Multi-server Marginal Allocation

With CVaR and Abandonment based QoS Measures

Per Enqvist

1

and G

¨

oran Svensson

1,2

1

Division of Optimization and Systems Theory, Kungliga Tekniska H

¨

ogskolan, Lindstedtsv 25, Stockholm, Sweden

2

Teleopti WFM, Teleopti AB, Sweden

Keywords:

Queueing, Queueing Networks, Marginal Allocation, Conditional Value-at-Risk, Abandonments.

Abstract:

Two multi-objective minimization problems are posed, one for Erlang-C queues and one for Erlang-A queues.

The objectives are to minimize the cost of added agents while also trying to optimize a quality of service

measure. For the Erlang-C system we propose using the Conditional Value-at-Risk measure with waiting time

as the loss function. We prove that this quality of service measure is integer convex in the number of servers.

For the Erlang-A system we use the fraction of abandoning customers and some rate based weighting function

as the service measure. Finally, a numerical comparison of the two system types is performed. The numerical

results show the similarities between the two systems in terms of optimal points.

1 INTRODUCTION

In this paper we investigate multi-class queueing net-

works and the optimal allocation of servers with re-

spect to a Quality of Service (QoS) measure. We con-

sider two types of queueing systems, one based on

the Erlang-C model with Conditional Value-at-Risk

(CVaR) (Rockafellar and Uryasev, 2000; Rockafellar

and Uryasev, 2002) on the waiting time as the QoS

measure. In the second system type we look at the

Erlang-A model and use the fraction of abandoning

customers as the basis of our QoS measure.

The basic model consists of a system of par-

allel server pools with a corresponding set of

agents(servers) that cater to customers(jobs), see Fig-

ure 1. Each server pool has a separate and inﬁnite

ﬁrst-come-ﬁrst-serve (FCFS) buffer. The separate and

parallel queueing systems are bound together by a

common budget constraint.

The optimization problem is formulated and

solved in terms of the marginal allocation (MA) al-

gorithm (Fox, 1966). When varying the budget con-

straint the whole efﬁcent front, consisting of efﬁcient

solutions, can be found. The MA algorithm depends

on the costs per agent and the improvements of the

QoS measure, for the different queues, to be sepa-

rable and (integer) convex functions. In the tradi-

tion of (Rolfe, 1971; Dyer and Proll, 1977; Weber,

1980) we proceed to prove that the QoS measure, de-

termined by CVaR, is decreasing and convex in the

number of agents. In (Parlar and Sharafali, 2014) the

authors summarize other proofs of convexity for dif-

ferent QoS measures. In a similar fashion the system

of queues where abandonments are allowed is posed

and solved, using a QoS measure based on the frac-

tion of abandoning customers. The convexity of the

measure is posed as a conjecture.

2 MODEL DESCRIPTION AND

THE QoS MEASURES

We consider a system of N ∈ N queues of either

M/M/c or M/M/c + M type (using the notation of

(Baccelli and Hebuterne, 1981)), i.e., Erlang-C or

Erlang-A models, each with its own inﬁnite queueing

buffer.

Introduce the index set I = {1,...,N}. Let c

i

, de-

note the number of servers in queue i, c = [c

1

... c

N

]

and let a

i

be the corresponding cost for each agent of

type i. The total cost of the agents is a separable and

integer convex function in c. Furthermore, assume

that the numbers of agents available for assignment to

the different demands are also limited. Let d

i

∈ N, be

the maximum number of available agents of type i.

The arrival process to queue i is a homogeneous

Poisson process with arrival rate parameter λ

i

. The

service rate of each server in pool i is denoted by µ

i

and service times are exponentially distributed. In the

case of the Erlang-A type systems the abandonment

Enqvist, P. and Svensson, G.

Multi-server Marginal Allocation - With CVaR and Abandonment based QoS Measures.

