Asymptotic behaviour of waiting times in queueing systems with accumulating priorities

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systems with accumulating priorities

Asymptotic behaviour of waiting times in queueing

Academic year 2019-2020

Master of Science in Industrial Engineering and Operations Research Master's dissertation submitted in order to obtain the academic degree of

Counsellor: Prof. dr. ir. Stijn De Vuyst

Supervisors: Prof. dr. ir. Sabine Wittevrongel, Prof. ir. Joris Walraevens

Student number: 01501578

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systems with accumulating priorities

Asymptotic behaviour of waiting times in queueing

Master of Science in Industrial Engineering and Operations Research Master's dissertation submitted in order to obtain the academic degree of

Counsellor: Prof. dr. ir. Stijn De Vuyst

Supervisors: Prof. dr. ir. Sabine Wittevrongel, Prof. ir. Joris Walraevens

Student number: 01501578

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”The author gives permission to make this master dissertation available for consultation and to copy parts of this master dissertation for personal use. In all cases of other use, the copyright terms have to be respected, in particular with regard to the obligation to state explicitly the source when quoting results from this master dissertation.”

Thomas Van Giel, June 2020

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Preface

This report is the final version my master’s dissertation, in order to obtain the academic de-gree of Master of Science in Industrial Engineering and Operations Research. In this report the asymptotic behaviour of a so called ”Accumulating priority queue” is researched. More specifically, it is compared to a regular FIFO queue and an absolute-priority queue. First of all I want to thank my supervisors: Prof. dr. ir. Sabine Wittevrongel and Prof. dr. ir. Joris Walraevens, for helping me in succeeding to bring the research to a good end. I also want to thank them for explaining some of the key concepts of queueing theory I needed to understand the literature and doing my own calculations and research. They were always ready to answer any questions I had, and helped a lot with the proofreading.

Secondly, I want to thank my counsellor Prof. dr. ir. Stijn De Vuyst, for showing me this opportunity, and peaking my interests in this subject. He did this during the thesis information moment on the ”industry day”.

At last I want to thank all my friends, parents, ... who, despite not always understanding what I was talking about, kept listening to my ramblings and reasoning, and helped me to find where they were wrong. Just by talking to them I made a lot of new breakthroughs. I especially want to thank Hans A. Kusmanto aiding me with some of these reasonings.

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ii

Asymptotic behaviour of waiting times in

queueing systems with accumulating

priorities

by

Thomas Van Giel Student number: 01501578

Master’s dissertation submitted in order to obtain the academic degree of Master of Science in Industrial Engineering and Operations Research

Supervisors: Prof. dr. ir. Sabine Wittevrongel, Prof. dr. ir. Joris Walraevens Counsellor: Prof. dr. ir. Stijn De Vuyst

Faculty of Engineering and Architecture Ghent university

Abstract

This thesis looks into the asymptotic behaviour of a queue with accumulating priorities in time, also called the accumulating priority queue (APQ). To find the asymptotic behaviour of this queue, the waiting-time distributions derived in Stanford et al. (2014) are used. In the first part of this thesis, that article is explained in a sometimes more intuitive manner, as the reader needs a good understanding of the article to understand the rest of this thesis. In the second part the derived models are used on two well known kinds of queues: a FIFO queue and an absolute priority queue. The application of our model onto these queues is then compared to already known results, as in Abate & Whitt (1997). This is to get a better understanding of the model and some parameters. The third part covers the location of the singularities and the asymptotic behaviour around these singularities is explored for the general model. The last part is a small part that looks into the influence of the different parameters on the different singularities.

Keywords

Priority queues · Time-dependent priority · Accumulating priority · Asymptotic waiting time

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Asymptotic behaviour of waiting times in queueing

systems with accumulating priorities

Thomas Van Giel

Supervisor(s): Joris Walraevens, Sabine Wittevrongel, Stijn De Vuyst

Abstract— This article tries to find the asymptotic be-haviour of the waiting-time distribution of the accumulating priority queue with 2 classes of customers.

Keywords— Priority queues · Time-dependent priority · Accumulating priority· Asymptotic waiting time

I. Introduction

Q

UEUES are found everywhere around us. From the cash registers in a shop, to roller coasters in a theme park. There exist multiple different queues, each with their own application. Most often we find FIFO-queues. The first customer to arrive in this queue, is the first to enter service. Another type of queue possibility is a multi-class absolute priority queue. Here, multiple classes of customers can arrive in the queue, and a higher class customer will always enter service before a lower class customer. In this paper, an accumulating priority queue is looked into. In this kind of queue, customers gain priority linearly based on how long they have waited in the queue. In this thesis, there are two classes of customers, each with their own rate of priority accumulation.

A. Base model

The used model is based on the model derived in [1]. In this article, a model is set up for an accumulating priority queue (APQ). An an APQ, customers gain priority based on how long they have stayed in the queue. This rise in pri-ority is linear in time. When the server becomes free, the customer with the highest accumulated priority is served. There are two classes of customers: class 1 and class 2, each with their own priority accumulation rate b1 and b2. Since

only the ratio between these two is of importance, b1 is set

to 1. Arrivals in the queue for both classes happen accord-ing to a Poisson process with rate λ1 for class-1 customers

and λ2for class-2 customers. The service times happen

ac-cording to general distributions B(1)_{(t) with average rate}

µ1 for class-1 customers and B(2)(t) with average rate µ2

for class-2 customers. This means we are working with an M/G/1 APQ. It is assumed that the queue is stable, i.e. ρ = λ1/µ1+ λ2/µ2 < 1. Figure 1 plots the accumulated

priorities of customers against time for a set of arrivals. The dotted vertical lines signal a departure, while the high-lighted lines are the priorities of the customer currently in service.

B. Goals

The goal of this article is to find the asymptotic be-haviour of the waiting times in the case of an M/G/1 APQ.

Fig. 1. Priority over time

We do this by finding the singularities of the Laplace-Stieltjes transforms(LST) of the distribution functions of the waiting times, and finding the behaviour of these LSTs around these points. These LSTs themselves have been drawn up in [1]. The most important part is the rightmost singularity, since that equates to the slowest decaying ex-ponential function in the time domain.

II. Waiting-time equations as described by [1] A. class-2 waiting-time equation

The first thing that is needed is an equation for the waiting-time distribution LSTs. This is given in [1]. Let

˜ Γ(1)(s) = ˜B(1)s + λ1(1− b2) (1− ˜Γ(1)(s)) , and let ˜ Γ(2)(s) = ˜B(2)s + λ1(1− b2) (1− ˜Γ(1)(s)) . Then these functions are called Accreditation periods started by a class-1 and class-2 customer respectively. They behave like a busy periods from a queue with service-time distribtuion ˜B(1)_{(s) and arrival rate λ}

1(1− b2) [2]. The

following function is the LST of the accumulated priority in an accreditation interval: ˜ V (s; b1, b2, λ1, B, B0) = κ (˜Γi(b2s)− ˜Bi(b1s)) (b1s− λ1(1− ˜B(b1s))) . Here κ is a normalising constant which causes ˜V (s = 0) = 1.

The class-2 waiting time LST is then given by: ˜

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with

˜

V(2)(s) = ˜V (s; b2, 0, λ∗, ˜Γ2, ˜Γ0),

where ˜V(2)_{(s) is the LST of the accumulated priority by a}

so called unaccredited customer at the start of its service. It is evident that ˜W(2)_{(s) will have the same singularities}

as ˜V(2)_{(s). This function has two possible singularities:} • −s(2)_denom∗ : the rightmost zero of the denominator of

˜ V(2)_(s/b

2),

• −τ1−1: the rightmost singularity of ˜Γ(1)(s).

