Multi-Server Queue, with Heterogeneous Service Valuations Induced by

Travel Costs

Itzhak Moshkovitz

, Irit Nowik

2 a

and Yair Shaki

Department of Applied Mathematics, Ariel University, Ariel, Israel

Department of Industrial Engineering and Management,

Jerusalem College of Technology, Jerusalem, Israel

Keywords:

Queuing, Travel Costs, Observable Queue, Social Welfare, Nash Equilibrium.

Abstract:

This work presents a variation of Naor’s strategic observable model (Naor, 1969) for a loss system M/G/2/2,

with a heterogeneous service valuations induced by the location of customers in relation to two servers, A,

located at the origin, and B, located at M. Customers incur a “travel cost” which depends linearly on the

distance of the customer from the server. Arrival of customers is assumed to be Poisson with a rate that is the

integral of a nonnegative intensity function. We ﬁnd the Nash equilibrium threshold strategy of the customers,

and formulate the conditions that determine the optimal social welfare strategy. For the symmetric case (i.e.,

both servers have the same parameters and the intensity function is symmetric), we ﬁnd the socially optimal

strategies; Interestingly, we ﬁnd that when only one server is idle, then under social optimality, the server also

serves far away consumers, consumers whom he would not serve if he was a single server (i.e., in M/M/1/1).

1 INTRODUCTION

Customers of a service system often have heteroge-

neous service valuations, and this heterogeneity may

be caused by various reasons. In this paper, we study

a model with two servers, each located at a differ-

ent site, therefore a consumer (in general) incures

different “travel costs” when arriving at each service

site. In such circumstances, customers need to decide

whether to arrive for service, and if so, to what service

point to arrive. A realistic example may be that of a

network of public schools, hospitals, etc., from which

an individual needs to choose. Of course “location”

may refer to a geographic location or it may serve as

a metaphoric way expressing different preferences on

the ideal type of service.

The performance of service systems with strate-

gic customers has attracted much attention in recent

years (see, for example, Hassin & Haviv, 2003; Has-

sin, 2016). Naor (1969), was the ﬁrst to introduce a

queueing model that describes customer rational de-

cisions. The model considers an FCFS M/M/1 sys-

tem with homogeneous customers, a ﬁxed reward as-

sociated with service completion, and linear waiting

costs. The Nash equilibrium solution in Naor’s model

https://orcid.org/0000-0002-9257-8349

is simple since there exists a dominant pure threshold

strategy n

, such that an arriving customer joins the

queue if and only if the observed queue upon arrival

is shorter than n

. This strategy maximizes the indi-

vidual’s expected welfare regardless of the strategies

adopted by the others. The socially-optimal behavior

is also characterized by a pure threshold strategy n

∗

such that n

∗

≤ n

Naor assumes that customers are homogeneous

with respect to service valuation, and much of the

literature on observable queues (i.e., assuming cus-

tomers know the queue length before joining it) fol-

low this assumption. Some exceptions are described

in Section 2.5 of Hassin & Haviv (2003). For ex-

ample, Larsen (1998) assumes that the service value

is a continuous random variable and proves that the

proﬁts and social welfare are unimodal functions of

the price. For the case of a loss system (where cus-

tomers join iff the server is idle), Larsen proves that

the proﬁt-maximizing fee exceeds the socially opti-

mal fee. Miller and Buckman (1987) consider an

M/M/s/s loss system with heterogeneous service val-

ues and characterize the socially optimal fee.

Some authors investigated the price of anarchy

(PoA) in various service systems (see, for example,

Koutsoupias & Papadimitriou, 1999; Mavronicolas

& Spirakis, 2001; Hassin, 2016). The price of an-

Moshkovitz, I., Nowik, I. and Shaki, Y.

Multi-Server Queue, with Heterogeneous Service Valuations Induced by Travel Costs.

DOI: 10.5220/0012419100003639

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 13th International Conference on Operations Research and Enterprise Systems (ICORES 2024), pages 133-145

ISBN: 978-989-758-681-1; ISSN: 2184-4372

133

archy measures the inefﬁciency of selﬁsh behavior.

It is deﬁned as the ratio of the social welfare un-

der optimum to the Social welfare in equilibrium.

In Gilboa-Freedman, Hassin and Kerner (2014), the

PoA in Naor’s model is shown to have an odd behav-

ior. It increases sharply (from 1.5 to 2) as the arrival

rate comes close to the service rate and becomes un-

bounded exactly when the arrival rate is greater than

the service rate, which is odd since the system is al-

ways stable.

Most relevant to our work is the work of Hassin,

Nowik and Shaki (2018), in which heterogeneity in

service valuation is introduced through a Hotelling-

type model where customers reside in a “linear city”

and incur “transportation costs” from their locations

to the location of the server. Similar models have

been investigated (e.g. D’aspremont & Jaskold,

1979; Dobson & Stavrulaki, 2007; Economides,

1986; Gallay, Olivier and Max-Olivier Hongler, 2008;

Hotelling, 1929; Kwasnica & Euthemia, 2008; Pang-

burn & Stavrulaki, 2008; Ray & Jewkes, 2004; §6.7

and §7.5 in Hassin, 2016) but they all assume a con-

stant density (possibly restricted to an interval). In

contrast, Hassin, Nowik and Shaki allow non-uniform

distributions of customer locations, and the potential

arrival of customers with distances less than x from

the service facility is assumed to be distributed ac-

cording to Poisson with rate λ(x) =

h(y)dy < ∞,

where h(y) is a nonnegative “intensity” function of

the distance y. The deﬁnition of λ(x) by an integral is

natural since the customers accumulate from location

0, to location x. The intensity function and (linear)

travel costs jointly generate the distribution of cus-

tomer service valuations. A simple example is a two-

dimensional city, in which the arrival of customers is

uniform. In this case the intensity function can be de-

ﬁned as h(x) = 2πx, and so the arrival of customers

with distances less than x is assumed to be a Poisson

process with rate λ(x) =

2πydy = πx

. In a loss

system M/G/1/1, Hassin, Nowik and Shaki (2018) de-

ﬁne the threshold Nash equilibrium strategy x

and the

socially-optimal threshold strategy x

∗

. They investi-

gate the dependence of the PoA on the parameter x

and the intensity function h. They develop an explicit

formula to calculate lim

→∞

PoA(h,x

) when it exists.

