M/M/1

347. M/M/1

Arrival rate ( $𝜆$ ): Average number of customers arriving per unit of time
Service rate ( $𝜇$ ): Average number of customers served per unit of time

Exponential Distribution:
Inter-arrival Times

Rate parameter λ = 2 (same as above)

Poisson Distribution:
Number of Arrivals in Time Period $𝑡 = 2$

Rate parameter $𝜆 = 2$ , so $𝜆 𝑡 = 4$

Key Relationship

Both distributions share the same rate parameter $𝜆$ = 2

If inter-arrival $\times \sim Exponential (𝜆)$ , then arrivals in time $𝑡 \sim Poisson (𝜆 𝑡)$
Mean inter-arrival time = $\frac{1}{𝜆}$ = 0.5
Mean arrivals per unit time = $𝜆$ = 2
Mean arrivals in time period 2 = $𝜆$ t = 4

Utilization ( $𝜌$ )

Fraction of time the server is busy

𝜌 = \frac{𝜆}{𝜇}

Average Number of Customers in the System ( $𝐿$ )

Average number of customers (both waiting and being served) in the system

𝐿 = \frac{𝜌}{1 - 𝜌}

Average Number of Customers in the Queue ( $𝐿_{𝑞}$ )

Average number of customers waiting in the queue

𝐿_{𝑞} = \frac{𝜌^{2}}{1 - 𝜌}

Average Time a Customer Spends in the System ( $𝑊$ )

Average time a customer spends in the system (from arrival until they are done being served)

𝑊 = \frac{1}{𝜇 - 𝜆}

Average Waiting Time in the Queue ( $𝑊_{𝑞}$ )

Average time a customer spends just waiting in line before being served

𝑊_{𝑞} = \frac{𝜌}{𝜇 - 𝜆}

Probability that the System is Empty ( $𝑃_{0}$ )

Probability that there are zero customers in the system (no one is being served and no one is waiting)

𝑃_{0} = 1 - 𝜌

Probability that n Customers are in the System ( $𝑃_{𝑛}$ )

Probability that there are n customers in the system (either waiting or being served)

𝑃_{𝑛} = (1 - 𝜌) \cdot 𝜌^{𝑛}

Probability the Queue is Full (if the Queue has Limited Capacity) ( $𝑃_{𝑛_{max}}$ )

Probability that the system is at full capacity

𝑃_{𝑛_{max}} = (1 - 𝜌) \cdot 𝑝^{𝑛_{max}}

System Throughput

Rate at which customers are served and leave the system

Throughput = 𝜆

Expected Time in Service ( $𝑊_{𝑠}$ )

Average time a customer spends actually being served (not including waiting time)

𝑊_{𝑠} = \frac{1}{𝜇}

Idle Time ( $1 - 𝜌$ )

Fraction of time that the server is idle (i.e., not serving any customers)

Idle Time = 1 - 𝜌

Probability of Having to Wait in the Queue ( $𝑃_{𝑤}$ )

Probability that an arriving customer will have to wait before being served, i.e., that the server is busy when the customer arrives

𝑃_{𝑤} = 𝜌

Variance of the Number of Customers in the System ( $Var (𝐿)$ )

Variance of the number of customers in the system

Var (𝐿) = \frac{𝜌}{{(1 - 𝜌)}^{2}}

Example

A bank with a single teller

Arrival rate ( $𝜆$ ): On average, 4 customers arrive every 10 minutes ( $𝜆 = 4$ custommers per 10 minutes)
Service Rate ( $𝜇$ ): The teller can serve 6 customers every 10 minutes ( $𝜇 = 6$ customers per 10 minutes)

Utilization ( $𝜌$ )

𝜌 = \frac{𝜆}{𝜇} = \frac{4}{6} = 0.67

The teller is busy 67% of the time. The remaining 33% of the time, the teller is idle, waiting for the next customer

Average Number of Customers in the System ( $𝐿$ )

𝐿 = \frac{𝜌}{1 - 𝜌} = \frac{0.67}{1 - 0.67} = 2

On average, there are 2 customers in the coffee shop at any given time, either being served or waiting in line

Average Number of Customers in the Queue ( $𝐿_{𝑞}$ )

𝐿_{𝑞} = \frac{𝜌^{2}}{1 - 𝜌} = \frac{{0.67}^{2}}{1 - 0.67} = 1.33

On average, about 1.33 customers are waiting in line at any time

Average Time a Customer Spends in the System ( $𝑊$ )