B. Class-1 waiting-time equation

The class-1 waiting time is more complex, and is given by ˜ W(1)(s) = (1− ρ) + ρ b2V˜(2)(s) +(1− ρ)(1 − b2) (1_{− σ}1) ˜ V(1,0)(s) + (ρ− σ1)(1− b2) (1_{− σ}1) ˜ V(2)(s) ˜V(1,1)(s) . It is again evident that ˜W(1)_{(s) will only have}

singular-ities from ˜V(2)_{(s), ˜}_V(1,0)_{(s) and ˜}_V(1,1)_{(s). ˜}_V(1,0)_{(s) and}

˜

V(1,1)_{(s) are given by:}

˜

V(1,0)(s) = ˜V (s; b1, b2, λ1, B(1), B(2)0 ),

˜

V(1,1)(s) = ˜V (s; b1, b2, λ1, B(1), B(2)2 ).

Togheter, these cause 4 possbile dominant singularities in ˜

W(1)_(s).

• −s(2)∗_denom/b2: the rightmost zero of the denominator of

˜ V(2)_(s),

• −τ1−1/b2: the rightmost singularity of ˜Γ(1)(b2s),

• −s(1)_denom∗ : the rightmost zero of the denominator of

˜

V(1,0)_{(s) and ˜}_V(1,1)_{(s) for s < 0,}

• −s∗_B2: the rightmost singularity of ˜B(2)(s), a pole of

˜

V(1,0)(s) and ˜V(1,1)(s).

For exponential service times, _−τ1−1/b2 will never be the

rightmost singularity as_−s(1)∗_denom will always dominate it. III. Known models

First of all these findings are compared to two already well-known models: the FIFO queue (b2= 1) and the

ab-solute priority queue (b2= 0). From this we can get more

information about the different singularities. −s(2)denom∗ is

the same pole as the pole of the class-2 waiting time in the absolute priority queue. It also turns into the pole of the FIFO queue for b2 = 1. −τ1−1 is the same as the possible

branch point in the class-2 waiting time in the absolute pri-ority queue. −s(1)denom∗ is the same as the pole in the class-1

waiting time pole in the absolute priority queue. −s∗ B2 is

the possible ˜B(2)_{(s) dominance in the class-1 waiting time}

in the absolute priority queue.

IV. asymptotic behaviour in the general model The asymptotic behaviour of the waiting times in the general model is similar to that in the absolute priority

queue in [3]. Using the findings in that article, the asymp-totic behaviours around the singularities and the time do-main could be determined. These formulas are very con-voluted. Because of this only the equation number in the article will be given.

For the class-2 waiting-time distribution,

• if−s(2)_denom∗ is dominant, the asymptotic waiting-time

be-haviour is as in equation (4.11).

• If −τ1−1 is the dominant singularity, the asymptotic

waiting-time behaviour is as equation (4.27). For the class-1 waiting-time distribution,

• if−s(2)_denom∗ /b2is the dominant singularity, the asymptotic

waiting-time behaviour is as in equation (5.11),

• if −τ1−1/b2 is the dominant singularity, the asymptotic

waiting-time behaviour is as in equation (5.24).

• if −s(1)_denom∗ is the dominant singularity, the asymptotic

waiting-time behaviour is as in equation (5.29),

• if −s∗B2 is the dominant singularity, the asymptotic

waiting-time behaviour is as in equation (5.31). V. influence of parameters

In the last section, the influence of the parameters on the location of the different singularities is looked into. No cut-off values where one singularity dominates over the others have been found for these parameters, but general trends have been found. This results in table 1.

TABLE 1

Influence of different parameters on dominance of the singularity b2 µ2 ρ1 ρ2 −s(2)denom∗ /b2 ++ - ++ ++ −τ1−1/b2 + / + / −s(1)denom∗ / / + / −s∗B2 / - - / / VI. conclusions

The class-1 waiting time’s singularities are a mix of the class-1 absolute priority singularities and the class-2 abso-lute priority singularities. Depending on the parameters, one of these may dominate.

References

[1] David A. Stanford, Peter Taylor, and Ilze Ziedins, “Waiting time distributions in the accumulating priority queue,” Queueing Syst, vol. 77, pp. 297–330, 2014.

[2] R.W. Conway, W.L. Maxwell, and L.W. Miller, Theory of Scheduling, Addison-Wesley, Reading, 1967.

[3] Joseph Abate and Ward Whitt, “Asymptotics for M/G/1 low-priority waiting-time tail probabilities,” Queueing Systems, vol. 25, pp. 173–233, 1997.

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Symbols

λi The mean arrival rate of class-i customers.

µi The mean service rate of class-i customers.

ρi λi/µi.

ρ ρ1+ ρ2.

b2 priority rate for class-2 customers.

B(i)(t) The service-time distribution for a class-i customer.

G(i)_j (t) The busy-period distribution for a class-i customer, started by a class-j customer.

˜

Γ(i)(s) The LST of the distribution of the duration of an accreditation period, started by a class-i customer.

W(i)(t) The waiting-time distribution of a class-i customer.

V(i)(t) The accumulated-priority distribution when a class-i customer enters service.

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Contents viii

Writing conventions

˜

F (s) The LST of F(t)

subscript i The LST of a busy/accreditation period started by a different class customer than all subsequent customers. subscript 0 at the start of a busy period.

subscript 2 in the middle of a busy period. superscript (i) references class-i customers.

superscript (1, 0) For accredited customers in an accreditation period at the start of a busy period.

superscript (1, 1) For accredited customers in an accreditation period in the middle of a busy period.

−s∗ _{denotes a singularity.}

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Contents ix

Important formulas

Sym-bol formula definition

equa-tion nr ˜ B₀(2) λ1B(1)+λ2B(2) λ1+λ2 Service-time distribution of an unaccredited customer at the start of a busy period. (2.3) ˜ B₂(2) λ1b2B(1)+λ2b1B(2) λ1b2+λ2b1 Service-time distribution of an unaccredited customer in the middle of a busy period.

(2.4) −ζ1 Solution to − ˜B(1)(s) = 1/(λ1(1 − b2)) (4.13) −τ₁−1 −ζ1− λ1(1 − b2) (1 − ˜B(1)_(−ζ 1))

The rightmost singularity of ˜Γ(1)(s) (4.14)

Abbreviations

LST Laplace-Stieltjes transform APQ Accumulating priority queue

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Chapter 1 Introduction

Queues can be found all around us, in a broad range of applications. Ranging from hospitals and theme parks, where the queues are human queues, to factories with queues of items and products, all the way to queues in data processing and communication technology, where queues consist of data packets. Because queues are omnipresent, engineers want to be able to model these queues, and analyse their behaviours with mathematical tools. For this, an entire discipline of queueing theory has been founded.

For each situation, a different type of queue can be used. The simplest of these queues is the First-In-First-Out queue (FIFO). The concept of this queue is that the first customer to arrive is the first customer to be served. Sometimes however, not every customer should be treated equally. One kind of queue that does this is the absolute-priority queue. This kind of queue has multiple classes of customers. If the server becomes free, the waiting customer of the highest class is taken. If multiple customers of that class are present, the FIFO principle will be used for these customers.

