Queueing Theory MCQ Quiz in తెలుగు - Objective Question with Answer for Queueing Theory - ముఫ్త్ [PDF] డౌన్‌లోడ్ కరెన్

Arrival rate, λ = 6/hr

Service rate, \(\mu = \frac{{60}}{8} = 7.5/hr\) 

Mean waiting time in queue \(= {W_q} = \frac{\lambda }{{\mu \left( {\mu - \lambda } \right)}}\) 

\(= \frac{6}{{7.5\left( {7.5 - 6} \right)}} = \frac{6}{{7.5 \times 1.5}} = \frac{4}{{7.5}}hrs\)

\(\therefore {W_q} = \frac{4}{{7.5}} \times 60 = 32\;minutes\)

Points to remember

i) Waiting time in queue \({W_q} = \frac{\lambda }{{\mu \left( {\mu - \lambda } \right)}}\)

Concept:

λ = Arrival rate (customer/time)

μ = Service rate (customer/time).

The probability that a customer has to wait or server is busy is given by \(\rho = \frac{\lambda }{\mu }\)

Calculation:

Arrival rode (λ) = 60/15 = 4 person per hour

Service rate (μ) = 60/5 = 12 person per hour.

Probability that a person has to wait \(\rho = \frac{\lambda }{\mu }\)

\(= \frac{4}{{12}} = 0.33\)

Explanation:

Monte Carlo Simulation for Queueing

• The Monte Carlo technique is quite useful for analyzing waiting line problems which are difficult or impossible to be analyzed mathematically.
• Simulated Sampling Method (SSM), helpful when first come, first served assumption is not valid for a particular queue.
• In many cases, the observed distribution of arrival times and service time can not be fitted to a certain mathematical distribution (Poisson and exponential distribution) and the Monte Carlo SSM is the only hope. 
• Multiple queues condition of arrival in one queue and departure from different queues is also well handled by the Monte Carlo SSM.
• SSM consist of replacing the actual universe of the item with its theoretical counterpart, which is the universe described by some assumed probability distribution.
• A random number table is then used for sampling from this theoretical population. Such SSM is called the Monte Carlo Method.

​Monte Carlo Method Advantages

• Monte Carlo simulation is a computerized mathematical technique that allows people to account for risk in quantitative analysis and decision making.
• Computerization use makes easy storage and retrieval of a large amount of data for months and years.
• Manipulation of those factors which can be controlled. (i.e. simulating)
• Monte Carlo simulation performs risk analysis by building models of possible results by uncertainty. 
• It then calculates results over and over, each time using a different set of random values from the probability functions.
• Monte Carlo simulation helps the decision-maker with a range of possible outcomes and the probabilities they will occur for any choice of action.

Queuing theory

• Queuing theory is the mathematical study of waiting for a queue.
• A queueing model is constructed so that queue lengths and waiting times can be predicted.
• 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.

Concept:

• In queueing theory, a discipline within the mathematical theory of probability, Little's result, the theorem is a theorem by John Little which states that the long-term average number L of customers in a stationary system is equal to the long-term average effective arrival rate λ multiplied by the average time W that a customer spends in the system.
• Expressed algebraically the law is

\(W=\frac{L_s}{λ }\)

• Where W is the average waiting time in the system,  L_s is the average number of customers in the system or length of the system, and arrival rate in the system.
• Hence little's law is the relation between waiting for time and length of queue or system

Explanation:

In queuing theory,

The arrival rate or arrival pattern follows the Poisson distribution service rate or service pattern follows Exponential distribution.

Poisson Probability Distribution for Arrivals Pattern:

The number of customer arrivals in a specific time period follows the Poisson Probability Distribution pattern and is expressed as follows:

\(P\left( X \right) = \frac{{{\lambda ^x}{e^{ - \lambda }}}}{{X!}}\)

X = the number of arrivals in the time period

λ = the average or mean number of customers arrivals per unit of the time period or mean arrival rate

Exponential Probability Distribution for Service Times

The service time for a unit or customer is variable and random. An exponential distribution function expresses the service time as follows:

f(t) = μ e-μt

μ is the rate that units or customers are served.