𝑊 = \frac{1}{𝜇 - 𝜆} = \frac{1}{6} - 4 = 0.5

On average, a customer spends 5 minutes ( $0.5 \times 10$ minutes) in the shop (including both waiting in line and getting served)

Average Waiting Time in the Queue ( $𝑊_{𝑞}$ )

𝑊_{𝑞} = \frac{𝜌}{𝜇 - 𝜆} = \frac{0.67}{6 - 4} = 0.33

On average, a customer waits 3.3 minutes ( $0.33 \times 10$ minutes) in line before being served by the teller

Probability that the System is Empty ( $𝑃_{0}$ )

𝑃_{0} = 1 - 𝜌 = 1 - 0.67 = 0.33

There is a 33% chance that the coffee shop is empty, meaning there is no customer in the queue or being served

Probability that n Customers are in the System ( $𝑃_{𝑛}$ )

𝑃_{𝑛} = (1 - 𝜌) \cdot 𝜌^{𝑛} = (1 - 0.67) \cdot {0.67}^{2} = 0.148

There is a 14.8% chance that exactly 2 customers are either in line or being served

Probability the Queue is Full (if the Queue has Limited Capacity) ( $𝑃_{𝑛_{max}}$ )

𝑃_{𝑛_{max}} = (1 - 𝜌) \cdot 𝑝^{𝑛_{max}} = (1 - 0.67) \cdot {0.67}^{5} = 0.028

There is a 2.8% chance that the system is full, and no new customers can enter

System Throughput

Throughput = 𝜆 = 4

The coffee shop serves 4 customers every 10 minutes, on average

Expected Time in Service ( $𝑊_{𝑠}$ )

𝑊_{𝑠} = \frac{1}{𝜇} = \frac{1}{6} = 0.167

On average, a customer spends 1.67 minutes being served by the teller

Idle Time ( $1 - 𝜌$ )

Idle Time = 1 - 𝜌 = 1 - 0.67 = 0.33

The teller is idle 33% of the time

Probability of Having to Wait in the Queue ( $𝑃_{𝑤}$ )

𝑃_{𝑤} = 𝜌 = 0.67

There is a 67% chance that a customer will have to wait when they arrive

Variance of the Number of Customers in the System ( $Var (𝐿)$ )

Var (𝐿) = \frac{𝜌}{{(1 - 𝜌)}^{2}} = \frac{0.67}{{(1 - 0.67)}^{2}} = {6.12}^{2}

The queue length varies significantly, with a variance of 6.12 customers

Costs

Cost per Waiting Customer per Unit Time ( $𝐶_{𝑤}$ ): cost incurred per customer for each unit of time they spend waiting in the queue

Total Waiting Cost = 𝐿_{𝑞} \times 𝐶_{𝑤} \times Unit Time

Cost per Idle Server per Unit Time ( $𝐶_{𝑠}$ ): cost incurred per unit time when the server is not serving customers

Total Idle Cost = (1 - 𝜌) \times 𝐶_{𝑠} \times Unit Time

Total Cost

Total Cost = Total Waiting Cost + Total Idle Cost

Example

Arrival Rate ( $𝜆$ ): 4 customers per 10 minutes
Service Rate ( $𝜇$ ): 6 customers per 10 minutes
Cost per Waiting Customer per Hour ( $𝐶_{𝑤}$ ): $10
Cost per Idle Server per Hour ( $𝐶_{𝑠}$ ): $20
Operational Time: 1 hour

Utilization

𝜌 = \frac{𝜆}{𝜇} = \frac{4}{6} = 0.67

Average Number of Customers in Queue ( $𝐿_{𝑞}$ )

𝐿_{𝑞} = \frac{𝜌^{2}}{1 - 𝜌} = \frac{{0.67}^{2}}{1 - 0.67} = 1.33 cusommers

Total Waiting Cost

Total Waiting Cost = 𝐿_{𝑞} \times 𝐶_{𝑤} \times Unit Time = 1.33 \times 10 \times 1 = $ 13.33

Idle Time

Idle Time = 1 - 𝜌 = 1 - 0.67 = 0.33

Total Idle Cost

Total Idle Cost = (1 - 𝜌) \times 𝐶_{𝑠} \times Unit Time = 0.33 \times 20 \times 1 = $ 6.67

Total Cost

Total Cost = Total Waiting Cost + Total Idle Cost = 13.33 + 6.67 = $ 20