One of the shortcomings of such an absolute-priority queue is that the service level for the high priority customers may be higher than necessary, while the service level for low priority customers may be inadequate. Therefore, a queue with less strict priorities can be used. One of the possibilities is a time-dependent priority queue, where the priority of a customer increases with the time spent in the queue. The proposal of such a model was in Kleinrock & Finkelstein (1967). In this article the queue is referred to as the time-dependent priority queue. Many others have followed and did research into the behaviour of such kind of queue, like in Stanford et al. (2014), or even more recently Mojalal et al. (2020). In the

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Chapter 1. Introduction 2

newer research to this kind of queue, the term accumulating priority queue (APQ) is used. We shall use this term to refer to this kind of queue as well.

Table 1.1: Canadian Triage and Acuity Scale Category Classification Access

1 Resuscitation immediate

2 Emergency 15 min

3 Urgent 30 min

4 Less urgent 60 min

5 Not urgent 120 min

One of the places where such a queue can be used is in the medical world. More specifically in the emergency waiting room. In many countries around the world, arriving patients are classified according to the severety and urgency of their issues. For example, the Canadian Triage and Acuity Scale Montfort (2020), as in table 1.1 is used by the canadian medical sector to categorise incomming patients. A goal for access times is also given. Another example is the Medical Priority Dispatch System (MPDS): the system used to dispatch appropriate aid to medical emergencies in the United Kingdom.

Table 1.2: MPDS categorisation

Category Classification Target Response Time

1 Life Threatening Illnesses or injuries Within 15 minutes 90% of the time 2 Emergency Calls Within 40 minutes 90% of the time

3 Urgent Calls within 2 hours 90% of the time

4 Less urgent Calls within 3 hours 90% of the time

5 Not urgent Calls N/A

In the MPDS system, a goal is specified for timeliness. These systems exist for the fire department (FPDS, 2020) and the police department (prioritydispatch, 2020) as well. In

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Chapter 1. Introduction 3

this thesis the 2-class case is treated. Moreover, arrivals are assumed to happen according to a Poisson process with a different rate for each class, and the service times happen according to general distributions with a different rate for each class. It is also assumed that the queue is stable.

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Chapter 2 Waiting-time distributions in the

accumulating priority queue

In this chapter, a summary is given of Stanford et al. (2014), and some quick, intuitive explanations are given for some of the most important parts. For the full proofs, and more information, the original article should be checked. The last subsections of this chapter will yield the Laplace-Stieltjes transforms (LST) of the waiting time distributions both for class-1 and class-2 customers. The type of queue that is researched is a 2-class queue with accumulating priority. Class-1 is the high-priority class. The arrivals happen according to a Poisson process, with rate λ1 for customers of class-1 and λ2for customers of class-2. The

service times are distributed according to a general distribution, with rate µ₁ for customers of class-1 and µ2 for customers of class-2. While waiting in the queue, customers gain

priority linearly in time. Customers gain priority with a constant rate b₁ and b₂ for class-1 and class-2 customers respectively, with b1 > b2. This means that a class-1 customer will

have the highest priority more often than a class-2 customer, but a long waiting class-2 customer may have priority over a class-1 customer. Whenever the server is done with its previous service, the customer with the highest priority will be taken into service.

In figure 2.1 an example of the priority in function of time for an accumulating priority queue is given. The different lines in the figure show the evolution of the accumulated priority for each of the customers in the system, from their arrival to their departure time. The parts where a customer is in service are highlighted. Arrivals occur at t=(1, 3, 10, 15, 17), and departures at t=(14, 21, 23, 26, 31). At t=14 we can see that even though there is a class-1 customer in the queue, the class-2 customer that arrived at t=3 has priority. In

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Chapter 2. Waiting-time distributions in the accumulating priority queue 5

Figure 2.1: Priority over time for b1 = 1, b2 = 0. (Stanford et al., 2014)

all other cases the class-1 customers have priority in this example.

2.1 Notation and assumptions

For the notations we will follow Stanford et al. (2014).

If there exists a variable X, distributed according to cumulative distribution function F (x), let the Laplace-Stieltjes transform (LST) of F (x) be ˜F (s) = E(e−sX) = R∞

0 e

−sx_{dF (x),}

defined for all s with Re(s) > 0, and for at least some s with Re(s) ≤ 0. Some important variables are:

• Let B(i)_{(t) be the service-time distribution of a customer of class i. Let the mean of}

B(i)(t) be 1/µi.

• Let G(i)_j be the busy-period distribution for a class-i customer, started by a class-j customer.

• Let X_n be the service time of customer n. X_n is distributed according to B(i)(t) for a class-i customer.

• Let b_i be the priority accumulation rate for a class-i customer. • Let Vn(t) be the accumulated priority of customer n.

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i when they start service.

• Let n(m) be the position of arrival of the mth customer to be served1_.

• Let C_n be the time when the nth arriving customer starts service.

• Let Dn be the time when the nth arriving customer ends service: Dn = Cn+ Xn.

This is called a stopping time.

In this entire paper it is assumed that the queue is stable. This means that ρ = λ1

µ1 +

λ2

µ2 <

1.

2.2 Maximum priority process

Firstly, a new function is defined called the Maximum priority process. This function M = {(M1(t), M2(t))} is an upper bound on the maximum priorities of both classes. M1(t) is

an upper bound on the priority for all class-1 customers at time t, and M₂(t) is an upper bound to the priority for all class-2 customers. Note that the Maximum priority process is a property of the entire queue, not for one customer.

Definition 2.1. The maximum priority process M(t) = {(M1(t), M2(t))} is defined as

follows:

1. If the queue is empty at time t:

M1(t) = M2(t) = 0

2. At a departure time (D_m, m = 0, 1, 2, ..): M1(Dn(m)) = max

n /∈{n(i):1≤i≤m}Vn(Dn(m)) (2.1)

M2(Dn(m)) = min{M1(Dn(m)), M2(Cn(m)) + b2Xn(m)} (2.2)

These times D_m are the stopping times of the system. 3. for a time t during service, or t ∈iCn(m), Dn(m)

h :

Mi(t) = Mi(Cn(m)) + bi(t − Cn(m))

1_{n is used to denote in which order customers arrive, m is used to denote in which order customers are}

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Figure 2.2 shows the maximum priority process. Here functions M1(t) and M2(t) are

high-lighted in blue and green respectively. Equation (2.1) can be interpreted as the maximum priority of customers that haven’t left service yet. The minimum-function in equation (2.2) causes the function to never make jumps upwards, only downwards. This is important at points such as t = 21 in figure 2.2. At this point there is a class-1 customer with a priority bigger than the maximum possible priority of any class-2 customer.

Figure 2.2: Maximum priority process (Stanford et al., 2014)

Due to the fact that arrivals are distributed as independent Poisson processes, the PASTA properties can be applied to the queue. With this, it can be reasoned that the accumulated priorities are also distributed according to independent Poisson processes.

Theorem 2.1. Let t ∈ [0, ∞[

1. Conditional on M(t), the accumulated priorities {V_ki(t), k = 1, 2, ..} of the customers still waiting from class i; i = 1, 2 are distributed as independent Poisson processes with rate λ_i/bi on the intervals [0, Mi(t)].

2. Conditional on M(t), the accumulated priorities {V_ki(t), k = 1, 2, ..} of all customers still present in the queue are distributed as a Poisson process with piecewise constant rates zero on the interval [M1(t), ∞], λ1/b1 on the interval [M2(t), M1(t)] and λ1/b1+

λ2/b2 on the interval [0, M2(t)].

