the Customev bumo
CefCrM,tsr isexpontniallyE
distributed)
er Vas Ct
ECtsar) How hat matche
weth L twhen
1-P
L +2
2C-P) fSer cexponentiolll,
distnibutedl..
BAD PoST OFECE
tca 20Sec O.9q prob
30min 0.01 prrb
Onecustomer amva por inor (a 1
CCUhat co be the vg. im Spontby the CUStome th he
post otcRi
On overage hoLo mony_CoStommre Co be hov nthe posS
L,w,us EL1se
Ex 2xP:-

39. classmateata
Page
2 0 x O.gq+_g00X 60 37.8See,
37-8
ser Ex2-(E)
0 0 x 0Qq (&O02 0.61 3 7 . 8 2
1367.6
78 0.626 (0.63)
12.9Y
CU
2.9 min-
A =_LcUstomer/min san 7-8Se
C (2.9 mn
0:63
M/D/
M-mamoyule arrrlal
t sskxecllcenSant Ddteministic gervce
2C1-)

40. lassmate
Date 2 202o
Page
LM/M) L (M/o)
O.11 O 06
O2s0 O.225o.2
O1S
2
49S
50
2PC-et
2-P)L 2C1L
2Ct-P) +
2
C-PLo
M/M2
LD
LqoCil D/D
MlCalL HDL
MM/I
Opproximates MM/

41. lssMAteData
318.0DPage
L="Aw HesVao
QUeUin
Sskem
M/M/
M/M
M/M/ N l
- pN
A
SelN/2
LaM/M/m a
-
(Pm+Pomnt

42. clAssmate- -
Date 3-8 2e
Poge 7-
20Mps 7OMbpSH u D e x e r c s e
2
s0Mbps ISDML- MM/L
toobps
PeOML
tTSDM&ps
tooMLpsM//
MA
1-h-6 & 2-6 _Strotagy
Ohich ane t beter
3 6 2-5-6 9raiag

43. clAssMAteDate 3.920
Page
ROUTNC
maka vilgo ata-
Vorse 33 Cwa'l_do our begt
Varse there is a uuanlda
A
to meven t DO dg
that isvrtualand.dikerent
uwalra upto he test
t can b So cold_
to tOrn mis D.orld ar.
makes us standl up pr uohats rignt-
Or hopR throluqh our ile
minichorusaaKge_reset ttto the start
mini-thorvs
Chonus hereLoe are goina-a
to Save oul tha we lave
giua, all wcv.got
minu-cheruswe LotlLMake it through
here uan, likea star
shininabright en yr ioerd
tocda Cmaka evil goaueay Noam Kania acals)
eode lyoka well,esetall
wnttencodeyeke b, there twhen you eall
Yga Amar
Fronck kellercdelyeko tronenatera
NISe2 o uorld omachines
AwORLD wITHOUT DANGER
on shadou homan natun eODE LYokO CTne 6 200)that twe need
e the wy to od the answen
ondcne thinq is onsore
chorus

44. lGsate
Dete 39 2020
RECAP
MIMImn
MMIm/m_ Quauing Sstem
MIal
M/DII
M/M/1
tabular SommaruWs Ltle Ls
70 Meps
(too ns
50
w
33-33
routing 3-6 & 2-S-6
-4-6 & 2--62
fouhng
,7o 2
10 07
ot delay fem23
o op s33:3.ns
30
7
3
2.C1-) 3x76 Xo

45. nte .
w:33:33nsA3 1oxto
Ch 2 33:33 ns
routing2
2-6
1 o 50
150
L 76 12.5 nS
70 C8o) xLo%
2- w _12.5 n
50
Met okela- 0 r wt O U2-Sns
olalays
loo
Control
outing
algorithm
altered
toad
fajected.
hroughpt oeedlload reected load

