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Operations Management
Lecture 5 slides
After class slides does not include the review
we cover at the beginning

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of the lectures and only includes the new materials. The additional
Materials(answers to questions,
are added to after class slides in pink)

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! There will be no classes or office hours during the reading week
! There will be no classes during the midterm week
! There will be extra office hour next Wednesday Feb 16th, 10AM-11AM on zoom (same zoom link as for the regular office hour)
! Midterm Exam: Feb 18, 7pm-9pm, )
– Ifyouhaveacademicortime-zoneconflictsyoushouldhave
emailed me.
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Read all slides, summary slides posted on Quercus, assignments,
sample problems, and sample past midterms (Class 1

Midterm Exam Info
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! Please make sure to upload your answers to Quercus by the end of the exam. If you fail to do so, the instructor reserves the right to refuse your answers.
! The final answers entered on Quercus should match your derivations on the paper.

Today’s Lecture
Review of Basic Queueing & the OM Triangle
– The Pollaczek-Khinchin (PK) Formula for average queue length and average system times
– Inventory, Capacity and Information
Queueing Models
– Single server queues: M/M/1, M/D/1, D/D/1
– Queues with multiple servers
– Pooling versus separate buffers

Example: Hospital
Suppose you are the system administrator of the emergency department for Toronto East General Hospital.

Review: Example: Hospital Point (a) – Extra Capacity
! To run an emergency room there is a high level of variability. Why?
– Inter-arrival and service times are random
! Assume expected demand for service is 2 patients/hour
! If you choose to position hospital at point (a)
– What do you need to do to achieve a utilization of 50%?
» Patients will not have to wait very long and there will be very few customers waiting in the buffer areas.
» This is indicated by a low-vertical position on the y axis.
4 patients/hour

Review: Example: Hospital Point (b) – High Inventory
! What happens if you can sustain a utilization of 0.9? – You can get by with much less capacity while still
accommodating the same demand of 2 patients/hour
! More capacity means more equipment, a bigger facility and more doctors and nurses (Costly!)
– This position may only appear to be much better
! You must tolerate a high level of inventory = many
patients waiting in the buffer areas
! You must consider whether this high inventory level (and long waiting time) is acceptable

Review: Example: Hospital Point (c) – Low Variability
! What happens if you reduce the unpredictable variability to an extremely low level (0.01)?
– Better information (or knowledge) is, more often than not, a key factor in reducing variability
Associate lower variability with better information!

Lecture 5:
1) Queue Representation 2) Multiple Servers

Attributes of a Queueing System
! Distribution of time between arrivals – Constant, Uniform, Exponential, etc.
! Distribution of service time
– Constant, Uniform, Exponential, etc.
! Number of servers (one/multi-server)
! Number of queues/buffers (i.e., waiting areas)
! Maximum queue length (buffer size) – “capacity”, e.g., waiting room area
! Service & Routing Policies
– First-Come, First-Served (FCFS), Priorities, Reservations, Service Time, Network Queueing Models

Representing Single Queues
Single queue is represented by
(inter-arr. distribution, service distribution, #of resources(servers))
Number of servers (c)
queue represents the queue length in a system with a single server where interarrival times have a general (meaning arbitrary) distribution and service times have another general distribution.
(A / S / c / K / N / D ) Queue
(A): Inter-arrival times distribution
Example: if interarrival times is from Exponential distribution and service time is also from exponential distribution and we have only one server we represent it as
M/ 1 queue
(S):Service time distribution
M: exponential distribution
D: deterministic (not variable) G: general distribution

Review: Justification of Exponential Assumptions
! In many situations, the exponential distribution assumption is a good approximation of what really happens
– Such an arrival process is called “Poisson process”
» Number of customers arriving per time unit is Poisson distributed
! Easy to analyze because coefficient of variation = 1 for the exponential distributionààà CX=s{X}/E{X}=1
! Recall the PK formula:
r2 C2 +C2 as

Single-Server Queues: M/M/1
The Simplest ‘Stochastic’ Queue
– The first “M” indicates the inter-arrival times are exponentially
distributed Þ Ca = 1
– The second “M” indicates the service times are exponentially
distributed Þ Cs = 1
– The last “1” indicates one single server
! For M/M/1 queue, the P-K formula is exact (=, not »):
Iq = ρ2 = λ2 1-ρ μ(μ-λ)
! Average waiting time in queue: Tq = Iq / l (Little’s Law)
! Self-test: I=Iq +Is =Iq +l/μ T=Tq+Ts=Iq /l+1/μ

Single-Server Queues: M/M/1
The Simplest ‘Stochastic’ Queue
Iq = ρ2 = λ2 1-ρ μ(μ-λ)
Average Arrival Rate
6 person/hour
Average Service
Time (per person) 5 min
μ= 1/5 person/min= 12 person/hour
! Ave. Number in Queue Iq = 36/(12*(12-6)) cust = 0.5 cust
! Ave. Waiting Time in Queue Tq = 0.5/6 hour = 5 min
! Ave.TimespentinSystem T=5min+5min=10min
! Ave. Number in System I = 0.5 cust + 6/12 cust = 1 customer

