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Time-dependent queue modelling

This outline discusses the motivation behind time-dependent queue modelling, various analytic methods including discrete time modelling, and the insights gained from such approaches. It explores the concept of time-dependent behavior in queueing systems and the limitations of steady-state models.

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Time-dependent queue modelling

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  1. Time-dependent queue modelling Dave Worthington Management Science Dept. Lancaster University d.worthington@lancaster.ac.uk

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

  3. Motivation • Many real-life queueing problems exhibit time-dependentbehaviour. • 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-dependentbehaviour, 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: PointwiseStationary 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 andFC(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. l2(t) For example: 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 RechercheOperationnelle; • 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. 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 Ptbe the vector of probabilities at time t, then: Pt= Pt-1P(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 Pt= Pt-1Pt(applied numerically & iteratively) gives probability distr’n of states after t time steps, when transition matrix Ptchanges 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. Select discrete time distribution by Matching mean and variance of actual service times • discrete distribution which takes the values v, 2v, .....mvwith probabilities p1, p2, .....pm • is chosen to match the mean 1/mand 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 - EuropeanJournal of Operational Research, 1991, vol 50, pp 310-324 • Queueing Models for Out-patient Appointment Systems: a Case Study - Journal of the Operational Research Society, 1991, vol 42, pp 733-746 • 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 103-113 • Using Discrete Distributions to Approximate General Service Time Distributions in Queueing Models, Journal of Operational Research Society, 1994, vol 45, pp 1398-1404 • Using the Discrete Time Modelling Approach to Evaluate The Time-Dependent Behaviour of Queueing Systems, Journal of Operational Research Society, 1999, 50(8), pp777-788. • A New Model for Call Centre Queue Management, Journal of Operational Research Society, 2004, Vol 55, pp1352-1357 • Time-Dependent Analysis of Virtual Waiting Time Behaviour in Discrete Time Queues, European Journal of Operational Research, 2007, vol 178, pp482-499

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

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