DOI: 10.5220/0006652602970303

In Proceedings of the 7th International Conference on Operations Research and Enterprise Systems (ICORES 2018), pages 297-303

ISBN: 978-989-758-285-1

Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

297

rate of each customer is included. The time to aban-

donment of a customer in queue i is exponentially dis-

tributed with rate parameter θ

i

. Hence, the customer

may leave the system due to either service completion

or impatience, whichever occurs ﬁrst. It is assumed

that customers being served do not defect.

Let b denote a budget constraint on the system,

meaning that the total cost of the assigned servers to

the queueing network must be within budget. The

model may be extended to include several budget con-

straints, affecting only a subset of the queues.

λ

1

.

.

.

λ

N

M/M/c

1

.

.

.

M/M/c

N

Figure 1: A system of N parallell M/M/c queues.

2.1 The Conditional Value-at-Risk

Measure

To promote a positive customer experience the queue-

ing system is endowed with a QoS measure. Typi-

cal QoS measures for contact centers include aver-

age speed of answer (ASA) and the telephone service

factor (TSF) (Gans et al., 2003), also known as ser-

vice level (SL). ASA measures the average time a cus-

tomer spends waiting on service while TSF considers

the acceptable waiting time (AWT) that a certain per-

centile of the customers have to wait before service.

The TSF type of QoS measure is also used in ﬁnance,

under the name of Value-at-Risk (VaR), to describe

the risk of (large) losses. The CVaR measure is of-

ten preferred to the VaR type measure due to its con-

vexity (Rockafellar and Uryasev, 2000; Rockafellar

and Uryasev, 2002) and coherent measure properties

(Artzner et al., 1999). In general CVaR offers a means

of controlling worst case outcomes, which might be of

great importance in ﬁelds like healthcare, as pointed

out in (Parlar and Sharafali, 2014). We consider the

QoS measure corresponding to CVaR where the loss

function is taken to be a customers waiting time in the

queue. We apply the CVaR measure to the system of

M/M/c queues.

First some preliminary results. The waiting time

distribution for a M/M/c queue is given by (Klein-

rock, 1996)

Pr(W

q

> t) = Π

q

(c,η)e

−(cµ−λ)t

= 1 − F

W

q

(t), (1)

where W

q

is the random variable representing the

waiting time and Π

q

(c,η) is the probability of delay,

i.e., of having to wait when there are c homogeneous

agents working under load η =

λ

µ

. This is given by

the Erlang-C formula (Kleinrock, 1996),

Π

q

(c,η) =

η

c

/c!

(1 − η/c)

c−1

∑

i=0

η

i

/i! + η

c

/c!

. (2)

It can be calculated efﬁciently via the following re-

cursion (Zeng, 2003)

Π

q

(c + 1,η) =

λ

µ

(cµ − λ)Π

q

(c,η)

(c + 1)µ − λ

c − λΠ

q

(c,η)

. (3)

This formula avoids numerical issues. We note that

the recursion is initiated by Π

q

(1,η) = η, and any

value larger than one should be interpreted as one,

since the larger values corresponds to unstable queues

when abandonments are excluded.

The different queues may have different QoS re-

quirements, thus let β

i

denote the quantile level for

queue i and t

i

be the VaR(AWT) value for queue i.

The QoS requirement for the VaR type measure is

then given by

Pr(W

q,i

≤ t

i

) ≥ β

i

, i ∈ I, (4)

and the corresponding β

i

-VaR is deﬁned as

t

i

(c

i

) = min{t|F

W

q,i

(t) ≥ β

i

,c

i

}

=

logΠ

q

(c

i

,η

i

) − log(1 − β

i

)

c

i

µ

i

− λ

i

. (5)

We use the notation of t

i

to underscore that the VaR

value is measured in units of time, in this case.

The corresponding β-CVaR is deﬁned by

φ

β

i

(c

i

) =

1

1 − β

i

∞

Z

t

i

(c

i

)

tdF

W

q,i

(t), i ∈ I, (6)

with explicit formulation for Π

q

(c

i

,η

i

) ≥ 1 − β

i

:

φ

β

i

(c

i

) =

1

c

i

µ

i

− λ

i

log

Π

q

(c

i

,η

i

)

1 − β

i

+ 1

= t

i

(c

i

) +

1

c

i

µ

i

− λ

i

, i ∈ I, (7)

where t

i

(c

i

) is given by (5).