As in Gilboa-Freedman, Hassin and Kerner

(2014), the number 2 arrises repeatadely in several

results of Hassin, Nowik and Shaki (2018), relating

to the limit of PoA when x

, goes to inﬁnity. For

instance, if h converges to a positive constant then

PoA converges to 2; if h increases (decreases) then

the limit of PoA is at least (at most) 2. In a system

with a queue they prove that PoA may be unbounded

already in the simplest case of uniform arrival.

The goal of this work is to extend Hassin, Nowik

and Shaki’s model to the case of two servers (instead

of a single server), where server A is located at the

origin and server B is located at a point denoted as

M. If the servers’ points are distsnce from each other

then the system is just a combination of two single-

server systems. It becomes more interesting when the

servers are closer, creating a dilemma for some con-

sumers regarding what service point to arrive at.

The value of information sharing between service

providers lies in its capacity to enhance coordination,

optimize resource allocation, and improve overall sys-

tem efﬁciency. When service providers have common

knowledge about each other’s status, they can collab-

orate more effectively, leading to a better distribution

of workloads and resources. This coordination of-

ten results in increased efﬁciency, reduced response

times, and improved service quality. The ability to

access real-time information about the status of other

providers allows for more informed decision-making,

enabling adaptive strategies that respond dynamically

to changing conditions. Ultimately, the value of such

information is reﬂected in its power to streamline op-

erations, enhance service delivery, and contribute to

a more resilient and responsive system. Think for

example of Air Trafﬁc Control Towers; In a situa-

tion where two air trafﬁc control towers manage ad-

jacent airspaces and are aware of each other’s work-

load, they can coordinate and optimize the allocation

of incoming ﬂights. If one tower is busy, the other

can efﬁciently handle additional aircraft to maintain

smoother air trafﬁc operations. Another example is

of a hospital with two emergency rooms, if each ER

is aware of the patient load and occupancy status of

the other, medical staff can coordinate patient assign-

ments. Deo and Gurvich (2011) consider a routing

problem motivated by the diversion of ambulances to

neighboring hospitals. These examples illustrate situ-

ations where the level of information sharing between

service providers can signiﬁcantly impact their abil-

ity to optimize resource allocation and overall system

efﬁciency.

2 THE M/G/1/1 MODEL:

NOTATIONS AND

FUNDEMENTAL RESULTS

In the model of one server, for all x ≥ 0, customers

with distances less than x, arrive to the system accord-

ing to a Poisson process with rate λ(x) =

h(y)dy,

where h(y) is an intensity function. The service dis-

ICORES 2024 - 13th International Conference on Operations Research and Enterprise Systems

134

tribution is general with average rate µ and the beneﬁt

from a service is R. There is a waiting cost c

per

unit time while in the system and a traveling cost of

per unit distance. The optimal (individual) strategy

of a customer located at a distance x from the origin,

is to enter service if the server is idle and the utility

is positive, namely: R ≥

+ c

x. This implies that

under individual optimization, a consumer located at

a distance x from the origin, is to enter service iff the

server is idle and

x ≤

R − c

/µ

Denote v =

R−c

/µ

. (Note that the optimal individ-

ual threshold strategy is denoted in Hassin, Nowik

and Shaki (2018), as x

, and in this paper as v.).

Hence there is a unique individual optimal strategy

(i.e., Nash equilibrium): v =

R−c

/µ

. Under this strat-

egy, a customer located at a distance x, enters service

iff the server is idle and x ≤ v.

The utility of a customer entering service from lo-

cation x is: R − c

/µ − c

x = c

(v − x). The balance

equation for the probability π

(x), of an idle server

satisﬁes:

(x)λ(x) = (1 − π

(x))µ.

This implies that:

(x) =

1 + ρ(x)

1 +

h(y)dy

where ρ(x) = λ(x)/µ.

The expected social beneﬁt per unit of time asso-

ciated with threshold x satisﬁes

(x) = c

(v − y)h(y)π

(x)dy =

(v − y)h(y)dy

1 +

h(y)dy

(1)

Let x

∗

be the threshold strategy that maximizes social

welfare. Under this strategy, only consumers with dis-

tances less than x

∗

, will enter the system. It is shown

in Hassin, Nowik and Shaki (2018), that;

∗

< v,

and that given v, the optimal threshold strategy x

∗

unique and satisﬁes,

v =

∗

− y)h(y)dy + x

∗

, (2)

(see Proposition 3.1 in Hassin, Nowik and Shaki

(2018)).

3 THE M/G/2/2 MODEL

3.1 Model Description

We consider two servers A and B on the interval

[0,M]. A is located at the origin, and B is located at

a point M. The model makes the following assump-

tions:

1. All customers reside on the interval [0,M].

2. The arrivals to the servers follow a Poisson

process with rates deﬁned according to a given

‘intensity function’ h, deﬁned over the inter-

val [0,M]. For any x, if consumers from in-

terval [0, x], turn to server A, then the arrival

rate from that interval is λ

(x) =

h(y)dy.

Similarily, for any x, if consumers from inter-

val [x,M], turn to server B, then the arrival rate

from that interval is λ

(x) =

h(y)dy.

