Assumptions of a M/M/1:
- Lambda MUST be SMALLER than µ → thus, lambda / µ < S
1st M = Arrival
2nd M = Service
1 = 1 Server
SIngle Server Queue Model Equations
Model Equations with s being the # of servers
Lq Table
The question can ask you what Lq is or Ws is. If it asks for Ws, find Lq first, then Wq to Ws,
where Wq = Lq / Lambda, and Ws = Wq + 1/µ.
Multi-Server Queueing System
Which server is more efficient?
System 1 is better
First City National Bank
● Verification of model assumptions (using data)
● Practical issues in determining model parameters
● Finding the number of servers
● Design issues
○ Single line or multiple lines?
● Staffing
Lambda = Avg. Arrivals per min = 142/30min = 4.73 customers per min @ 12pm
µ = 60/45.5 = 1.32 customer per min
Common Queue or Separate Queues
First step: Identify arrival rate and service rate for a given time interval
→ Use M/M/s spreadsheet to analyze Alternatives 1 and 2
Lambda: Arrival rate to the system
µ: Service rate of each server
S: Number of servers
Rules:
Lambda < µs
S > Lambda/µ
Class Example: Alternative 1: Common Queue; Alternative 2: 4 Separate Queues
Observations
1. Same utilization
2. Lower waiting time in the common queue
In Alt 2 (Separate Queue), some people are waiting in line, while some servers are idle, while in
Alt 1 (Common Queue) this is impossible → Pooling resources is more efficient
Pro & Con of Common Queue
● Pro: Lower waiting time
● Con: A longer queue
Summary
● Waiting lines form due to variability
● Basic tradeoff between cost and quality
● Decisions:
○ Service Capacity: service rate, number of servers
○ System configuration
● Performance measures:
○ Under Exponential service times and interarrival times
●
○ Use of M/M/s Spreadsheet to examine system performance
Common queue is better than separate queues in terms of reducing waiting time