Single-Server Queues: M/D/1
The Simplest ‘Semi-Stochastic’ Queue
Assumptions: (A/S/c/K/N/D)
– Inter-arrival times (A) follow an Exponential distribution – M » M stands for memoryless
» Arrival process follows a Poisson Distribution
– Service times (S) are constant (i.e., Deterministic) – D » There is no service variability!
– Number of servers (c) – 1
! Other technical assumptions:
» There is a single buffer that serves the entire queue.
» There is no limit on the length the buffer can grow to (K).
» The population the queue can service is unlimited (N).
» Service discipline (D) is First Come – First Served (FCFS).
» All units that arrive enter the queue (no balking)
» Any unit entering the system stays in the queue till served (no reneging)
» All units arrive independently of each other (no batching or correlation). 16

Single-Server Queues: M/D/1
The Simplest ‘Semi-Stochastic’ Queue
– The “M” indicates the inter-arrival times are exponentially
distributed Þ Ca = 1
– The “D” indicates the service times are a constant Þ Cs = 0 – The “1” indicates single server
! First come first served (FCFS)
! For M/D/1 queue, the P-K formula gives us:
I=ρ2æ1ö= λ2
q 1 – ρ çè 2 ÷ø 2 μ ( μ – λ )
! Average waiting time in queue: Tq = Iq / l (Little’s Law)
! Self-test: I=Iq +Is =Iq +l/μ T=Tq+Ts=Iq /l+1/μ

Single-Server Queues: M/D/1
The Simplest ‘Semi-Stochastic’ Queue
ρ2 æ1ö 1-ρçè2÷ø
Average Arrival Rate 6 person/hour
2μ(μ-λ) Service Time (per person)
5 min and is NOT RANDOM
! l=6cust/hour,1/μ=5minÞμ=12cust/hour
! Ave. Number in Queue, Iq = 36/(2*12(12-6)) cust = 0.25 cust
! Ave. Waiting Time in Queue, Tq = 0.25/6 hour = 2.5 min
! Ave.TimespentinSystem, T=2.5min+5min=7.5min
! Ave. Number in System, I = 0.25 cust + 6/12 cust = 0.75 cust

Single-Server Queues: D/D/1
The Simplest ‘Non-Stochastic’ Queue
Assumptions: (A/S/c/K/N/D)
– Inter-arrival times (A) are constant (i.e., Deterministic) – D
» There is no arrival variability!
– Service times (S) are constant (i.e., Deterministic) – D » There is no service variability!
– Number of servers (c) – 1
! Other technical assumptions:
» There is a single buffer that serves the entire queue.
» There is no limit on the length the buffer can grow to (K).
» The population the queue can service is unlimited (N).
» Service discipline (D) is First Come – First Served (FCFS).
» All units that arrive enter the queue (no balking)
» Any unit entering the system stays in the queue till served (no reneging)
» All units arrive independently of each other (no batching or correlation). 19

Single-Server Queues: D/D/1
The Simplest ‘Non-Stochastic’ Queue
l<μ Arrival Rate(person/min)Inter-arrival timeService Rate (persons/min)Throughp ut?Service Time (min)Service time Waiting Time ? Single-Server Queues: D/D/1The Simplest ‘Non-Stochastic’ Queuel>μ Arrival Rate
(person/min)
Inter-arrival time
Service Rate (persons/min)
Throughp ut?
Service Time (min)
Customers waiting:
Waiting Time ? , the queue size “blows-up” and the average
waiting time in the long run goes to infinity and average number in queue also goes to infinity.
Service time

Single-Server Queues: D/D/1
The Simplest ‘Non-Stochastic’ Queue
! This is queueing under a Deterministic scenario – There is no uncertainty or variability in the process.
– l and μ are constants (not even average rates!)
! Customers in queue (Iq) and time spent in queue (Tq) – If l < μ, each customer (job) is processed before the next arrival.The average waiting time and the average queue size is 0.– If l > μ, the queue size “blows-up” and the average waiting time in the long run goes to infinity and average number in queue also goes to infinity.

Today’s outline
Queue Analysis:
! We learned so far:
– Single resources: » PK Formula
» OM Triangle
! We learn today:
– Multiple Identical Resources (e.g., multiple ATMs): » Updated PK formula for multiple resource
» Pooling versus separate queues

Multiple Servers
Separate Queue
Pooled Queue

Multi-Server Queueing Models c servers
Arrival rate (average input rate)
Average throughput l=1/E[a] persons/min
c servers,
capacity utilization
l=1/E[a] persons/min inter-arrival time
distribution a Ca = s[a]/E[a]
τ = l / ( c x μ) Service rate (per server)
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