3. The intensity function h may be any nonnega-

tive function for which

h(y)dy is ﬁnite for all

x ≥ 0.

4. Customers know their distance from each of

the two servers.

5. The status of the servers is observable.

6. Customers are risk neutral, maximizing ex-

pected net beneﬁt.

7. The service distribution of servers A and B,

is exponentially with rate µ

, and µ

, respec-

tively. The system is a loss system.

8. The beneﬁt from a completed service is R.

9. The waiting cost is c

per unit time (while in

the system).

10. The traveling cost to servers A and B are c

and

, respectively, per unit distance, and travel-

ing is instantaneous.

11. All processes are mutually independent.

12. The decision of the customer is whether to en-

ter to service and if so then which of the servers

to turn to.

The states of the system are denoted with (i, j), i, j ∈

{0,1}, where i = 0 means that server A is free, and

i = 1 means that Server A is busy. The same for j and

server B.

For State (0,0), let x

A00

, and x

B00

, be the arrival

thresholds, for servers A and B respectively. Namely,

if both servers are idle, then consumers with loca-

tions closer to the origin than x

A00

, (i.e., with loca-

tions x, s.t., x ≤ x

A00

), turn to server A, and similar-

ily, consumers with locations closer to M than x

B00

Multi-Server Queue, with Heterogeneous Service Valuations Induced by Travel Costs

135

(i.e., with locations x, s.t., x ≥ x

B00

), turn to server

B. For State (0,1), let x

A01

be the arrival threshold,

from which we allow consumers to arrive to Server A,

when Server B is busy, and similarly for state (1,0),

let x

B10

be the arrival threshold, from which we al-

low consumers to arrive to Server B, when Server A

is busy.

Every strategy is described by 4-dimensional vec-

tor

−→

x = (x

A00

A01

B00

B10

For the strategy to be well deﬁned, it is necessary

that,

A00

≤ x

B00

, (3)

since if x

B00

< x

A00

, then when both servers are idle,

consumers that are located between x

B00

and x

A00

should turn to server A according to x

A00

, but accord-

ing to x

B00

they should turn to server B.

Given

−→

x , denote π

i j

(

−→

x ), i, j ∈ {0,1}, as the

probability that the system is in state (i, j), i, j ∈

{0,1}.

In the following sections, we consider individual

and socially optimal strategies.

4 INDIVIDUAL OPTIMIZATION

Assume for the moment that only one server is idle

(i.e., state (0, 1) or state (1, 0)). Then the optimal

strategy of a customer located at a distance x from

server A (at a distance M − x from server B), is to ar-

rive to server A (server B) if server A (server B) is idle

and R ≥

+ c

x (R ≥

− c

(M − x

)). In other

words, the threshold strategies are:

R − c

/µ

, and v

= M −

R − c

/µ

. (4)

For state (0,0), we need to relate separately to two

cases; v

≤ v

, (case 1), and v

> v

, (case 2).

4.1 Case 1. v

≤ v

In this case, which is illustrated by Figure 1, every

customer between the origin and v

will turn to server

A, if he is idle, and every customer between v

and M

will turn to server B, if he is idle. The customers be-

tween v

and v

will not turn to any server (as their

utility is negative when turning to either server). In

fact, since the intervals [0,v

] and intervals [v

,M]

are disjoint, our system is equivalent to two indepen-

dent service systems. Therefore, the individual opti-

mal strategy (i.e., Nash equilibrium) is:

−→

= (x

A00

A01

B00

B10

) = (v

Server A

Server B

Figure 1: v

≤ v

4.2 Case 2. v

> v

In this case, which is illustrated by Figure 2, if only

server A is idle, consumers located between the ori-

gin and v

will turn to server A. If only server B is

idle consumers located between v

and M will turn

to server B. However, if both servers are idle, con-

sumers located between the origin and v

will turn to

server A, and customers located between v

and M

will turn to server B. But customers located in the in-

terval [v

] may potentially go to either server A or

B, (since the utility by going to either server, is posi-

tive). Thus, the optimizing individual strategy would

be turnning to the server which yields the greatest util-

ity for the consumer.

Server A

Server B

Figure 2: v

< v

If the customer turns to A, his beneﬁt will be

R −

+ c

x = c

− x), whereas if he turns to B,

his beneﬁt will be R −

− c

(M − x) = c

(x − v

Consequently, if

(x − v

) < c

− x), (5)

the customer will turn to A. The above is equivalent

x <

+ c

)

+ c

)

. (6)

Substituting v

and v

, from (4), in (6), we get

x <

M + c



−



+ c

)

Denote v

M+c



−



)

, then a customer lo-

cated at x, and observes that the two servers are idle,

will turn to A if his location x, satisﬁes x < v

, and

otherwise, will turn to B. Hence, the individual opti-

mal strategy in this case is,

−→

= (x

A00

A01

B00

B10

) = (v

Note that in the special case, in which c

= c

, and

= µ

, then the individual optimal strategy is:

−→

= (

,M − v

). (7)

ICORES 2024 - 13th International Conference on Operations Research and Enterprise Systems

136

5 SOCIALLY OPTIMAL

STRATEGY

Under social optimality, the mutual inﬂuences of ac-

tions chosen by the players must be taken into con-

sideration. These inﬂuences are not trivial. For exam-

ple, given

−→

x = (x

A00

A01

B00

B10

), if we increase

A00

, to include consumers that are further away from

server A but still have positive utility (namely their

location x, satisﬁes x

A00

< x < v

), then, on one hand

it may increase the social welfare function since con-

sumers that are further away from server A will now

get service. But on the otherhand, we may lose some

of the closer, (thus more valuable), consumers that

may ﬁnd the server busy more often than before and

this may reduce the social welfare function.