46. aSMAtr
ROUTINCn delays
algol
ensuredelays ar minimized
ocxèmize throughputs pocr=
protocalsreuttna atge2
algorithms
gcod
CStathio NS dummio hroughput
lat vS. hienarchical SSet
C-ttinta-donMainvS 0ter-clomaun
distribuied vs. Centralized
ingle VS. multipath
liok stiate vsdustonca Necton
nead toknoa Chieranhia bellman-ord
|whole netoonk routi
Cdjstra
RoOTNG ALOORITHMS DLJKSTRA
CShotestpath algorithms)
ChCV,E V et oh vertic8
0oiKshra Set o elges
ob 2
G)bellmarn focc 5
s(
2
Ve Step w add nocde
Shoctst pat e

47. ASSMAte
ute 20
nitia za tion O
P}
ALCORITHM
PD mi D
PPO Si
DD minD, Dt dij
2 + 3
gcto until Dz G
D 0 Dniki alazhio D,:o D n : 2
P:i
. P: 23
D,2, D2 D,o D Peoo
.P 2,31
D 2 D 2
P ,2, 3,s
D D 2 D S D-5S
GPa,2,s, s41
done

48. elassmate
Data 3.9 2AL
Pnga
rUnning time v) Nlv terati ons N-
OCM)n each deration
o) o(NlogM)OCIEI+V)loqtvl)
di Ksira disadlvaniago
)-Ve edges are no1 allowod
BELLMAN-FORD
ve edger ora allow d ho riesf path k no odekined
bvt, no-v w1 Cuclec
0
D i 2 -
Mgorithm D min oltD, Ct_D
turminate i D : D"
min Cest ot reaching usgatmost n edgei
min Sd +D"D
JF
averyStep shortestpoth with edge
is alczlatedto each node

49. elasSMate
Date 3.4.e
Page
2
5
Do D: D D = D co
Yonning am
oCIvllEL) oCN
v: N
O (mIE1
# terotion equred
m N
ElN

50. alassMAte
Data3,9.200
Page 8.
PROFOSITION
a D s aegeneated hyhe aloarthmare qual to
he ShoctestpaBh ramifu 1_o t ecoes h.
healgarithm terminates k alculesNOt_Cotamng nccle
Lare having non--Le costk. elgoferminates ttSo dn
hNe at tecmination De 's he cof of horteai path
romtolor to
HwKh prove htL
D. D 4 K h._
FLOYD-wARSHALL
all-pair sortest path
d Or h:0..N
h
minD P PeK
terahion-h:
te have the shortest path costromi fo, 'vsin9 0d-
2 h (h=O wlo any Yoole)

51. elassmate
Date 7.9.2020
Page
KoUTING AND VPN
RECAP
TOUhg-algorithms
hartest path)
olkstra-gre0elu
bellman-tprel
goydwarshall
progfaongUntast TOUHng-
broadast
min(s,1)
3
Kruskal's
rmin. Cs) MST
prims
mu tcast
33,53
s
M St M Spans 3,55
Cost C)> tW ts minimiaedeeM
min coSt Steiner tree NP (omplete
PIM

52. elassmate
Data7.9.2020
Paga
ARPANEL C96a
o DD 2
D DaD
2
5)D-S3
2 0
BELLMAN FORD
h D":o D,
D min d
e NG)
d des dd Cwhat ih oach Oocde sqnchronov2
sy sBaft
Ais htkengin-
nDde
D
h_node e=
h
D h-
min d +
jeNG)
t algprithm Coi olso Convege
m _ _ol
JEN C)

53. clAsSMAte
Date7:9.2
Page
825ms CoiL braodcast Ds fo Detghbaurs.25 ms
DYNAMIC RoUTING.
inkStatestanco vector?
YOute shoulol noto NOUierS Should nOO
only i neiahboor whole_networkK.
o kaRps only olit gov ka0p cwnole netewor
fo nerahbour ) Lno alL Rink Sates
delays-bellman-kool
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54. elasSMAte
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55. elassmAte
Dote 7:9-2
Page.
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56. elassate
Data 7- 202
Page 6.
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57. elasSMAte
Date 7.428R0
Page
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58. classmwte
Date7.202
Page
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