1. Time-dependent queue modelling
Dave Worthington
Management Science Dept.
Lancaster University

2. Time-dependent queue modelling -Outline
Motivation
What is time-dependent behaviour?
Analytic methods
Discrete time modelling (DTM) approach
DTM-based task

3. Motivation
Many real-life queueing problems exhibit time-dependent behaviour.
Steady-state models can be very misleading
Simulation is always an option, but can be time-consuming, difficult to use analytically, and lacking in insights.
Some analytic queue modelling approaches for time-dependent behaviour, and some insights.

4. What is time-dependent behaviour?
Transient
‘Fully’ time-dependent

5. A classic analytic approach: Chapman-Kolmogorov equ’ns for M(t)/M(t)/c system

6. Analytic Approaches 0: Numerical solution of C-K equations
1: Newell’s Graphical approach
2: Simple Stationary Approximation (SSA)
3: Pointwise Stationary Approximation (PSA)
4: M(t)/G/∞ system
5: Network of M(t)/G/∞ systems
6: Discrete Time Modelling (DTM) Approx.

7. 1: Newell’s Graphical approach
If arrivals exceed service capacity, ignore random variation and approximate using ‘business flow process’ approach (i.e. Newell’s Graphical approach).
This approach can also be used for networks of queues.

8. Newell’s Graphical Method (‘Business Flow’ Approach)
E[cumulative arrivals]
Predicted no. of customers in system at time t
No. of customers
E[cumulative departures] n
Predicted time in system for customer n
time t

9. Newell’s method can be applied to ‘Business Flows’
A ‘complicated’ queueing network ….
…. is represented by a simplified input/output process

10. 2: Simple Stationary Approximation (SSA)

11. 3: Pointwise Stationary Approximation (PSA)

12. SSA v PSA

13. SSA v PSA v Correct

14. 4A: M(t)/G/∞ system Can be analysed exactly!
Number in system N(t) has a time-dependent Poisson distribution with mean E[N(t)] given by:
where l(u) is the arrival rate at time u
and FC(t-u) = prob(service time > t-u)
And by working in discrete time (e.g. round to nearest hour/minute/.....)

15. 4B: M(t)/GD/∞ system
And by working in discrete time (e.g. round to nearest hour/minute/...) this is easy to evaluate in a spreadsheet, i.e.:
Number in system N(t) has a time-dependent Poisson distribution with mean E[N(t)] given by:
Where: E[A(i) ] is average arrivals in time interval i
and: FC(t-i) = prob(service time >= t-i)

16. 5: Network of M(t)/G/∞ systems
Can also be analysed exactly!
Number in system at node k has a time-dependent Poisson distribution with mean E[Nk(t)] calculated using the same logic as for single node M(t)/G/∞ system.
For example:
l2(t)
l1(t)
N1(t)
N2(t)
(r1)
E[N2(t)]= E[N1(t) +N2(t)]- E[N1(t)]
where E[N1(t)]= ……..
and

17. 6: Discrete Time Modelling (DTM)
Origins of Discrete Time Modelling
Meisling (1958), Operations Research;
Dafermos & Neuts (1971), Cahiers du Centre Recherche Operationnelle;
Neuts (1973), NRLQ;
Alfa (1990s).

18. Formulate as a Finite Markov Chain: e.g.: M/D2/1
Transition matrix Pt
e.g.: 1:1 to 1:2 if one arrival and new service time=2
States: {[0]&[n:i]} where n= no in sys,
i= residual service time (rounded up);.
Service time = 1 or 2 with probs = s1, s2.
State at time (t+1)
1:1
1:2
2:1
2:2
Etc
State at time t: a0
a1s1
a1s2
a2s1
a2s2 … a1
a0s1
a0s2
etc
Arrivals in unit time = 0, 1, 2, …. with probs a0, a1, a2, ..
e.g. for random arrivals rate l,

19. Markov Chain ‘solution’ for M(t)/GD/c systems:
Let P t be the vector of probabilities at time t, then:
P t= P t-1 P (applied numerically & iteratively)
gives probability distr’n of states after t time steps, when transition matrix P does not change with t; (in this case steady-state may exist as well).
and
P t= P t-1 Pt (applied numerically & iteratively)
gives probability distr’n of states after t time steps, when transition matrix Pt changes with t.

20. BUT: Dimensional Challenge
No. of states =
E.g. When c=4 (servers), L=30 (max no. in system) and m=10 (no. of values taken by service time)
Transition matrix will be 19,591 x 19,591!
Whereas when c=4, L=30 and m=3
Transition matrix will be 425 x 425
Can we reduce m whilst maintaining the accuracy of the discrete time model?

21. Well-known Pollaczek-Khintchine result for M/G/1 queues
INSIGHTS:
For given l, mean service time (1/m) has 1st order effect;
And Var(st) has 2nd order effect.
Maybe higher moments of service time can be ignored.
Where:
l= arrival rate;
1/m = mean service time;
Var(st) = variance of service time.

22. 3 equations in (m+1) unknowns
Select discrete time distribution by Matching mean and variance of actual service times
discrete distribution which takes the values v, 2v, .....mv with probabilities p1, p2, .....pm
is chosen to match the mean 1/m and variance Var(S) by solving:
3 equations in (m+1) unknowns

23. e.g. Two discrete service time distributions which match in the first two moments.

24. Time-dependent behaviour of two M(t)/G/4 queues whose service times match in the first two moments

25. ‘Correct’= Discrete time modelling (DTM)

26. DTM Results
The Finite Capacity Multi-Server Queue with Inhomogeneous Arrival Rate and Discrete Service Time Distribution: and its Application to Continuous Service Time Problems - European Journal of Operational Research, 1991, vol 50, pp
Queueing Models for Out-patient Appointment Systems: a Case Study - Journal of the Operational Research Society, 1991, vol 42, pp
Practical Methods for Queue Length Behaviour for Bulk Service Queues of the form M/Go,c/1 and M(t)/Go,c/1, European Journal of Operational Research, 1994, Vol. 74, pp
Using Discrete Distributions to Approximate General Service Time Distributions in Queueing Models, Journal of Operational Research Society, 1994, vol 45, pp
Using the Discrete Time Modelling Approach to Evaluate The Time-Dependent Behaviour of Queueing Systems, Journal of Operational Research Society, 1999, 50(8), pp
A New Model for Call Centre Queue Management, Journal of Operational Research Society, 2004, Vol 55, pp
Time-Dependent Analysis of Virtual Waiting Time Behaviour in Discrete Time Queues, European Journal of Operational Research, 2007, vol 178, pp

27. DTM Software
DTM software has been developed to solve the M(t)/G/S(t) to sufficient accuracy for most/all practical problems.
Basis of method is to approximate service time distribution with an ‘equivalent’ discrete distribution;
Method has computational constraints, but they continue to recede.

28. Queue Modelling Task: The Scenario
NHS Direct is a 24-hour telephone helpline which members of the public can call for advice from specially trained health advisors for advice about health problems and their treatment
Use the DTM queue modelling software (instructions attached) to answer the following questions

29. Running DTM 1. Decide on suitable time unit 2. Prepare: 3. Prepare:
Filenam1.DAT
by editing ES.DAT in Notepad
3. Prepare:
Filenam2.SER
by running front end of DTM.EXE
3 cont’d. Run: DTM.EXE
4. Output 1:
Filenam1.QLS
(Queue Length Stats)
Open & analyse in Excel
?. Output 2:
Filenam1.SUS
(Server Utilisation Stats)
Open & analyse in Excel

30. e.g. mean no. in queue at each time
Filenam1.QLS in Excel
e.g. mean no. in queue at each time