344. M/M/1
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Arrival rate (): Average number of customers arriving per unit of time
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Service rate (): Average number of customers served per unit of time
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Exponential Distribution: Rate parameter λ = 2 (same as above) |
Poisson Distribution: Rate parameter , so |
Key Relationship
Both distributions share the same rate parameter = 2
- If inter-arrival , then arrivals in time
- Mean inter-arrival time = = 0.5
- Mean arrivals per unit time = = 2
- Mean arrivals in time period 2 = t = 4
- Utilization ()
Fraction of time the server is busy
- Average Number of Customers in the System ()
Average number of customers (both waiting and being served) in the system
- Average Number of Customers in the Queue ()
Average number of customers waiting in the queue
- Average Time a Customer Spends in the System ()
Average time a customer spends in the system (from arrival until they are done being served)
- Average Waiting Time in the Queue ()
Average time a customer spends just waiting in line before being served
- Probability that the System is Empty ()
Probability that there are zero customers in the system (no one is being served and no one is waiting)
- Probability that n Customers are in the System ()
Probability that there are n customers in the system (either waiting or being served)
- Probability the Queue is Full (if the Queue has Limited Capacity) ()
Probability that the system is at full capacity
- System Throughput
Rate at which customers are served and leave the system
- Expected Time in Service ()
Average time a customer spends actually being served (not including waiting time)
- Idle Time ()
Fraction of time that the server is idle (i.e., not serving any customers)
- Probability of Having to Wait in the Queue ()
Probability that an arriving customer will have to wait before being served, i.e., that the server is busy when the customer arrives
- Variance of the Number of Customers in the System ()
Variance of the number of customers in the system
Example
A bank with a single teller
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Arrival rate (): On average, 4 customers arrive every 10 minutes ( custommers per 10 minutes)
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Service Rate (): The teller can serve 6 customers every 10 minutes ( customers per 10 minutes)
- Utilization ()
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 ()
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 ()
On average, about 1.33 customers are waiting in line at any time
- Average Time a Customer Spends in the System ()
On average, a customer spends 5 minutes ( minutes) in the shop (including both waiting in line and getting served)
- Average Waiting Time in the Queue ()
On average, a customer waits 3.3 minutes ( minutes) in line before being served by the teller
- Probability that the System is Empty ()
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 ()
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) ()
There is a 2.8% chance that the system is full, and no new customers can enter
- System Throughput
The coffee shop serves 4 customers every 10 minutes, on average
- Expected Time in Service ()
On average, a customer spends 1.67 minutes being served by the teller
- Idle Time ()
The teller is idle 33% of the time
- Probability of Having to Wait in the Queue ()
There is a 67% chance that a customer will have to wait when they arrive
- Variance of the Number of Customers in the System ()
The queue length varies significantly, with a variance of 6.12 customers
Costs
- Cost per Waiting Customer per Unit Time (): cost incurred per customer for each unit of time they spend waiting in the queue
- Cost per Idle Server per Unit Time (): cost incurred per unit time when the server is not serving customers
Total Cost
Example
- Arrival Rate (): 4 customers per 10 minutes
- Service Rate (): 6 customers per 10 minutes
- Cost per Waiting Customer per Hour (): $10
- Cost per Idle Server per Hour (): $20
- Operational Time: 1 hour
- Utilization
- Average Number of Customers in Queue ()
- Total Waiting Cost
- Idle Time
- Total Idle Cost
- Total Cost