Recall that the arrivals to server A from [0,x] fol-

low a Poisson process with rate of λ

(x) =

h(y)dy,

and the arrivals to server B from [x,M] follow a Pois-

son process with rate of λ

(x) =

h(y)dy, where h

is the intensity function. Also recall that π

i j

(

−→

x ), is

the probability that the system is in state (i, j) when

−→

x = (x

A00

A01

B00

B10

). Given

−→

x , the transition

diagram is presented in Figure 3.

0,1

1,1

0,0 1,0

𝜆

ሺ

𝑥

B00

ሻ



𝜆

𝐴

ሺ

𝑥

𝐴00

ሻ



𝜆

ሺ

𝑥

B10

ሻ



𝜆

𝐴

ሺ

𝑥

A01

ሻ

Figure 3: Transition Diagram.

In order to ﬁnd the steady-state probabilities, we

need to solve the following balance equation system:

1) µ

+ µ

= [λ

A00

) + λ

B00

)]π

2) µ

+ λ

A00

)π

= [λ

B10

) + µ

]π

3) λ

B10

)π

+ λ

A01

)π

= [µ

+ µ

]π

4) π

+ π

= 1.

(8)

Let S(

−→

x ) = S(x

A00

A01

B00

B10

) denote the social

welfare function. We have

−→

x ) = π

(

−→

x )

A00

− y)h(y)dy+

(

−→

x )

A01

− y)h(y)dy+

(

−→

x )

B00

(y − v

)h(y)dy+

(

−→

x )

B10

(y − v

)h(y)dy.

(9)

We wish to ﬁnd:

−→

∗

= (x

∗

A00

∗

A01

∗

B00

∗

B10

) that

maximizes the social Welfare function S. Recall ﬁrst

the model with a single server (see Section 2). Ac-

cording to (2), if server A was a single server, located

at the origin, then the optimal threshold strategy x

∗

unique and satisﬁes,

∗

− y)h(y)dy + x

∗

. (10)

Similarly, if server B was a single server, located at

M, then the optimal threshold strategy x

∗

is unique

and satisﬁes,

= x

∗

−

∗

(y − x

∗

)h(y)dy. (11)

The values of x

∗

and x

∗

depend on the parameters

of the model and on the intensity function h. Under

some conditions, x

∗

≤ x

∗

, (Case A), and under other

conditions, x

∗

> x

∗

, (Case B).

As we show in the sequal, x

∗

and x

∗

, although

originated in the single server mode, nevertheless play

a signiﬁcant role in the model with two servers.

For simplicity, we assume from now on that

servers have the same capacity. Additionally, We nor-

malize all other parameters according to this common

capacity hence

= µ

= 1.

Lemma 5.1. For all 0 < x < v,

• If x < x

∗

, then

(v − y)h(y)dy < (λ

(x) + 1)(v − x)

• If x = x

∗

, then

(v − y)h(y)dy = (λ

(x) + 1)(v − x)

• If x > x

∗

, then

(v − y)h(y)dy > (λ

(x) + 1)(v − x)

Proof. Note that,

(v − y)h(y)dy =

(v − x + x − y)h(y)dy =

(v − x)λ

(x) +

(x − y)h(y)dy.

(12)

Multi-Server Queue, with Heterogeneous Service Valuations Induced by Travel Costs

137

It follows that,

(v − y)h(y)dy =

(v − x)λ

(x) − x +

x +

(x − y)h(y)dy

. (13)

By (10), x

∗

satisﬁes,

v = x

∗

− y)h(y)dy. (14)

Thus, substituting x = x

∗

in (13), we get:

∗

(v − y)h(y)dy = (v − x

∗

)λ

∗

) − x

∗

+ v

= (λ

∗

) + 1)(v − x

∗

), (15)

proving the second statement of the lemma. Note that

x +

(x − y)h(y)dy, appearing in the square brackets

at the right hand side of (13) is increasing in x, hence

it follows from (14) that:

• For x < x

∗

(v − y)h(y)dy < (v − x)λ

(x) − x + v

= (λ

(x) + 1)(v − x), (16)

and,

• For x > x

∗

(v − y)h(y)dy < (v − x)λ

(x) − x + v

= (λ

(x) + 1)(v − x), (17)

proving the ﬁrst and last statements of the lemma.

A similar lemma, regarding server B is:

Lemma 5.2. For all v < x < M,

• If x < x

∗

, then

(y − v)h(y)dy > (λ

(x) + 1)(x − v).

• If x = x

∗

, then

(y − v)h(y)dy = (λ

(x) + 1)(x − v).

• If x > x

∗

, then

(y − v)h(y)dy < (λ

(x) + 1)(x − v).

5.1 Socially Optimal Strategies in the

Symmetric Case

In this section we assume that the model is completely

symmetric with regards to the two servers. We will

show that in case A (namely when x

∗

≤ x

∗

), con-

sumers with distances that are less than x

∗

from server

A, turn to server A in any case (wheather server B

is avilable or not). Similarly, consumers with dis-

tances that are less than x

∗

from server B (i.e., their

location is M − x

∗

and beyond) turn to server B. In

Case B (namely when x

∗

> x

∗

), we will show that

if both servers are available, then server A serves

consumers with locations between 0 and M/2, and

server B serves consumers from that point on (until

the end of the interval [0,M]). Interestingly, if only

one server is available, say server A, then she serves

consumers with distances that are beyond x

∗

, which

was the service-threshold when server A was the only

server in a single-server system.

We assume that,

• A1. h(M − y) = h(y), ∀0 ≤ y ≤ M.

