25700722037-(HM-HU601).pdf mechanical engineering

Project

1. DEPARTMENT OF MECHANICAL ENGINEERING
TOPIC-(M|M|1):(∞|FCFS|∞)

NAME :ANANTA KUMAR NANDI

ROLL NO.:25700722037

REG. NO.:222570120277 OF 2022-23

SUBJECT :(OPERATIONS RESEARCH) {HM-HU601}

YEAR :3RD

SEM :6TH

2. INTRODUCTION-:

M/M/1 denotes a queueing system with one server and a Poisson distribution
for customer interarrival times and service times. The notation /FCFS
indicates that a first-come-first-served (FCFS) service discipline is being used,
which means that customers are served in the order in which they arrive.

3. Inthismodel,
1.
DistributionofarrivalisPoissonwitharrivalrateλ,
2.
DistributionofdepartureisPoissonwithservicerateμ(λ<μ),
3.
Distributionofinter-arrivaltimeisexponentialwithmean
arrivaltime(1/λ),
4.
Distributionofservicetimeisexponentialwithmeanservice
time(1/μ),
5.
Systemhassingleserver,
6.
Queuelengthisunrestricted,
7.
QueueDisciplineisfirstcomefirstserve.
MODEL -(M|M|1):(∞|FCFS|∞)

4. 1.SteadyStateDistribution:
Thesteadystatedistributionforthemodelisobtainedunderfollowing
axioms:
Axiom1:Theno.ofarrivalsaswellasdeparturesinnon-overlapping
intervalsoftimearestatisticallyindependent.
Axiom2:Theprobabilitythatanarrivaloccurswithinaverysmalltime
intervalΔtisgivenby:
P
1
[Δt]=λΔt+o(Δt)
Axiom3:Theprobabilitythatandepartureoccurswithinaverysmalltime
intervalΔtisgivenby:
P
1
[Δt]= μΔt+o(Δt)
Axiom4:Theprobabilityofmorethanonearrivalsormorethanone
departuresduringtimeintervalΔtisnegligiblysmall,i.e.,o(Δt)
Axiom5:Thearrivalsanddeparturesarestatisticallyindependent.
CHARACTERISTICSOF(M|M|1):(∞|FCFS|∞)

5. LetP
n
[t]denotestheprobabilitythattherearencustomersinthesystemat
timet.
Let’sconsiderthecasewhenthereisatleastonecustomerinthesystemat
timet(n >0)
P
n
[t+Δt]
=P[ncustomersattimet,no arrivalandnodepartureduringΔt]
+P[(n–1)customersattimet,onearrivalandno departureduringΔt]
+P[(n+1)customersattimet,no arrivalandonedepartureduringΔt]
+P[ncustomersattimet,onearrivalandonedepartureduringΔt]+…
=P
n
[t].P[noarrivalduringΔt].P[nodepartureduringΔt]
+P
(n–1)
[t].P[onearrivalduringΔt].P[nodepartureduringΔt]
+P
(n+1)
[t].P[noarrivalduringΔt].P[onedepartureduringΔt]
+P
n
[t].P[onearrivalduringΔt].P[onedepartureduringΔt]+o(Δt)

6. Onsolving theequation,itwouldbeobtainedthat:
TakinglimitasΔt→0,itwouldbeobtainedthat:
Now,Applyingthesteadystateconditionitwould beobtainedthat:
0= –λP
0
+μP
1
(2)Or, P
1
=(λ/μ)P
0
=ρP
0
Now,puttingn=1inequation (1),itisobtainedthat:
P
2
=(1+ρ)P
1
–ρP
0
=(1+ρ)ρP
0
–ρP
0
=ρ
2
P
0
(fromequation2)
(3)

7. Now,puttingn=2inequation (1),itisobtainedthat:
P
3
=(1+ρ)P
2
–ρP
1
=(1+ρ)ρ
2
P
0
–ρ
2
P
0
(Fromequation2and3)
=ρ
3
P
0
LettherelationP
n
=ρ
n
P
0
istrueforalln≤m,nowputtingn=mis
equation(1),itisobtainedthat:
P
(m+1)
=(1+ρ)P
m
–ρP
(m–1)
=(1+ρ)ρ
m
P
0
–ρ
m
P
0
=ρ
(m+1)
P
0
Hence,bymathematicalinduction,P
n
=ρ
n
P
0
,holdsforalln.
Itisalsoknownthatthetotalprobabilityalwaysbe1,i.e.,

8. Or, P
0
=(1–ρ)
Or, P
n
=(1–ρ)ρ
n
Itisthesteadystatedistribution,forthemodel(M|M|1):(∞|FCFS),which
givestheprobabilitythattherearencustomersinthesystemattimet.