Remark 1 (Probability atoms and CVaR). If β is

small then a probability atom might have to be han-

dled. To avoid such an atom it is required that Π

q

≥

1 − β holds. For a complete treatment of CVaR in

cases where there is a probability atom see (Rockafel-

lar and Uryasev, 2002). However, for most realistic

choices of parameters this is not an issue and will thus

be ignored throughout the rest of the paper.

The QoS measure given by this deﬁnition of the β-

CVaR is integer convex and decreasing in the number

of agents, c. The use of the MA algorithm rests on

this fact. This is formalized in Proposition 1.

ICORES 2018 - 7th International Conference on Operations Research and Enterprise Systems

298

Proposition 1 (Integer Convexity of β-CVaR). Con-

sider a M/M/c queue with constant rate parameters

µ,λ > 0 such that cµ > λ and that K < β ≤ 1, where

K is large enough that there is no probability atom.

Then

φ

β

i

(c

i

) = t

i

(c

i

) +

1

c

i

µ

i

− λ

i

, i ∈ I,

is decreasing and integer convex in c ∈ N.

The proof is given in the Appendix.

2.2 Probability of Abandonment

For a system of multi-server Erlang-A queues other

QoS measures are of interest. In (Garnett et al., 2002;

Mandelbaum and Zeltyn, 2007) different measures

for queueing systems with abandonments are consid-

ered. Erlang-A queues are stable for arbitrary loads

while Erlang-C queues are only stable when cµ > λ.

Here we consider systems in steady state, with con-

stant (random) demand. Perhaps the most obvious

measure is the probability that a customer will aban-

don (defect) the queue before recieving service. This

may occur if the arriving customer ﬁnds all servers

occupied and thus have to wait in queue. Then the

customer will abandon if his/her patience runs out be-

fore a server becomes available.

Let π

(i)

j

denote the probability of queue i being in

state j ∈ N, where the states are given by the number

of customers in queue i. Furthermore, let E

(i)

1,c

i

denote

the Erlang blocking formula of the i:th queue with c

i

servers. The incomplete Gamma function is deﬁned

as

γ(x,y) =

y

Z

0

t

x−1

e

−t

dt, (8)

then, in accordance with (Mandelbaum and Zeltyn,

2007), let

A(x,y) =

xe

y

y

x

γ(x,y), x > 0,y ≥ 0. (9)

The probability of an arrival ﬁnding all servers busy

is given by

Pr(W

q

> 0) =

A(

cµ

θ

,

λ

θ

)E

1,c

1 +

A(

cµ

θ

,

λ

θ

) − 1

E

1,c

. (10)

The fraction of customers abandoning, conditioned

on having to wait on arrival, is given by

Pr(Ab|W

q

> 0) =

1

ρA(

cµ

θ

,

λ

θ

)

+ 1 −

1

ρ

, (11)

where ρ =

λ

cµ

, the offered load per agent. Then us-

ing the deﬁnition of conditional probability yields the

fraction of customers abandoning the queue.

The QoS measure given by the fraction of cus-

tomers abandoning is decreasing and integer convex

in the number of servers according to numerical tests

we performed. We are convinced it holds in general

and may be known. At this stage we pose it as a con-

jecture.

Conjecture 1 (Integer Convexity of Pr(Ab)). Con-

sider a M/M/c + M queue with constant rate param-

eters µ,λ > 0. Then

Pr(Ab) = Pr(Ab|W

q

> 0)Pr(W

q

> 0)

=

1

ρA(

cµ

θ

,

λ

θ

)

+

ρ − 1

ρ

!

A(

cµ

θ

,

λ

θ

)E

1,c

1 +

A(

cµ

θ

,

λ

θ

) − 1

E

1,c

(12)

is decreasing and integer convex in c ∈ N.