• A2. c

= c

, thus travel cost is simply c

The following 4 properties follow,

• P1. v

= M − v

. We denote v = v

• P2. x

∗

= M − x

∗

• P3. x

∗

B10

= M − x

∗

A01

• P4. x

∗

B00

= M − x

∗

A00

Recall that in all cases (including the general non-

symmetric case), x

A00

≤ x

B00

, (see (3)). This together

with P4 above gives in the symmetric case

∗

A00

≤

. (18)

Lemma 5.3. In the symmetric case, for all 0 ≤ x ≤ M,

(M − x) = λ

(x).

Proof. By A1,

(M − x) =

M−x

h(u)du =

h(M − u)du =

h(u)du = λ

(x). (19)

Denote a = 1 + λ

A00

)(2 + λ

A01

)).

Proposition 5.4. In the symmetric case,

, π

= π

A00

)

and,

A00

)λ

A01

)

The proof follows from Lemma 5.3, Assumptions

A1-A3 and Properties P1-P4 above.

Proposition 5.5. In the symmetric case,

−→

x ) =



A00

(v − y)h(y)dy + λ

A00

A01

(v − y)h(y)dy



(20)

ICORES 2024 - 13th International Conference on Operations Research and Enterprise Systems

138

Proof. By Proposition 5.4, the assumptions and prop-

erties, we obtain by (9) that:

−→

x ) =

A00

− y)h(y)dy+

A00

)

A01

− y)h(y)dy+

M−x

A00

(y − M + v

)h(y)dy+

A00

)

M−x

A01

(y − M + v

)h(y)dy.

(21)

Now, by A1 and by substituting t = M − y in

M−x

A00

(y − M + v

)h(y)dy, we obtain

M−x

A00

(y−M +v

)h(y)dy =

A00

−t)h(t)dt.

(22)

Similarly,

M−x

A01

(y−M +v

)h(y)dy =

A01

−t)h(t)dt.

(23)

Substituting (22) and (23) in (21) gives

−→

x ) =

A00

− y)h(y)dy+

A00

)

A01

− y)h(y)dy+

A00

− y)h(y)dy+

A00

)

A01

− y)h(y)dy =



A00

− y)h(y)dy + λ

A00

)

A01

− y)h(y)dy



(24)

As Propositions 5.4 and 5.5 show, because of the

symmetry, S(

−→

x ) can be presented as a function of the

parameters of server A only, namely; x

A00

and x

A01

The values of x

B00

and x

B10

are then derived according

to properies P1-P4 above. Hence in this section we

abbreviate the notations so that

v = v

, x

= x

A00

, x

= x

A01

= λ

A00

), λ

= λ

A01

In the new notations, we obtain from Proposi-

tion 5.5:

S(x

) =



(v − y)h(y)dy +

(v − y)h(y)dy



(25)

We wish to ﬁnd (x

∗

), that maximizes

S(x

). (x

)

∗

must satisfy that both deriva-

tives of S, (with respect to x

, and x

) equal zero.

Recall that x

∗

is the socially optimal strategy in the

case of a single server A (see Section 2). We wish to

prove ﬁrst that (x

∗

) is the unique maximum point

of S(x

Proposition 5.6. Given x

> 0, the ˜x

= ˜x

that satisﬁes

∂

∂x

S(x

) = 0,

is the unique local maximum point of S(x

,•). If

̸= x

∗

, then ˜x

) > x

∗

Proof. Note that the derivatives of π

and π

with

respect to x

are,

′

= −

h(x

)

(1 + λ

(2 + l

))

= −

h(x

)

, (26)

and,

′

= −

h(x

)

(1 + λ

(2 + λ

))

= −

h(x

)

. (27)

It follows from (25), (26) and (27) that

∂

∂x

S(x

) =

h(x

)

{−

(v −y)h(y)dy + (v − x

)(1 +λ

(2 +λ

))−

(v − y)h(y)dy} (28)

Thus, for a given x

, the ˜x

= ˜x

) for which

∂

∂x

S(x

) = 0, solves:

−

(v−y)h(y)dy+(v−x

)(1+λ

(2+λ

))−

(v − y)h(y)dy = 0, (29)

where λ

h(y)dy.

Now, if we differentiate the left hand side of (29)

with respect to x

, we arrive at

− λ

(v − x

)h(x

) − (1 + λ

(2 + λ

))+

(v − x

)h(x

) = −a < 0. (30)

Hence the left hand side of (29) is decreasing in

. It follows that,

∂

∂x

S(x

) > 0, ∀x

< ˜x

∂

∂x

S(x

) = 0, for x

= ˜x

Multi-Server Queue, with Heterogeneous Service Valuations Induced by Travel Costs

139

and

∂

∂x

S(x

) < 0, ∀x

> ˜x

Thus, ˜x

is indeed the unique local maximum of

S(x

,•).

Denote λ

∗

= λ

∗

) =

∗

h(y)dy. For any given

, (29) presents the condition for

∂

∂x

S(x

) =

0. Subtituting x

= x

∗

in the left hand side of (29)

gives

−

(v − y)h(y)dy−

∗

(v − y)h(y)dy + (v − x

∗

)(1 + λ

(2 + λ

∗

))

= −

(v − x

∗

+ x

∗

− y)h(y)dy−

∗

(v − y)h(y)dy + (v − x

∗

)(1 + λ

(2 + λ

∗

)) =

− (v − x

∗

)λ

−

∗

− y)h(y)dy−

∗

(v − y)h(y)dy+

(v − x

∗

)(1 + 2λ

+ λ

∗

(31)

By (12),

∗

(v −y)h(y)dy = (v −x

∗

)λ

∗

−y)h(y)dy.