9. 2.Averagenumberofcustomersinthesystem:
AveragenumberofcustomersinthesystemaredenotedbyE(n).Itis
definedby:
Hencetheaveragenumberofcustomersinthesystemisρ/(1–ρ).

10. 3.AveragequeueLength:
Thecustomersinthesysteminvolvethecustomersinqueueaswellasthe
customerwhoisattheservicecounter(server)andgettingservice.For
obtainingtheaveragequeuelengththecustomerattheserverisnotconsidered.
ItisdenotedbyE(m),anddefinedby:
Hencetheaveragequeuelengthisρ
2
/(1–ρ).

11. 4.Probabilitythat thereareatleastkcustomersinthesystem:
Hencetheprobabilitythatthereareatleastkcustomersinthesystemisρ
k
.

12. 5.Waitingtimedistribution:
Waitingtimeofthecustomerisacontinuousvariableexceptthatthereisa
nonzeroprobabilitythatuponarrivalthecustomerisservedimmediately,
i.e.,waitingtimeiszero.Thewaitingtimeisdenotedbywandthe
cumulativedensityfunctionofwaitingtimeisdenotedbyψ
w
(t).
Now,
Ψ
w
(0)=P[w=0]
= P[thereisnocustomerinthesystem]
=P
0
=(1–ρ)
Nowconsider,

13. Hencethewaitingtimedistributionisgivenby:
Theprobabilitydensity functionofwaitingtimeψ(t)isgiven by:

14. 6.AverageWaitingtime:
Theaveragewaitingtimeistheaveragetimespentbyacustomerinthe
queue.ItisdenotedbyE[w]andgivenby:
Taking{–μ(1–ρ)t}= z,itisobtainedthat:
Hence,theaveragewaiting timeisρ/{μ(1–ρ)}.

15. 7.ProbabilityDensityFunctionforthetimespentinthesystemby
customer:
Itisdenotedbyψ(t\t>0)and givenby:
Ψ(t\t>0)= ψ(t)/P(t>0)
=μρ(1–ρ)e
–μ(1–ρ)t
/1–(1–ρ)
=μ(1–ρ)e
–μ(1–ρ)t
Hence,thep.d.f.forthetimespentinthesystembycustomer is:
μ(1–ρ)e
–μ(1–ρ)t

16. 8.Averagetimespentbythecustomerinthesystem:
Thetimespentbythecustomerinthesystemisdenotedbyv.Theaverage
timespentbythecustomerinthesystemisgiven by:
Taking{–μ(1–ρ)t}= z,itisobtainedthat:
HencetheAveragetimespentby thecustomer inthesystemis1/{μ(1–ρ).

17. 9.Little’sFormula:
It was given by the John D.C. Little, That’s why it is termed as Little’s
formula.Itprovidestherelationbetweentheaveragewaitingtimeinthe
systemandaveragenumberofcustomersinthesystem.
Itisknownthat:
E[n]=λ/(μ–λ)and
E[v]=1/(μ–λ)
Whichgivesriseto therelation,
E[n]=λE[v]
Similarly itcanbeeasilyshownthat
E[m]=λE[w]

18. CONCLUSION-:

Using queuing theory, the bottleneck of the systems can be identified.
Scenario and software-based simulations provide solutions to the problem of
queues. The study is an application of Queuing Theory with a focus on
efficient resource utilization.

19. REFERENCES-:

edition;PearsonEducationInc.
3.
SwarupK.,GuptaP.K.&Manmohan;OperationsResearch;11
th
edition;
SultanChand&Sonspublication.
4.
SharmaS.D.,SharmaH.;OperationsResearch:Theory,Methodsand
Applications;15
th
edition;KedarNathRamNathPublishers.
1.
HillerS.H.,LiebermanG.J.;IntroductiontoOperations
Research;7
th
edition,McGrawHillPublications
2.
Taha,H.A.;OperationsResearch:AnIntroduction;8
th