3 OPTIMIZATION

FORMULATION

Here we pose optimization formulations based on the

systems and QoS measures of Subsections 2.1 and

2.2. Since costs, CVaR and the probability to abandon

are integer convex functions in the number of servers,

the multi-objective solutions given by the MA algo-

rithm will be in terms of an efﬁcient front.

First we will formulate the multi-objective prob-

lem that minimizes the total agent cost and β-CVaR

for all M/M/c queues. Then the following Optimiza-

tion problem can be formulated,

min

c

ϕ

N

∑

i=1

a

i

c

i

+ψ

N

∑

i=1

φ

(i)

β

i

(c

i

),

Sub. to

N

∑

i=1

a

i

c

i

≤ b,

c

i

≤ d

i

, ∀i ∈ I,

c

i

∈ N, a

i

,b,ϕ,ψ > 0, ∀i ∈ I,

(13)

where φ

(i)

β

i

(c

i

) denotes the QoS measure, for the i:th

queue, based on CVaR with waiting time as the loss

function. The weights ϕ and ψ are constant and de-

termine the relative importance of the two objectives.

Note that both functions in the objective are separable

functions.

In a similar fashion, we pose an optimization

problem minimizing the agent costs and the probabil-

ities to abandon modiﬁed by a weighting function for

the M/M/c + M queues. The optimization problem

Multi-server Marginal Allocation - With CVaR and Abandonment based QoS Measures

299

for a queue with abandonments can be formulated as

min

c

ϕ

N

∑

i=1

a

i

c

i

+ψ

N

∑

i=1

ω(λ

i

,µ

i

)Pr

i

{Ab|c

i

},

Sub. to

N

∑

i=1

a

i

c

i

≤ b,

c

i

≤ d

i

, ∀i ∈ I,

c

i

∈ N, a

i

,b,ϕ,ψ > 0, ∀i ∈ I,

(14)

where Pr

i

{Ab|c

i

} denotes the probability of abandon-

ment, for the i:th queue, given that there are c

i

servers

at work and ω(λ

i

,µ

i

) is some weight function that cor-

respond to some notion of the population of queue i.

We will use the offered load. Note that both functions

in the objective are separable functions.

The optimization problem formulated in (14) can

be extended to include a waiting time based QoS mea-

sure. Ideally we would like said condition to be the

CVaR measure with loss function in terms of the wait-

ing time under the condition that the waiting customer

gets served. This measure needs to be deﬁned as of

yet. Instead we settle for average waiting time(mean

delay) under the condition that the demand eventu-

ally gets served. This result can be found in (Rior-

dan, 1962, p. 112, eq. 88). This condition forms

a lower bound on the number of agents and it is in-

cluded in the marginal allocation algorithm as the ini-

tiating number of agents of each type, thus guarantee-

ing that this QoS measure will be met.

3.1 The Marginal Allocation Algorithm

An efﬁcient algorithm for ﬁnding the efﬁcient points

for two integer convex and separable functions is the

MA algorithm, described in (Fox, 1966; Svanberg,

2009).

In general when minimizing the multi-objective

optimization problem for functions f ,g : N

N

→ R

where

∆ f

j

(x

j

) ≤ ∆ f

j

(x

j

+ 1) < 0 ∀ j,x

j

∈ N,

0 < ∆g

j

(x

j

) ≤ ∆g

j

(x

j

+ 1) ∀ j, x

j

∈ N,

(15)

the optimal vector x

∗

∈ N

N

minimizes ϕg(x) + ψ f (x)

if and only if the following conditions are satisﬁed for

each j = 1,...,N:

−∆ f

j

(x

∗

j

)

∆g

j

(x

∗

j

)

≤

ϕ

ψ

≤

−∆ f

j

(x

∗

j

−1)

∆g

j

(x

∗

j

−1)

if x

∗

j

> 0,

−∆ f

j

(0)

∆g

j

(0)

≤

ϕ

ψ

if x

∗

j

= 0.