3. A waiting customer with priority V ∈ [0, M₂(t)] is of class-1 with probability λ1b2

λ1b2+λ2b1

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4. The statements 1-3 above also hold at any random time T that is a stopping time with respect to M(t).

The proof of theorem 2.1 can be found in Stanford et al. (2014, Theorem 3.2). The important part is that the accumulated priorities are also distributed according to independent Poisson processes.

2.3 Accreditation

Definition 2.2. An accredited customer n is a customer where V_n(t) ∈]M₂(t), M₁(t)].

Remark 1. The open bracket indicates that when the queue is idle, the first customer to

arrive is not an accredited customer.

Note that only class-1 customers can become accredited, since M2(t) is an upper bound on

the priority of all class-2 customers. Whenever a class-1 customer has become accredited, it will stay accredited until it leaves the queue.

An unaccredited customer who enters service can either be of class 1 or class 2. This can be followed by a sequence of accredited customers. We call such an interval, started by an unaccredited customer and followed up by a series of accredited customers, an accreditation interval. This is, an accreditation interval is started when an unaccredited customer enters service, and stops when the first unaccredited customer after that starts service. An unac-credited customer m that isn’t followed by acunac-credited customers is an accreditation interval of duration X_m. If an unaccredited customer m is followed by a series of exactly k accredited customers, the total duration of the accreditation interval equals Pm+k

c=mXc.

The maximum priority process during an accreditation interval is depicted in figure 2.3. The accreditation interval starts with an unaccredited customer with priority Vinit. The

maximum priority M2(t) rises with slope b2. class-1 customers become accredited at times

t = s1 and t = s2. Note that the start of a busy period is also the start of an accreditation

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Figure 2.3: The maximum priority process during an accreditation interval

What follows are some important theorems and lemmas to work out the waiting times. All proofs can be found in Stanford et al. (2014). A brief intuitive interpretation will be provided.

Lemma 2.2. During an accreditation interval, the time points sk at which customers

be-come accredited occur according to a Poisson process with rate λ₁(1 −b2

b1).

Intuitive interpretation. Only class-1 customers can become accredited. They only become accredited when their accumulate priority becomes higher than the maximum possible pri-ority M₂(t) of class-2. The class-2 maximum priority rises with rate b₂, and the class-1 customer priority rises with rate b1. From there it can be reasoned that the rate at which

customers become accredited is λ₁(1 − b2

b1). Since all arrivals are Poisson processes, it can

also be reasoned that the accreditation times are Poisson processes. From this, we can look at accredited customers as arriving in a different queue with rate λ1(1 −b_b2₁).

Definition 2.3. Let B₀(2)= λ1B (1)_{+ λ} 2B(2) λ1+ λ2 , (2.3) B₂(2)= λ1b2B (1)_{+ λ} 2b1B(2) λ1b2+ λ2b1 . (2.4)

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B₀(2)(t) and B(2)₂ (t) are the service-time distributions of an unaccredited customer at the start and in the middle of a busy period respectively. B(2)₀ (t) is a weighted sum of all customers and their service-time distributions. B(2)₂ (t) is a weighted sum of all unaccredited customers and their service-time distributions.

Lemma 2.3. The durations of the accreditation intervals are independent random variables

whose distributions depend on V_init only via I(V_init> 0).

Intuitive interpretation. Lemma 2.3 states that the only dependence that an accreditation interval has on V_initis whether V_initis zero (start of busy period), or V_initis bigger than zero (middle of a busy period). This is because a customer starting an accreditation period at the start of a busy period has a service-time distribution B₀(2)(t), while a customer starting an accreditation period in the middle of a busy period has service-time distribution B(2)(t).

Remark 2. Theorem 2.1(2) tells us that, at time t during a busy period, the priorities

of customers in the interval [0, M₂(t)[ are distributed according to a Poisson process with rate λ1/b1+ λ2/b2. These priorities are generated by a mixture of class-1 customers that

have been arriving at rate λ₁ over the time interval ]t − M₂(t)/b₁, t], and class-2 customers that have been arriving at rate λ₂ over the time interval ]t − M₂(t)/b₂, t]. However, the distribution of priorities is the same as if customers had arrived with a rate of λ2+ λ1b2/b1,

while all gaining priority at a rate b₂. This will be important when finding the waiting times for an unaccredited customer.

2.4 LST of Accreditation interval distribution

As mentioned in a lot of literature about queuing systems, such as Conway et al. (1967), the LST of the distribution of the length of a busy period ˜G(s) in an M/G/1 queue with arrival rate λ and service-time distribution LST ˜B(s) can be obtained by solving the functional equation

˜

G(s) ≡ ˜G(s; λ, B) = ˜B(s + λ(1 − ˜G(s))). (2.5) The LST of the duration of a busy period initiated by a customer with service-time distri-bution B_i(t) and followed up by customers with service-time distribution B(t) is given by the functional equation

˜

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where ˜G(s) is the solution to (2.5).

In Stanford et al. (2014), it is observed that an accreditation interval behaves the same way as a FIFO busy period, as described by (2.5) and (2.6), but with an arrival rate of λ1(1 −b_b2₁). This means that the LST of the accreditation interval duration can be obtained

by solving the functional equation ˜ Γ(s) ≡ ˜Γ(s; b₁, b2, λ1, B) = ˜B s + λ1 1 −b2 b1 (1 − ˜Γ(s)) (2.7)

when the starting service-time distribution is B(t), and

˜ Γ_i(s) ≡ ˜Γ_i(s; b₁, b2, λ1, B, Bi) = ˜Bi s + λ1 1 −b2 b1 (1 − ˜Γ(s)) (2.8) when the starting service-time distribution is Bi(t), followed by service time distributions

B(t).

Intuitive interpretation. Since the accreditation times sk are distributed according to a

Poisson process with rate λ₁(1 − b₂/b1) , the accredited individuals can be seen as a queue

with arrival rate λ1(1 − b2/b1).

Now the actual length of the APQ accreditation interval can be defined. The LST of distribution of the duration of an accreditation interval initiating a busy period and an interval in the middle of a busy period equal

˜

Γ0(s) = ˜Γi(s; b1, b2, λ1, B(1), B0(2)) (2.9)

˜

Γ2(s) = ˜Γi(s; b1, b2, λ1, B(1), B2(2)) (2.10)

respectively, with ˜Γi as in (2.8).

2.5 LST of waiting times and priorities

Let W(1)(t) and W(2)(t) be the class-1 and class-2 waiting-time distributions respectively, and let V(i)(t) be the accumulated-priority distribution of a class-i customer starting service. Since priority rises linearly with the waiting time, i.e. W(i)(t) = V(i)(t)/bi; i = 1, 2, the

LST of the waiting-time distribution for class-1 ˜W(1)(s) can be expressed quite easily as a function of ˜V(1)(s), and the LST of the waiting-time distribution for class-2 as a function of ˜V(2)(s). This means that, if the LST for the distribution of the priority has been found,

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the LST of the distribution of the waiting time can be deduced. The different priority LSTs are given in the following theorems.

Theorem 2.4. For an accreditation interval with parameters b1, b2, λ1 and B, that starts

at time t₀ with initial priority level V_init, let ˆV = Vinit+ V denote the accumulated priority

of customers at the point that their service starts.

1. The distribution of V , conditional on V_init= v has LST

˜ V∗(s; b₁, b2, λ1, B) = µ − λ1 1 −b2 b1 (˜Γ(b2s) − ˜B(b1s)) 1 −b2 b1 (b1s − λ1(1 − ˜B(b1s))) , (2.11)

2. The random variable V is independent of Vinit.

With ˜Γ(s) as in (2.7). This denotes the priority during an accreditation interval.

