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. In this model,
1. Distribution of arrival is Poisson with arrival rate λ,
2. Distribution of departureis Poisson with service rate μ (λ<μ),
3. Distribution of inter-arrival time is exponential with mean
arrival time (1/λ),
4. Distribution of service time is exponential with mean service
time (1/μ),
5. System has single server,
6. Queue length is unrestricted,
7. Queue Discipline is first come first serve.
MODEL - (M|M|1):(∞|FCFS|∞)
4. 1. Steady State Distribution:
The steady state distribution for the model is obtained under following
axioms:
Axiom 1: The no. of arrivals as well as departures in non-overlapping
intervals of time are statistically independent.
Axiom 2: The probability that an arrival occurs within a very small time
intervalΔt is given by:
P1[Δt] = λΔt + o(Δt)
Axiom 3: The probability that an departure occurs within a very small time
intervalΔt is given by:
P1[Δt] = μΔt + o(Δt)
Axiom 4: The probability of more than one arrivals or more than one
departures during time intervalΔt is negligibly small, i.e., o(Δt)
Axiom 5: The arrivals and departures are statistically independent.
CHARACTERISTICS OF (M|M|1):(∞|FCFS|∞)
5. Let Pn[t] denotes the probability that there are n customers in the system at
time t.
Let’s consider the case when there is at least one customer in the system at
time t (n > 0)
Pn[t+Δt]
= P[n customers at time t, no arrival and no departure during Δt]
+P[(n–1) customers at time t, one arrival and no departure duringΔt]
+P[(n+1) customers at time t, no arrival and one departure duringΔt]
+ P[n customers at time t, one arrival and one departure during Δt] + …
=Pn[t].P[no arrival during Δt].P[no departure duringΔt]
+P(n–1)[t].P[one arrival duringΔt].P[no departure duringΔt]
+P(n+1)[t].P[no arrival during Δt].P[one departure duringΔt]
+Pn[t].P[one arrival during Δt].P[one departure duringΔt] + o(Δt)
6. On solving the equation, it would be obtained that:
Taking limit as Δt→0, it would be obtained that:
Now,Applying the steady state condition it would be obtained that:
0 = – λP0 + μP1
(2)
Or, P1 = (λ/μ)P0 = ρP0
Now, putting n = 1 in equation (1), it is obtained that:
P2 = (1+ρ)P1 – ρP0
= (1+ρ)ρP0 – ρP0
= ρ2P0
(from equation 2)
(3)
7. Now, putting n = 2 in equation (1), it is obtained that:
P3 = (1+ρ)P2 – ρP1
= (1+ρ)ρ2P0 – ρ2P0 (From equation 2 and 3)
= ρ3P0
Let the relation Pn = ρnP0 is true for all n ≤ m, now putting n = m is
equation (1), it is obtained that:
P(m+1) = (1+ρ)Pm – ρP(m–1)
= (1+ρ)ρmP0 – ρmP0
=ρ(m+1)P0
Hence, by mathematical induction, Pn = ρnP0, holds for all n.
It is also known that the total probability always be 1, i.e.,
8. Or, P0 = (1 – ρ)
Or, Pn = (1 – ρ)ρn
It is the steady state distribution, for the model (M|M|1):(∞|FCFS), which
gives the probability that there are n customers in the system at time t.
9. 2. Average number of customers in the system:
Average number of customers in the system are denoted by E(n). It is
defined by:
Hence the average number of customers in the system is ρ/(1 – ρ).
10. 3. Average queue Length:
The customers in the system involve the customers in queue as well as the
customer who is at the service counter (server) and getting service. For
obtaining the average queue length the customer at the server is not considered.
It is denoted by E(m), and defined by:
Hence the average queue length is ρ2/(1 – ρ).
11. 4. Probability that there are at least k customers in the system:
Hence the probability that there are at least k customers in the system is ρk.
12. 5. Waiting time distribution:
Waiting time of the customer is a continuous variable except that there is a
non zero probability that upon arrival the customer is served immediately,
i.e., waiting time is zero. The waiting time is denoted by w and the
cumulative density function of waiting time is denoted by ψw(t).
Now,
Ψw(0) = P[w = 0]
= P[there is no customer in the system]
=P0
= (1 – ρ)
Now consider,
13. Hence the waiting time distribution is given by:
The probability density function of waiting time ψ(t) is given by:
14. 6.Average Waiting time:
The average waiting time is the average time spent by a customer in the
queue. It is denoted by E[w] and given by:
Taking {– μ(1 – ρ)t} = z, it is obtained that:
Hence, the average waiting time is ρ/{μ(1 – ρ)}.
15. 7. Probability Density Function for the time spent in the system by
customer:
It is denoted by ψ(t t>0) and given by:
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, the p. d. f. for the time spent in the system by customer is:
μ(1–ρ)e–μ(1– ρ)t
16. 8.Average time spent by the customer in the system:
The time spent by the customer in the system is denoted by v. The average
time spent by the customer in the system is given by:
Taking {–μ(1 – ρ)t} = z, it is obtained that:
Hence the Average time spent by the customer in the system is 1/{μ(1 – ρ).
17. 9. Little’s Formula:
It was given by the John D.C. Little, That’s why it is termed as Little’s
formula. It provides the relation between the average waiting time in the
system and average number of customers in the system.
It is known that:
E[n] = λ/(μ – λ) and
E[v] = 1/(μ – λ)
Which gives rise to the relation,
E[n] = λE[v]
Similarly it can be easily shown that
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-:
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4. Sharma S.D., Sharma H.; Operations Research: Theory, Methods and
Applications; 15th edition; Kedar Nath Ram Nath Publishers.
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Research; 7th edition, McGraw Hill Publications
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