(16)

x

∗

j

is an efﬁcient solution if and only if there are con-

stants ϕ,ψ > 0 such that the conditions (16) are satis-

ﬁed for each j = 1,...,N.

Marginal Allocation Algorithm

Step 0: Generate a table with N columns, ﬁll the

columns with the quotients

−∆ f

j

(n)/∆g

j

(n) for n = 0,1,2,... where

n starts at zero. Set k = 0, x

(0)

= (0,...,0),

g(x

(0)

) = g(0) and f (x

(0)

) = f (0).

Step 1: Select the largest uncancelled quotient in

the table. Cancel it and let l be the

corresponding column number.

Step 2: Let k := k + 1 then let x

(k)

l

= x

(k−1)

l

and

x

(k)

j

= x

(k−1)

j

,∀ j 6= l.

Let f (x

(k)

) = f (x

(k−1)

) + ∆ f

l

(x

k−1

l

) and

g(x

(k)

) = g(x

(k−1)

) + ∆g

l

(x

k−1

l

).

Terminate algorithm if g(x

(k)

) ≥ g

max

;

otherwise go to Step 1.

We identify our budget constraint b to be g

max

and

the constraint on available agents, d

i

,i ∈ I, to be the

number of quotients to calculate for column i. This

algorithm can now be applied to (13) and (14), where

g(x) =

N

∑

i=1

a

i

c

i

(17)

f (x) =

N

∑

i=1

f

(i)

CVaR

(c

i

) for (13),

N

∑

i=1

ω(λ

i

,µ

i

)Pr

i

{Ab|c

i

} for (14).

(18)

4 NUMERICAL EXAMPLES

The main beneﬁt of using the MA approach is that

huge systems can be optimized almost effortlessly.

An example of the efﬁcient front of a system of 100

M/M/c queues, with a (maximal) budget constraint

of b = 4500, is shown in Fig 2. The input parame-

ters in terms of arrivals, service rates and more were

randomly generated. The optimal stafﬁng calculation

took less than a second to perform on a laptop.

To compare the Erlang-C and the Erlang-A based

systems we look at a three class multi-server system

(i.e., three queues). First, ﬁnd the efﬁcient points for

the Erlang-C type system with CVaR as the service

measure. Then use those points in the solution of the

Erlang-A type system with fraction of abandonments

as basis for the service measure. These points are then

compared to the optimal points, shown in Figure 3.

The parameters used were β

β

β = [.95 .95 .95]

T

,λ

λ

λ =

[15 10 20]

T

,µ

µ

µ = [0.5 0.6 0.7]

T

and a = [12 15 18]

T

.

The procedure was repeated for two sets of im-

patience rates, θ

θ

θ = [0.25 0.25 0.25]

T

and θ

θ

θ =

[10 10 10]

T

, respectively. The weight function,

ω(λ

i

,µ

i

), in (14) is deﬁned to be the offered load, η

i

.

ICORES 2018 - 7th International Conference on Operations Research and Enterprise Systems

300

Figure 2: Showing the efﬁcient front for a system of 100

M/M/c-queues, using CVaR as QoS measure and with

a budget constraint of 4500. The parameters where ran-

domly generated. In the bottom ﬁgure some speciﬁc efﬁ-

cient points are shown.

With respect to the patience parameter the two so-

lutions do not differ signiﬁcantly. The given solutions

are very similar, which seem to hold for larger sys-

tems and for a wide range of parameter values. Even

though the probability of abandonment is very sim-

ilar for the two approaches, the optimal number of

agents may be different. Therefore, we consider the

optimal distributions in Table 1 to illustrate this for

the example seen in Figure 3. With abandonments

the ﬁrst agent pool is prioritized over the second and

third, and recieves one or two more agents compared

to the CVaR assignment.