Theorem 2.5. If the initial service time distribution Bi differs from the service time

distri-bution of the subsequent customers within the accreditation interval, the LST of the priority accumulated during the interval is

˜ V (s; b1, b2, λ1, B, Bi) = µ0 1 − λ₁1 −b2 b1 /µ(˜Γ_i(b₂s) − ˜Bi(b1s)) 1 −b2 b1 (b₁s − λ1(1 − ˜B(b1s))) . (2.12)

We know that the initial service-time distribution Bi is the service-time distribution of an

unaccredited customer. These are B₀(2) or B(2)₂ as in (2.3) and (2.4) respectively, depending if it is at the start or in the middle of a busy period.

An accreditation interval behaves like a busy period. The inverse is also true. If accredi-tation happens whenever a customer enters the queue, we have a busy period that is also an accreditation interval. We do this by setting the slope of M₂(t) equal to 0. If we use the observations from remark 2 where the arrival rate from non-accredited customers can be transformed to λ2+ λ1b2/b1≡ λ∗, with every unaccredited priority rate equal to b2 and

service time distributions ˜Γ₂ and ˜Γ₀, then we can observe that the busy period has LST ˜

Γi(s; b2, 0, λ∗, ˜Γ2, ˜Γ0).

Intuitive interpretation. Each accreditation interval starts with a non-accredited customer. This means that an accreditation interval can be seen as one “service”, with service-time distribution equal to the distribution of an accreditation interval (˜Γ₂ and ˜Γ₀), with an arrival rate equal to the equivalent rate of non-accredited customers: λ∗.

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The LST of the stationary accumulated priority of the non-accredited customers at the time they enter service, conditional on it being positive, also follows from the above observations, i.e. the parameters for the priority are also b2, 0, λ∗, ˜Γ2 and ˜Γ0. This helps us to arrive at

the LST of the priority distribution for non-accredited customers: ˜

V(2)(s) = ˜V (s; b2, 0, λ∗, ˜Γ2, ˜Γ0) (2.13)

for ˜V as in (2.12).

A class-2 customer with priority v has been waiting in the queue for time v/b2. Thus the

LST of the stationary waiting time for class-2 customers is given by the weighted sum of the LSTs of zero and ˜V(2)(s/b₂).

˜

W(2)(s) = (1 − ρ) + ρ ˜V(2)(s/b₂) (2.14)

For a class-1 customer, multiple scenarios are possible. 1. It arrives to an empty queue.

2. It arrives to a non-empty queue, and is not accredited when it enters service. In this case the LST of its stationary accumulated priority on entering service is ˜V(2)(s), since the class of a non-accredited customer is independent of its priority (theorem 2.1(3)). 3. It enters service during the first accreditation interval of the busy period, in which

case its stationary priority has LST ˜

V(1,0)(s) = ˜V (s; b1, b2, λ1, B(1), B0(2)). (2.15)

4. It enters service during an accreditation interval which is started during a busy period. This means that V_init > 0, which means that the arriving customer accumulates priority V_init which has LST ˜V(2)_{, and on top of that priority with LST}

˜

V(1,1)(s) = ˜V (s; b1, b2, λ1, B(1), B2(2)). (2.16)

This means that the total priority in this case has LST V(2)(s)V(1,1)(s)

The full LST of the priority of a class-1 customer conditional on it being positive is then a weighted sum (to see probability calculations, see Stanford APQ paper Stanford et al. (2014)), of the LSTs of the different cases.

˜ V(1)(s) = b2 b1 ˜ V(2)(s) +(1 − ρ)(b1− b2) b1(1 − σ1) ˜ V(1,0)(s) +(ρ − σ1)(b1− b2) b1(1 − σ1) ˜ V(2)(s) ˜V(1,1)(s), (2.17)

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with σ1 = ρ1(b1− b2)/b1.

The waiting-time LST becomes ˜

W(1)(s) = (1 − ρ) + ρ ˜V(1)(s/b1). (2.18)

2.6 Further improvements

For the asymptotic expansion, it is important to look at the rightmost singularity −s∗. Since waiting times are completely dependent on variations of the accumulated priority, the only singularities of the waiting time are those of the priority. The equation for the accumulated-priority LST when service-time distribution of the first customer in an accreditation interval differs from the subsequent customers is repeated here for convenience. ˜Bi(s) is the LST of

the initial service-time distribution. ˜ V (s; b1, b2, λ1, B, B0) = µi 1 − λ1 1 −b2 b1 /µ(˜Γi(b2s) − ˜Bi(b1s)) 1 −b2 b1 (b1s − λ1(1 − ˜B(b1s))) (2.12) The term µi 1−λ1 1−b2_b1 /µ 1−b2 b1

is a constant factor, only dependent on the parameters and not on s. If b2 = b1, the denominator becomes zero. This would just result in a regular M/G/1

FIFO queue with arrival rate λ1+ λ2, and a service time distribution B0(2) ≡ λ1B

(1)_+λ 2B(2)

λ1+λ2 .

This case has been expanded on in section 3.1. From now on we will call this term κ, and we will assume that b2 < b1, unless stated differently.

Something that can be considered is that setting b₁ = 1, and only varying b₂ between 0 and 1 can be done without loss of generality. From now on, it will always assumed that b₁= 1, and 0 < b2 < 1. By doing this, parameter b1 can be dropped.

Equation ˜V (s) can be reduced to ˜ V (s; b1 = 1, b2, λ1, B, Bi) ≡ ˜V (s; b2, λ1, B, Bi) = κ (˜Γi(b2s) − ˜Bi(s)) (s − λ₁(1 − ˜B(s))) (2.19) with κ = µi(1−λ1(1−b2)/µ) (1−b2) .

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2.6.1 Simplifying ˜V(i)_(s)

Simplifying ˜V(2)(s) We deduce from (2.13) that

˜ V(2)(s) = µ(acc)₀ 1 − λ∗/µ(acc)_{(1 − ˜}_Γ 0(s)) s − λ∗(1 − ˜Γ2(s)) . (2.20)

We know that µ(acc)₀ is the rate of ˜Γ₀(s), and µ(acc) is the rate of ˜Γ₂(s). After taking the derivative, and setting s = 0, we arrive at the following equation for µ(acc):

µ(acc)= µ2(µ1− λ1) µ1

= µ₂(1 − ρ₁).

To calculate µ(acc)₀ we need 1/µ_Idle: the mean duration of the service for a customer starting a busy period: µidle= (λ1+ λ2)µ1µ2 λ1µ2+ λ2µ1 = λ1+ λ2 ρ .

From this, we can calcultate µ(acc)₀ : µ0 = µidle(µ1− λ1) µ1 = (λ1+ λ2)(µ1− λ1) ρµ1 Since ˜Γ₀(s) = ˜B0 s + λ1(1 − b2)(1 − ˜Γ(s)) , and ˜B0(s) = λ1 ˜ B(1)_(s)+λ 2B˜(2)(s) λ1+λ2 , ˜Γ0(s) = λ1Γ˜(1)(s)+λ2Γ˜(2)(s) λ1+λ2 , where ˜Γ

(i)_{(s) is the LST of an accreditation period started by a}

class-i customer. The same reasonclass-ing can be used to express ˜Γ2(s) = λ1b2 ˜

Γ(1)(s)+λ2Γ˜(2)(s)

λ1b2+λ2 . Using

this information, ˜V(2)(s) can be transformed to

˜ V(2)(s) = κ(2) λ1(1 − ˜Γ (1)_(b 2s)) + λ2(1 − ˜Γ(2)(b2s)) b2s − λ1b2(1 − ˜Γ(1)(b2s)) − λ2(1 − ˜Γ(2)(b2s)) , (2.21) where κ(2) = 1 − ρ ρ . (2.22)

Since ˜V(i)(s) is the LST of a probability function, ˜V(i)(0) = 1, i = (2), (1, 0), (1, 1). Checking this gives us the confirmation that our constant κ(2) is correct.