Substituting this in (31) gives

− (v −x

∗

)λ

−

∗

− y)h(y)dy−



(v − x

∗

)λ

∗

− y)h(y)dy



+ (v −x

∗

)(1 + 2λ

+ λ

∗

) = (v − x

∗

)(−λ

− λ

∗

1 + 2λ

+ λ

∗

) + (−λ

− 1 +1)

∗

− y)h(y)dy−

∗

− y)h(y)dy =

(32)

= (λ

+ 1)



v −



∗

− y)h(y)dy



∗

− y)h(y)dy.

Note that by (10), the expression in the square brakets

above equals 0, hence we get

∗

− y)h(y)dy, (33)

which is positive for all x

̸= x

∗

, as now explained:

For x

∗

≥ x

this is obvious. For x

∗

< x

, the left

hand side of (33) equals

−

∗

− y)h(y)dy =

∗

(y − x

∗

)h(y)dy > 0.

So we proved that for x

̸= x

∗

, the left hand side

of (29), is positive for x

= x

∗

. Now, in comparis-

sion to that, if we substitute x

= ˜x

) in the left

hand side of (29), then by deﬁnition of ˜x

), this

equals 0. Earlier (see (30)) we showed that the left

hand side of (29) is decreasing in x

. Hence it fol-

lows that ˜x

) > x

∗

Similarly,

Proposition 5.7. Given x

> 0, the ˜x

= ˜x

that satisﬁes

∂

∂x

S(x

) = 0,

is unique and it is the local maximum point of

S(•,x

). If x

̸= x

∗

, then ˜x

) < x

∗

The proof of Proposition 5.7 is very similar to

the proof of Proposition 5.6. and involves proving

that for a given x

, the ˜x

= ˜x

) for which

∂

∂x

S(x

) = 0, solves:

− (2 + λ

)

(v − y)h(y)dy+

(v − y)h(y)dy + (v − x

)a = 0. (34)

Theorem 5.8. The point (x

∗

) is the unique global

maximum point of S(x

), and it satisﬁes that

S(x

∗

) = 2c

(v − x

∗

Proof. Substituting x

= x

∗

in (20) and uti-

lizing Lemma 5.1 indeed gives 2c

(v − x

∗

). In order

to prove that (x

∗

) is the unique global maximum

point of S(x

), we ﬁrst, prove that (x

∗

) is

the only point in which both derivatives of S(x

(with respect to x

and with respect to x

), equal 0.

A point in which both derivatives equal 0, must satisfy

both (29) and (57).

Recall that, Equation (29) is:

−

(v−y)h(y)dy+(v−x

)(1+λ

(2+λ

))−

(v − y)h(y)dy = 0, (35)

and Equation (57) is:

− (2 + λ

)

(v − y)h(y)dy+

(v − y)h(y)dy + (v − x

)a = 0. (36)

ICORES 2024 - 13th International Conference on Operations Research and Enterprise Systems

140

From (29) we have:

(v − y)h(y)dy = (v − x

)a − λ

(v − y)h(y)dy.

Subtituting this in (57) gives:

−(2+λ

)



(v − x

)a − λ

(v − y)h(y)dy



(v − y)h(y)dy + (v − x

)a = 0. (37)

This implies that:



v − x

− (2 + λ

)(v − x

)





1 + λ

(2 + λ

)



(v − y)h(y)dy = 0, (38)

which is equivalent to:



v − x

− (2 + λ

)(v − x

)



(v − y)h(y)dy = 0. (39)

From this we arrive at:

− x

(v − y)h(y)dy − (1 + λ

)(v − x

)

= 0. (40)

We ﬁrst show that x

must equal x

∗

: If on the con-

trary, x

̸= x

∗

, then we proved in Proposition 5.6 that

˜x

) > x

∗

. But in that case, by Lemma 5.1, the ex-

pression in the square brackets above is positive. Ad-

ditionally, x

− x

> 0, (since, x

≤ M/2 < x

∗

˜x

)). Thus the left hand side of (40) is strictly

positive and thus cannot satisfy (40). Hence,

= x

∗

. (41)

Now, if ˜x

̸= x

∗

, then by Proposition 5.7, x

< x

∗

which contradicts (41), thus,

˜x

∗

) = x

∗

. (42)

Hence, we proved that (x

∗

) is the only point

in which both derivatives of S equal 0. To prove that

∗

) is also the unique global maximum point, we

need to check the value of S at the borders, namely

at the boundaries of the rectangular [0,M/2] ×[0, M] :

If x

= 0, then λ

= 0. Hence by Proposition 5.4,

= π

= 0.

Hence from the the balance equation for state

(1,1) (see equation 3 in the balance equation system

appearing in (8)), we get:

0 = [µ

+ µ

]π

which implies that π

= 0, hence π

= 1. Thus the

service-system is always empty and thus S(0,x

) =

0, which is smaller than S(x

∗

) = 2c

(v − x

∗

If x

= 0, then (25) gives

S(x

,0) = 2c



1 + 2λ

(v − y)h(y)dy



which implies:

S(x

,0) < 2c



1 + λ

(v − y)h(y)dy



. (43)

Recall that in M/G/1/1, (see Section 2), the prob-

ability that the system is empty when the service-

threshold is x

, (and µ = 1), is

) =

1 + λ

In M/G/1/1, we denoted the social welfare function,

when the service-threshold is x, as S

(x). By (1):

(x) = c

(v − y)h(y)π

(x)dy. (44)

Thus it follows from (43) that:

S(x

,0) < 2c

(v − y)h(y)π

)dy =

) < 2S

∗

) = 2c

∗

(v − y)h(y)π

∗

)dy

= 2c

∗

(v − y)h(y)

1 + λ

∗

dy =

1 + λ

∗

(v − y)h(y)dy. (45)

By Lemma 5.1, the right hand side of (45) equals

(v − x

∗

), which by Theorem 5.8 equals S(x

∗

Hence,

S(x

,0) < S(x

∗

For the cases of x

= M/2, or x

= M, we can

use the symmetry of the model with regards to the

servers, and thus, for example, x

= M, implies that

for server B, x

B00

= M, as well (see (3)) which yields

B00

= 0, and from that point to continue like in the

case where λ

= 0.