5 SUMMARY AND

CONCLUSIONS

The main objectives of this paper is to ﬁnd an opti-

mal distribution of agents between N queues, bound

by budget constraints, via the MA algorithm. The

QoS measures used are CVaR, with waiting time as

the loss function, for the systems of M/M/c queues

and a weighted probability of abandonment for the

M/M/c + M queues. The advantage of using the MA

algorithm is that it can ﬁnd the efﬁcient points ex-

tremely fast. As can be seen from the numerical ex-

amples studying two different queueing networks give

similar solutions. The two QoS measures have rather

different shape, consider the curvature of the graphs

in Figure 3, but generate similar optimal points. We

Table 1: Showing the partial agent distributions for the

CVaR solution (left ﬁgure) and for the abandonment solu-

tion where θ = 0.25 (middle ﬁgure).

Agents

CVaR Ab

77 31 17 29 32 17 28

78 31 18 29 33 17 28

79 31 18 30 33 17 29

80 32 18 30 33 18 29

81 32 19 30 34 18 29

82 33 19 30 34 18 30

83 33 19 31 34 19 30

84 33 20 31 35 19 30

85 34 20 31 35 19 31

86 34 20 32 36 19 31

87 35 20 32 36 19 32

88 35 21 32 36 20 32

89 36 21 32 36 20 33

90 36 21 33 37 20 33

91 36 22 33 37 21 33

believe this is partly a consequence of the exponential

distributions of the service times. It may well diverge

for other service time distributions, e.g. lognormal

distributions.

ACKNOWLEDGEMENTS

The authors would like to thank Teleopti AB, Stock-

holm Sweden, for their support of our work.

REFERENCES

Artzner, P., Delbaen, F., Eber, J., and Heath, D. (1999).

Coherent measures of risk. Mathematical Finance,

9(3):203–228.

Baccelli, F. and Hebuterne, G. (1981). On queues with im-

patient customers. Research report, RR-0094, INRIA.

Dyer, M. and Proll, L. (1977). On the validity of marginal

analysis for allocating servers in M / M / c queues.

Management Science, 23(9).

Fox, B. (1966). Discrete optimization via marginal analysis.

Management Science, 13(3):210–216.

Gans, N., Koole, G., and Mandelbaum, A. (2003). Tele-

phone call centers: Tutorial, review, and research

prospects. Manufacturing Service Oper. Management.

Garnett, O., Mandelbaum, A., and M.Reiman (2002).

Manufacturing and Service Operations Management,

4:208–227.

Kleinrock, L. (1996). Queueing systems : problems and

solutions. Wiley, New York.

Mandelbaum, A. and Zeltyn, S. (2007). Service Engineer-

ing in Action: The Palm/Erlang-A Queue, with Appli-

Multi-server Marginal Allocation - With CVaR and Abandonment based QoS Measures

301

Figure 3: The left ﬁgure shows the CVaR solution for a system of three queues. In the middle and right ﬁgures the solution

under abandonments are shown, for θ = 0.25 and θ = 10. The MA solution for abandonments is compared to the CVaR

solution for abandonments.

cations to Call Centers, pages 17–45. Springer Berlin

Heidelberg, Berlin, Heidelberg.

Parlar, M. and Sharafali, M. (2014). Optimal design of multi

server markovian queues with polynomial waiting and

service costs. Applied Stochastic Models in Business

and Industry.

Riordan, J. (1962). Stochastic Service Systems. Wiley.

Rockafellar, R. and Uryasev, S. (2000). Optimization of

conditional value-at-risk. Journal of risk, 2.

Rockafellar, R. and Uryasev, S. (2002). Conditional value-

at-risk for general loss distributions. Journal of bank-

ing & Finance, 26:1443–1471.

Rolfe, A. J. (1971). A note on marginal allocation in

multiple-server service systems. Management Sci-

ence.

Svanberg, K. (2009). On marginal allocation. Dept. of

mathematics, KTH, Stockholm.

Weber, R. (1980). On the marginal beneﬁt of adding servers

to G/GI/m queues. Management Science, 26(9).

Zeng, G. (2003). Two common properties of the Erlang-

B function, Erlang-C function, and Engset blocking

function. Mathematical and Computer Modelling,

37(12):1287–1296.