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Chapter 2. Waiting-time distributions in the accumulating priority queue 16 Simplifying ˜V(1,0)(s) and ˜V(1,1)(s) ˜ V(1,0) is given by: V(1,0)(s) = µ (2) 0 (1 − λ1(1 − b2)/µ1)(˜Γ0(b2s) − ˜B(2)0 (s)) (1 − b₂)(s − λ₁(1 − ˜B(1)_(s))) . (2.23)

Here, µ(2)₀ is the rate of ˜B0(s) as in (2.3): the service-time distribution of a customer at the

start of a busy period. After some quick calculations, we find µ(2)₀ = λ1+ λ2 ρ . ˜ V(1,1) is given by: V(1,1)(s) = µ (2) 2 (1 − λ1(1 − b2)/µ1)(˜Γ0(b2s) − ˜B(2)2 (s)) (1 − b₂)(s − λ₁(1 − ˜B(1)_(s))) . (2.24)

Here, µ(2)₂ is the rate of ˜B0(s) as in (2.3): the service-time distribution of a customer in the

middle of a busy period. After some quick calculations, we find µ(2)₂ = b2λ1+ λ2

b2ρ1+ ρ2

Using the same reasoning as for ˜V(2)(s), these can be rewritten as: ˜ V(1,0)(s) = κ(1,0)λ1(˜Γ (1)_(b 2s) − ˜B(1)(s)) + λ2(˜Γ(2)(b2s) − ˜B(2)(s)) (s − λ1(1 − ˜B(1)(s))) , (2.25) with κ(1,0) = 1 − ρ1(1 − b2) ρ(1 − b2) , (2.26) and ˜ V(1,1)(s) = κ(1,1)λ1b2(˜Γ (1)_(b 2s) − ˜B(1)(s)) + λ2(˜Γ(2)(b2s) − ˜B(2)(s)) (s − λ1(1 − ˜B(1)(s))) , (2.27) with κ(1,1) = 1 − ρ1(1 − b2) (ρ1b2+ ρ2)(1 − b2) . (2.28)

2.7 Final Equations of ˜

W

(i)

(s)

Now a summary will be given of the previous findings to be able to follow the subsequent sections.

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2.7.1 Final equations of ˜W(2)_(s)

We start by the simplest of the waiting times: ˜W(2)(s).

˜ W(2)(s) = (1 − ρ) + ρ ˜V(2)(s/b₂) (2.29) = (1 − ρ) + (1 − ρ) λ1(1 − ˜Γ (1)_{(s)) + λ} 2(1 − ˜Γ(2)(s)) s − λ1b2(1 − ˜Γ(1)(s)) − λ2(1 − ˜Γ(2)(s)) . (2.30) Here • ˜Γ(1)(s) = ˜B(1)s + λ1(1 − b2) (1 − ˜Γ(1)(s))

: The duration of an accreditation period started by a class-1 customer,

• ˜Γ(2)(s) = ˜B(2)s + λ1(1 − b2) (1 − ˜Γ(1)(s))

: The duration of an accreditation period started by a class-2 customer,

• ˜B(i)(s) is the service-time distribution of a customer of class i.

2.7.2 Final equations of ˜W(1)_(s)

Now the final equations of ˜W(1)(s) is repeated. ˜ W(1)(s) = (1−ρ)+ρ b2V˜(2)(s) + (1 − ρ)(1 − b₂) (1 − σ1) ˜ V(1,0)(s) +(ρ − σ1)(1 − b2) (1 − σ1) ˜ V(2)(s) ˜V(1,1)(s) , (2.31) Where • ˜V(2)(s) = κ(2) λ1(1−˜Γ(1)(b2s))+λ2(1−˜Γ(2)(b2s)) b2s−λ1b2(1−˜Γ(1)(b2s))−λ2(1−˜Γ(2)(b2s))

: The accumulated-priority distribu-tion LST by an unaccredited customer,

• ˜V(1,0)(s) = κ(1,0) λ1(˜Γ(1)(b2s)− ˜B(1)(s))+λ2(˜Γ(2)(b2s)− ˜B(2)(s))

(s−λ1(1− ˜B(1)(s)))

: The accumulated-priority dis-tribution LST by an accredited customer during an accreditation interval at the start of a busy period,

• ˜V(1,1)(s) = κ(1,1) λ1b2(˜Γ(1)(b2s)− ˜B(1)(s))+λ2(˜Γ(2)(b2s)− ˜B(2)(s))

(s−λ1(1− ˜B(1)(s))) : The accumulated-priority

distribution LST by an accredited customer during an accreditation interval in the middle of a busy period,

• κ(2)₌ 1−ρ

ρ ,

• κ(1,0)₌ 1−ρ1(1−b2)

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• κ(1,1)₌ 1−ρ1(1−b2)

(ρ1b2+ρ2)(1−b2),

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Chapter 3 FIFO and absolute priority

queue

Two special cases of the APQ are when b2 = 1, and when b2 = 0. When b2 = 1, the

priorities of class-1 and class-2 customers increase with the same pace. This means that the customer with the highest priority is always the customer who was in the queue the longest. When this is the case, the queue is a regular FIFO queue.

When b₂ = 0, every class-1 customer is accredited, and so, whenever a customer of class-1 is in the queue, it has priority over all class-2 customers. In this case, the queue is an absolute priority queue.

3.1 FIFO queue

In the case when b2 = 1, the queue transforms in a FIFO queue, with arrival rate λ1+ λ2,

and service-time distribution

B₀(2)(t) = λ1B

(1)_{(t) + λ}

2b1B(2)(t)

λ1+ λ2

.

From Daigle (2005) it is known that the distribution of the waiting times should have LST

˜

W (s) = (1 − ρ)s

s − λ + λ ˜B(s). (3.1)

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Chapter 3. FIFO and absolute priority queue 20

In the model of the APQ, if b2 = 1, the class-2 waiting time is given by ˜W(2)(s) = (1 − ρ) +

ρV(2)(s/b₂). By filling in b₂= 1, this equation should become the same as (3.1).

Since b1 = b2, accreditation never happens. ˜Γ(1)(s) and ˜Γ(2)(s) denote the duration of an

accreditation interval started by a class-1 and class-2 customer respectively, which is a series of one unaccredited + k accredited customers. Since there are no accredited customers, ˜Γ2

is exactly the same as the serving time LST of the one unaccredited customer: ˜Γ(1)(s) = ˜

B(1)_{(s) and ˜}_Γ(2)_{(s) = ˜}_B(2)_{(s). The LST of the priority for a class-2 customer in the FIFO}

queue becomes: ˜ V_{F IF O}(2) (s) = 1 − ρ ρ λ1(1 − ˜B(1)(s)) + λ2(1 − ˜B(2)(s)) s − λ1(1 − ˜B(1)(s)) − λ2(1 − ˜B(2)(s)) (3.2) = (λ1+ λ2)(1 − B (2) 0 (s)) s − (λ1+ λ2)(1 − B0(2)(s)) . (3.3)

Filling this in into the class-2 waiting-time distribution gives us ˜

W_{F IF O}(2) (s) = (1 − ρ)s

s − (λ1+ λ2)(1 − ˜B0(2)(s))

, (3.4)

Which is the expected outcome for the FIFO queue.