Corollary 5.9. Given x

S(x

, ˜x

)) = 2c

(v − ˜x

)).

Proof. By (25) we have:

S(x

, ˜x

)) =



(v − y)h(y)dy + λ

˜x

)

(v − y)h(y)dy



(46)

Multi-Server Queue, with Heterogeneous Service Valuations Induced by Travel Costs

141

Additionally, ˜x

) satisﬁes (29), implying

that:

˜x

)

(v − y)h(y)dy =

−

(v − y)h(y)dy + a



v − ˜x

)



. (47)

Subtituting (47) in (46), gives:

S(x

, ˜x

)) =

(v − y)h(y)dy−

(v − y)h(y)dy + a(v − ˜x

))

= 2c

(v − ˜x

)). (48)

Recall that Case A is deﬁned by x

∗

≤ x

∗

, and

Case B is deﬁned by x

∗

> x

∗

. Theorem 5.10 ahead

claims that under social optimality, in Case A, each

server serves consumers that are x

∗

or less, away

from the servic point, and that in Case B: When both

servers are idle, then each server serves consumers

that are M/2 or less, away from the service point.

In case only one server is available, then she serves

consumers with distance that exceeds x

∗

, where the

service- threshold is determined according to (49)

ahead.

Formally, going back to our earlier notations; re-

call that x

was an abbreviation for x

A00

the service-

threshold of server A when both servers are idle, and

B00

was the service-threshold of server B when both

servers are idle. Similarly, x

A01

( x

B10

) was the service

threshold of A, (B) when only he was available.

Theorem 5.10. In the symmetric model, the strat-

egy

−→

∗

= (x

∗

A00

∗

A01

∗

B00

∗

B10

) that maximizes so-

cial welfare S(

−→

x ), is the following

• Case A, namely when x

∗

≤ x

∗

, then:

∗

A00

∗

A01

∗

B00

∗

B10

) = (x

∗

,M − x

∗

,M − x

∗

and S = 2c

(v − x

∗

• Case B, namely when x

∗

> x

∗

1. If both servers are available then (x

∗

A00

∗

B00

) =

(M/2,M/2), and S = 2c

(v − M/2).

2. If only server A is available then, x

∗

A01

˜x

(M/2), and if only server B is available

then, x

∗

B10

= M − ˜x

(M/2). In both cases, S =

(v − ˜x

(M/2)), where ˜x

(M/2), is the

unique solution for:

(v − y)h(y)dy −

(v − x

)



1 + λ

(M/2)(2 + λ

)



+ λ

(M/2)

(v − y)h(y)dy = 0. (49)

Proof. Because of the symmetry between servers A

and B, we have by properties P3 and P4, that x

∗

B10

M − x

∗

A01

, and x

∗

B00

= M − x

∗

A00

, hence we only need

to prove the theorem for server A. Recall that in the

symmetric case x

A00

must satisfy

∗

A00

≤

(see (18)). Now, by Theorem 5.8, (x

∗

) = (x

∗

)

is the unique global maximum of the social welfare

function S, hence whenever this solution is possible

(i.e., x

∗

≤ M/2), then this will be the strategy of

server A. This proves the statement of the theorem

regarding case A, (since in that case, x

∗

≤ M/2.).

In Case B, x

∗

> M/2, hence x

∗

is not an option

for x

∗

which must satisfy x

∗

A00

≤

Recall that (29) deﬁnes ˜x

) for a given x

where (29) is

−

(v − y)h(y)dy + (v − x

)(1 + λ

(2 + λ

))

− λ

(v − y)h(y)dy = 0. (50)

Deﬁne the left hand side of (29), as a function

f (x

). By deﬁnition of ˜x

f (x

, ˜x

)) = 0. (51)

We showed, (see (30)), that f (x

) is decreasing

in x

Below we show that f (x

) is also decreasing

in x

for all x

̸= x

∗

. This implies that ˜x

) is

decreasing as a function of x

, since if we increase

to x

+ ε then since f (x

) is decreasing in

, then f (x

+ ε, ˜x

)) < f (x

, ˜x

)) = 0,

and since f is also decreasing in x

, then f (x

ε, ˜x

+ ε)) = 0 implies that ˜x

+ ε) <

˜x

To see that indeed f (x

) is decreasing in x

for all x

̸= x

∗

, note that the derivative of f with

respect to x

, is:

− (v − x

) + (v − x

)(2 + λ

)h(x

)

− h(x

)

(v − y)h(y)dy, (52)

which equals:

h(x

)

(1 + λ

)(v − x

) −

(v − y)h(y)dy

−

h(x

). (53)

By Lemma 5.1, the expression in the square brackets

is negative, (since by Proposition 5.6, if x

̸= x

∗

, then

˜x

) > x

∗

Now, by Corollary 5.9, S(x

, ˜x

)) = 2c

(v −

˜x

)), which is maximized for the smallest

ICORES 2024 - 13th International Conference on Operations Research and Enterprise Systems

142

˜x

). Since we showed that ˜x

) is decreas-

ing as a function of x

, then 2c

(v − ˜x

)) is

maximized for the largest x

possible, which is M/2,

(see (18)). Thus x

= M/2 and x

= ˜x

(M/2).

The value of S in case A and case B1 in Theo-

rem 5.10, follow from (25). The value of S in case B2

follow from Corollary 5.9.