Concept:

Waiting time in Queue is \(W_q = \frac{L_q}{λ}\)

The average number of customers in queue \(L_q = \frac{ρ^2}{1\ -\ ρ }\)

System Utilisation or utilisation factor or average utilisation, ρ: is the ratio of arrival and service rate \(ρ = \frac{λ}{μ}\)

Calculation:

Given:

Arrival rate, λ" role="presentation" style="display: inline; position: relative;" tabindex="0">λλ" id="MathJax-Element-1-Frame" role="presentation" style="display: inline; position: relative;" tabindex="0"><span style="font-weight: 700;">λ</span> = 5 /hr 

Time is taken to wash a car = 10 min per car = \(\frac{1}{μ}\)

Service rate, μ = \(\frac{1\ }{10}\) = 0.1 car per min = 6 car per hr

Average utilisation \(ρ = \frac{λ}{μ} = \frac{5}{6} = \frac{5}{6}\) 

Average waiting time \(W_q = \frac{L_q}{λ}\)

\(L_q = \frac{ρ^2}{1\ -\ ρ } = \frac{(\frac{5}{6})^2}{1\ - \frac{5}{6}} = \frac{25}{6}\)

Average waiting time \(W_q = \frac{\frac{25}{6}}{5} = \frac{25}{30}\times 60=50~mins \) 

Concept:

Waiting time in a queue \(w_q = \frac{\lambda }{{\mu \left( {\mu - \lambda } \right)}}\) 

Calculation:

Given:

that: arrival rate (λ) = 6 (per hour),

service rate (μ) = 10 (per hour)

\(w_q = \frac{\lambda }{{\mu \left( {\mu - \lambda } \right)}} = \frac{6}{{10\left( {10 - 6} \right)}} = \frac{3}{{20}}\)

Concept:

The utilization factor for the services,

\(\rho = \frac{\lambda }{\mu }\)        ....(1)

Where,

λ = letter arriving rate

μ = Service rate

The waiting on the part of the typewriter means typewriter is idle so the average cost due to waiting,

⇒ C = C_o × T × P₀

⇒ C = C_o × T × (1 – ρ)      …(2)

Where,

P₀ = 1 - ρ = The probability when nobody in the system and the system is idle

C₀ = The typewriter cost (Rs per hour)

T = Working time (hour per day)

Calculation:

Given:

C_o = 120 Rs. per hour, λ = 6 letter per hour, μ = 12 letter per hour, t = 8 hour per day

Using equation (1),

⇒ \(\rho = \frac{6}{{12}} = 0.5\)

Using equation (2),

⇒ C = 120 × 8 × (1 - 0.5)

⇒ C = 480 Rs. per day

Concept:

λ = Arrival rate (customer/time)

μ = Service rate (customer/time).

Average arrival time and the time spent in the system (Waiting time in system) = \({W_s} = \frac{1}{{\mu - \lambda }}\)

Average arrival time and the time spent in the queue (before being served) (Waiting time in queue) = \({W_q} = \frac{\lambda }{\mu }.\frac{1}{{\mu - \lambda }}\)

Calculation:

λ =12 customer/hr

μ = 24 customer/hr

\({W_q} = \frac{\lambda }{{\mu \left( {\mu - \lambda } \right)}} = \frac{{12}}{{24\left( {24 - 12} \right)}}\)

\(= \frac{1}{{24}}hrs = \frac{1}{{24}} \times 60 = 2.5\;minutes\)

Explanation:

For steady-state μ > λ i.e. service rate should be greater than arrival rate.

where μ = service rate and λ = arrival rate.

As λ = 12.

So for finite queue length μ = 24 is the best option.