Now we do this for ˜W(1)(s) as in (2.31). For this we need to calculate ˜V(1,0)(s) and ˜V(1,1)(s). However, since ˜B0(s) = ˜B(s), only one of these needs to be calculated. Even more, on

further inspection, ˜V(1,0)(s) and ˜V(1,1)(s) are the priority LSTs upon entering service for accredited customers. When b1 = b2, no accreditation occurs at all. This can also be seen

in (2.31), where their respective factors become equal to 0 when b₁ = b₂. This means that ˜

W_{F IF O}(1) (s) is reduced to the class-2 waiting-time distribution ˜W_{F IF O}(2) (s).

3.1.1 Singularities in the FIFO queue

It is clear that the only singularities in a FIFO queue waiting-time LST are the zeros of s − (λ1+ λ2)(1 − ˜B0(2)(s)). For a queue containing only class-1 customers, this is reduced

to

s − λ1(1 − ˜B(1)(s)) = 0, s < 0. (3.5)

It can be seen that this is also a singularity of ˜V(1,0)(s) and ˜V(1,1)(s). From now on, let us call this singularity −s(1)∗_denom, denoting that this singularity is from the denominator of

˜

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3.2 Absolute priority queue

When b2 = 0, every class-1 customer that doesn’t start a busy period is automatically an

accredited customer. This means that, whenever a class-1 customer is in the queue, it will enter service before any class-2 customers in the queue. This makes the queue an absolute priority queue. The behaviour of this kind of queue has been described in Abate & Whitt (1997).

First, the class-2 waiting times will be looked at. After that, the class-1 waiting times will be looked at, more notably for exponential service-time distributions.

3.2.1 Class-2 waiting times

To find the class-2 waiting-time distribution, we first of all repeat the equation of its LST: ˜ W(2)(s) = (1 − ρ) + ρ ˜V(2)(s/b₂) (2.29) = (1 − ρ) + (1 − ρ) λ1(1 − ˜Γ (1)_{(s)) + λ} 2(1 − ˜Γ(2)(s)) s − λ1b2(1 − ˜Γ(1)(s)) − λ2(1 − ˜Γ(2)(s)) . (2.30)

Since b₂ = 0, the second term in the denominator is omitted. The only other dependencies on b₂are in ˜Γ(1)_{(s) and ˜}_Γ(2)_{(s). These are the LSTs of the accreditation periods started by a}

class-1 and class-2 customer respectively. If b2 = 0, every class-1 customer is automatically

accredited if it is not the first customer of a busy period. Every unaccredited customer is therefore also a class-2 customer if it is not at the start of a busy period. This means that an accreditation period is a class-1 busy period started either by a class-1 or class-2 customer.

If ˜G(1)₁ (s) is the LST of a busy period of class-1 started by a class-1 customer, then ˜G(1)₁ (s) is the solution to the functional equation ˜G(1)₁ (s) = ˜B(1)(s+λ₁(1− ˜G(1)₁ (s))). As we know from (2.6), the busy period of class-1 customers initiated by a class-2 customer is as follows:

˜

G(1)₂ (s) = ˜B(2)(s + λ₁(1 − ˜G(1)₁ (s))). (3.6) Since accreditation periods have become class-1 busy periods started by either a class-1 or class-2 customer, the accreditation periods can be rewritten as:

˜

Γ(1)(s) ≡ ˜G(1)₁ (s) ˜

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Note that this is only the case for b2= 0.

Using these findings, the waiting-time LST can be rewritten as:

lim b2→0 ˜ W(2)(s) = (1 − ρ) + (1 − ρ) λ1(1 − ˜G(1)1 (s)) + λ2(1 − ˜G(1)2 (s)) s − λ2(1 − ˜G (1) 2 (s)) (3.7) = (1 − ρ)s + λ1(1 − ˜G (1) 1 (s)) s − λ2(1 − ˜G(1)2 (s)) . (3.8)

By bringing s + λ1(1 − ˜G(1)1 (s)) into the denominator, and adding and subtracting 1 to the

denominator, the following equation can be reached:

lim b2→0 ˜ W(2)(s) = 1 − ρ 1 −λ1(1− ˜G(1)₁ (s))+λ2(1− ˜G(1)₂ (s)) s+ρ1(1− ˜G(1)1 (s)) .

By choosing units so that µ₁ = 1, so that λ₁ = ρ₁, the waiting-time equation as in Abate & Whitt (1997, eq. (2.14)) is obtained.

From (3.7) it can be seen that there are 2 possibilities for singularities for ˜W(2)(s): • The rightmost singularity of ˜G(1)₁ (s),

• the rightmost zero of s − λ₂(1 − ˜G(1)₂ (s)) = 0 for s < 0.

Both of these singularities will be expanded on for the general case of the APQ further on.

3.2.2 Class-1 waiting times

The class-1 waiting-time LST is given by ˜ W(1)(s) = (1−ρ)+ρ b2V˜(2)(s) + (1 − ρ)(1 − b₂) (1 − σ₁) ˜ V(1,0)(s) +(ρ − σ1)(1 − b2) (1 − σ₁) ˜ V(2)(s) ˜V(1,1)(s) . (2.31) In the case of absolute priority, b2 = 0. This means that this equation is reduced to:

˜ W_absolute(1) (s) = (1 − ρ) + ρ _{(1 − ρ)} (1 − ρ₁) ˜ V(1,0)(s) +(ρ − ρ1) (1 − ρ₁) ˜ V(2)(s) ˜V(1,1)(s) . (3.9) This means that the rightmost singularity of the waiting time is either the rightmost sin-gularity of ˜V(2)_{(s), ˜}_V(1,0)_{(s) or ˜}_V(1,1)_{(s). The rightmost singularity of either ˜}_V(1,1)_{(s) or}

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Chapter 3. FIFO and absolute priority queue 23 ˜

V(2)(s) may coincide with a zero of the other function, which may result in a non-singular point.

The priorities ˜V(1,0)(s) and ˜V(1,1)(s) in the absolute priority case are given by: lim b2→0 ˜ V(1,0)(s) = 1 − ρ1 ρ λ1(1 − ˜B(1)(s)) + λ2(1 − ˜B(2)(s)) s − λ1(1 − ˜B(1)(s)) , (3.10) lim b2→0 ˜ V(1,1)(s) = (1 − ρ1) ρ2 λ2(1 − ˜B(2)(s)) s − λ1(1 − ˜B(1)(s)) . (3.11)

Simplifying ˜V(2)(s) by setting b₂ = 0 causes the indeterminate fraction 0₀ to occur. By applying L’Hˆopitale’s rule by differentiating denominator and numerator with respect to b2, and rearranging the terms, it is seen that limb2→0V˜

(2)_{(s) is just constant:} lim b2→0 ˜ V(2)(s) = 1, (3.12) even if lim_b₂→0V˜(2) s b2

is not a constant function. To see this isn’t a contradiction, the following function can be looked at: f (s) = b₂s+1. It can easily be seen that limb2→0f (s) =

1: a constant function, while limb2→0f (s/b2) = s + 1: a function that is dependent on

s.