6 CONCLUSIONS

In this study we establish that if the model is sym-

metric with regards to the servers, then under social

optimality, when the service points are distant from

each other, each server behaves as he would if he was

the sole server. But when the service points are within

close proximity, then when only one server is idle, so-

cial optimality dictates that the available server also

caters to distant customers, a behavior it would not

exhibit if it were the sole server in the service sys-

tem (i.e., in M/M/1/1). These results apply when

both servers are informed about each other’s status

(idle/busy). While sharing information about server

status is beneﬁcial, there are situations where it is not

feasible. Take, for instance, two ride-sharing drivers

operating in the same area without real-time knowl-

edge of each other’s current ride status. In this sce-

nario, each driver independently accepts ride requests

without knowing whether the other is currently occu-

pied. This lack of information may lead to subop-

timal resource allocation as both drivers might end

up serving nearby locations simultaneously, poten-

tially reducing overall efﬁciency. For future research,

comparing between the two models; the model where

servers are informed about each other’s status to the

model where servers are ignorant of each other’s sta-

tus would be intriguing. How signiﬁcant is this infor-

mation? If there is a substantial difference between

the outcomes, it may warrant consideration for inter-

vention by authorities.

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Multi-Server Queue, with Heterogeneous Service Valuations Induced by Travel Costs

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APPENDIX

Proof of Proposition 5.7

Proof. Note that the derivatives of π

and π

with

respect to x

are,

′





′



1 + λ

(2 + λ

)



′

= −

(2 + λ

)h(x

)

(1 + λ

(2 + λ

))

= −

(2 + λ

)h(x

)

(54)

and,

′





′



1 + λ

(2 + λ

)



′

h(x

)(1 + λ

(2 + λ

)) − λ

h(x

)(2 + λ

)

(1 + λ

(2 + λ

))

h(x

)(1 + 2λ

+ λ

− 2λ

− λ

)

(1 + λ

(2 + λ

))

h(x

)

(1 + λ

(2 + λ

))

h(x

)

(55)

It follows from (54), (55) and (25), that

∂

∂x

S(x

) = 2c

h(x

)

[−(2 + λ

)

(v − y)h(y)dy + (v − x

)a+

(v − y)h(y)dy]. (56)

Thus, for a given x

, the ˜x

= ˜x

) for which

∂

∂x

S(x

) = 0,

solves:

− (2 + λ

)

(v − y)h(y)dy+

(v − y)h(y)dy + (v − x

)a = 0, (57)

where λ

h(y)dy, and a = 1 + λ

(2 + λ

Now, if we differentiate the left hand side of (57)

with respect to x

, we arrive at

− (2 + λ

)(v − x

)h(x

) − (1 + λ

(2 + λ

))+

(2 + λ

)(v − x

)h(x

) = −a < 0. (58)

It follows that,

∂

∂x

S(x

) > 0, ∀x

< x

∗

∂

∂x

S(x

) = 0, for x

= x

∗

and

∂

∂x

S(x

) < 0, ∀x

> x

∗

We now prove that if x

̸= x

∗

, then x

∗

) <

∗

Recall that λ

∗

= λ

∗

) =

∗

h(y)dy. For

any given x

, (57) presents the condition for

∂

∂x

S(x

) = 0. Subtituting x

= x

∗

in the left

hand side of (57) gives

− (2 + λ

)

∗

(v − y)h(y)dy+

(v − y)h(y)dy + (v − x

∗

)(1 + λ

∗

(2 + λ

)).

(59)

By Lemma 5.1:

− (2 +λ

)

∗

(v − y)h(y)dy +

(v − y)h(y)dy+

(v − x

∗

)(1 + λ

∗

(2 + λ

)) =

− (2 +λ

)



(v − x

∗

)λ

∗

− y)h(y)dy



(v − x

∗

+ x

∗

− y)h(y)dy + (v − x

∗

)(1 + λ

∗

(2 + λ

))

= −(2 + λ

)



(v − x

∗

)λ

∗

− y)h(y)dy



(v − x

∗

)λ

∗

− y)h(y)dy+

(v − x

∗

)(1 + λ

∗

(2 + λ

)) =

(v − x

∗

)(−2λ

∗

− λ

∗

+ λ

+ 1 +2λ

∗

+ λ

∗

)

− (1 +λ

+ 1)

∗

− y)h(y)dy+

∗

− y)h(y)dy =

(1 + λ

)



v −



∗

− y)h(y)dy



−

∗

− y)h(y)dy.

(60)

Note that by (10), the expression in the square

brakets above equals 0, hence we get that

−

∗

− y)h(y)dy,

which is negative for all x

̸= x

∗

Now, in comparission to that, if we substitute

= ˜x

) in the left hand side of(57), then by

deﬁnition of ˜x

), this equals 0. Earlier (see (58))

we showed that the left hand side of (57) is decreasing

in x

. Hence it follows that ˜x

) < x

∗

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144

Proof of Lemma 5.2

Proof. Note that,

(y − v)h(y)dy = (x − v)λ

(x) + x−

x −

(y − x)h(y)dy

. (61)

It follows from (61) and (11) that for x = x

∗

(y − v)h(y)dy = (x

∗

− v)λ

∗

+ x

∗

− v =

∗

− v)(λ

∗

+ 1), (62)

proving the second statement of the lemma. Note that

x −

(y− x)h(y)dy, appearing in the square brackets

at the right hand side of (61) is increasing in x, hence

it follows from (62) that:

• If x < x

∗

, then

(y−v)h(y)dy > (x−v)λ

(x)+

x − v = (λ

(x) + 1)(x − v),

• If x > x

∗

, then

(y−v)h(y)dy < (x−v)λ

(x)+

x − v = (λ

(x) + 1)(x − v),

proving the ﬁrst and last statements of the lemma.

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