APPENDIX

Proof of Proposition 1

First note that β-CVaR satisﬁes φ

β

(c,λ,µ) =

(1/λ)Φ

β

(c,1,µ/λ), so we can without loss of gener-

ality assume that λ = 1 and µ < 1.

The forward difference ∆φ

β

(c) = φ

β

(c + 1)− φ

β

(c) is

given by

1

cµ + µ − 1

log

Π

c+1

Π

c

− µφ

β

(c)

≤ 0 (19)

since Π

c

is non-increasing, hence φ

β

(c) is non-

increasing.

The second forward difference is

∆

2

φ

β

(c) =

1

C

1

C

2

2µ

2

φ

β

(c) + G

(20)

where C

k

= (c + k)µ − 1, and

G = C

1

log

C

1

C

0

C

1

c − Π

c

C

2

(c + 1) − Π

c+1

−2µlog

C

0

(C

1

c − Π

c

)µ

. (21)

Convexity of β-CVaR follows by showing that G ≥ 0.

Using that x/(1 + x) ≤ log(1 + x) ≤ x we have that

log

C

1

C

2

(c + 1) − Π

c+1

≥ −log(c + 1) (22)

−

1

c + 1

µ(c + 1) − Π

c+1

C

1

, (23)

and

log

C

1

c − Π

c

C

0

≥ log(c) +

cµ − Π

c

cC

0

+ µc − Π

c

. (24)

Then

G ≥ C

1

log

c

c + 1

+ 2µ log(cµ) − µ

+

Π

c+1

c + 1

+

C

3

(cµ − Π

c

)

cC

0

+ (cµ − Π

c

)

. (25)

From (3), it follows that

G ≥

C

−1

c

+

Π

c

C

0

(c+1)µ

−C

2

+ 2µ

2

c

(cC

0

+ (cµ − Π

c

)

. (26)

Note that c(cµ − 1) + cµ − Π

c

> 0 since cµ > 1. Us-

ing that Π

c

≤

1

cµ+1

1

(c−1)µ+1

, G(c(cµ − 1) + cµ − Π

c

)

is bounded from below by

(cµ − 1)

2

+ µ

2

(2c − 1) −

1

cµ + 1

1

(c − 1)µ + 1

·

(cµ − 1)(

c + 1

c

−

1

(c + 1)µ

) + µ(2 −

1

c

)

(27)

ICORES 2018 - 7th International Conference on Operations Research and Enterprise Systems

302

Introduce ε =

cµ−1

µ

, and eliminate c in (27) to ob-

tain

µ

2

(ε

2

+ 2ε +

2

µ

− 1) −

H

(2 + µε)(2 + µε − µ)

, (28)

where H = µ(2 + ε) + µ

2

ε−1

1+µε

−

1

1+µ+µε

.

Consider two cases, ε ≥ 1 and ε < 1.

If ε ≥ 1, then (28) is greater than

µ

2

(ε

2

+ 2ε +

2

µ

− 1) − H/4 (29)

≥ (εµ +

7µ − 2

8

)

2

+

12 + 11µ + 97µ(1 − µ)

64

which is non-negative and we used that

1

1+µε

≤ 1 and

1

1+µ+µε

≥ 1 − µ − µε.

If ε < 1, then we write (28) on common denomi-

nator and use

1

1+µε

≥ 1 − µε and

1

1+µ+µε

≥ 1 − µ − µε,

we can show that the numerator is bounded below by

(1 − µ)

1 − εµ + µ

3

+ 5µ(1 − µ) + µ(1 − ε)

+4εµ

2

(13/4 − µ + ε(1 + µ)))

+µ

3

ε

3

(4ε + µε

2

+ 7) ≥ 0 (30)

for all ε,µ ∈ (0, 1).

Then ∆

2

φ

β

(c,1,µ) ≥

2µ

2

φ

β

(c,1,µ)

C

2

C

1

≥ 0, and φ

β

is

integer-convex.

Multi-server Marginal Allocation - With CVaR and Abandonment based QoS Measures

303