Combining these equations for the priorities gives the rather elegant equation for the class-1 waiting-time LST in the absolute priority queue:

˜

W_absolute(1) (s) = (1 − ρ)s + λ2(1 − ˜B

(2)_(s))

s − λ1(1 − ˜B(1)(s))

. (3.13)

It can easily be seen that two possibilities exist for the rightmost singularity of ˜W(1)(s): • The rightmost zero of s − λ₁(1 − ˜B(1)(s)) for s < 0,

• The rightmost singularity of ˜B(2)(s).

This equation has a lot of similarities with the FIFO waiting-time equation if the queue exists only of customers of class 1, but with the extra dependency on ˜B(2)(s).

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Chapter 4 Asymptotic behaviour of ˜

W

(2)

(s)

The upcoming sections will all be about the general model, where b2 ∈ [0, 1]. From this

point onward, we will assume that ˜B(1)(s) and ˜B(2)(s) are class-I distribution

as described by Abate & Whitt (1997, section 3). This means that the rightmost

singularity of ˜B(1)(s) and of ˜B(2)(s) are both poles to the right of s = 0. Let’s call these singularities −s∗_B1 and −s∗_B2 respectively. In short, the 3 classes of distributions will be repeated. If the LST of a probability function has rightmost singularity −s∗, then the class is as follows:

• class I: −s∗ < 0, and ˜B(−s∗) = ∞, • class II: −s∗ _{< 0, and 1 < ˜}_B(−s∗_{) < ∞,}

• class III: −s∗= 0, and ˜B(−s∗) = 1.

4.0.1 Singularities

In this chapter, different possibilities of singularities of ˜W(2)(s) will be looked at, and the behaviour of the function around those singularities. This behaviour will then be extended from the Laplace domain to the time domain.

Once again, we start by repeating the equation for ˜W(2)(s): ˜ W(2)(s) = (1 − ρ) + ρ ˜V(2)(s/b2) (2.29) = (1 − ρ) + (1 − ρ) λ1(1 − ˜Γ (1)_{(s)) + λ} 2(1 − ˜Γ(2)(s)) s − λ1b2(1 − ˜Γ(1)(s)) − λ2(1 − ˜Γ(2)(s)) . (2.30) 24

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Chapter 4. Asymptotic behaviour of ˜W(2)(s) 25

It is clear that only the singularities of ˜V(2)(s) need to be accounted for, as W(2)(s) will have the same singularities. From this equation, the possibilities for the rightmost singularity −s∗ can be seen on sight:

• −s∗ _{is the rightmost zero of s − λ}

1b2(1 − ˜Γ(1)(s)) − λ2(1 − ˜Γ(2)(s)) for s < 0. Let’s

call this −s(2)∗_denom.

• −s∗ _{is the rightmost singularity of ˜}_Γ(1)_{(s). Let’s call this −τ}−1

1 , after the rightmost

singularity of the busy period in Abate & Whitt (1997). • −s∗ _{is the rightmost singularity of ˜}_Γ(2)_{(s). Let’s call this −τ}−1

2 .

Remark 3. If −τ₁−1 or −τ₂−1 are singularities of ˜V(2)(s), then these are branch points of Γ(1)(s) or Γ(2)(s) respectively. This is because, if they were regular poles, they would be poles in both the numerator and denominator, and they would cancel each other out. This also means that these cause branch-points in the accumulated priority, and as such, the waiting time.

Remark 4. it can be seen that, since the rightmost singularity of ˜V(2)(s) is negative, if it exists, the rightmost zero of s − λ₁b2(1 − ˜Γ(1)(s)) − λ2(1 − ˜Γ(2)(s)), s < 0 will always be to

the right of −τ₁−1 and −τ₂−1: the rightmost singularity of the numerator of ˜V(2)(s). This means that −s(2)∗_denom will always be the dominant singularity if it exists.

4.1 Existence of −s

(2)∗_denom

From remark 4, it follows that, if −s(2)∗_denom exists, it is always the dominant singularity of ˜W(2)(s). This means that, if we can find out when −s(2)∗_denom exists, we know for what singularity we must look.

From now on let’s call the denominator of ˜V(2)(s/b2) f(2)(s).

f(2)(s) := s − λ1b2(1 − ˜Γ(1)(s)) − λ2(1 − ˜Γ(2)(s)) (4.1)

Lemma 4.1. For some > 0, the function f(2)(s) = s − λ1b2(1 − ˜Γ(1)(s)) − λ2(1 − ˜Γ(2)(s))

is negative ∀s ∈ [−, 0[.

Proof. Since ˜B(1)(s) and ˜B(2)(s) are assumed to be class-I distributions, ˜Γ(1)(s) is a class-II distribution (Abate & Whitt, 1997, Corollary 7.1.). From Abate & Whitt (1997, Theorem 7.3 c)), we can deduce that ˜Γ(2)(s) is either a class-I or a class-II distribution. The function

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Chapter 4. Asymptotic behaviour of ˜W(2)(s) 26

f(2)(s) is a continuous function to the right of the rightmost singularity. Setting s = 0 gives f(2)(s) = 0. Also, f(2)0(0) = 1 + λ1b2Γ˜(1)0(0) + λ2˜˜Γ(2)0(0) (4.2) = 1 − λ1b2 µ1− λ1(1 − b2) − λ2µ1 µ2(µ1− λ1(1 − b2))

(Stanf ordet al., 2014, eq.(19)) (4.3)

= 1 − ρ

1 − ρ₁(1 − b₂). (4.4)

Equation (4.4) is always strictly positive, since ρ < 1 and ρ₁(1 − b₂) < 1. This means that, for some > 0, f (s) < 0, ∀s ∈ [−, 0[.

Lemma 4.2. (Abate & Whitt, 1997, Theorem 7.3 b) The value of −τ₂−1 depends on on the solution of (4.13): −ζ₁ and the rightmost singularity of ˜B(2)_{(s) : −s}∗

B2: − τ₂−1=    −τ₁−1; −ζ1 > −s∗B2, s∗_B2+ λ1(1 − b2) 1 − ˜B(1)(−s∗_B2); −ζ1 < −s∗_B2. (4.5) With s∗_B2+ λ1(1 − b2) 1 − ˜B(1)(−s∗_B2)to the right of −τ₁−1.

Corollary 4.2.1. Filling in the second case into (2.8) gives that ˜Γ(2)(−τ₂−1) = ˜B(2)_(−s∗

B2) =

∞.

Lemma 4.3. If −ζ1 < −s∗B2, then −s

(2)∗

denom exists.

Proof. If −ζ1 < −s∗_B2, then the rightmost singularity in the numerator is −τ2−1, which is

only a singularity of ˜Γ(2)(s). From lemma 4.2, we know that lim_s→−τ−1 2

˜

Γ(2)(s) = ∞. This means that lim_s→−τ−1

2 f

(2)_{(s) = ∞. Since f}(2)_{(s) is both negative (lemma 4.1) and postitive}

between ] − τ₂−1, 0[ and it is continuous, it has a zero in ] − τ₂−1, 0[.

This lemma tells us that −τ₂−1 is of no importance, because if it is to the right of −τ₁−1, then −s(2)∗_denom exists, which is always the rightmost singularity of ˜V(2)(s/b2).

Lemma 4.4. Let

−γ∗= λ₁b2(1 − ˜Γ(1)(−τ1−1)) + λ2(1 − ˜Γ(2)(−τ1−1)) (4.6)

= λ₁b2(1 − ˜B(1)(−ζ1)) + λ1(1 − ˜B(2)(−ζ1)) (4.7)

Then